Cayley Graph

 

Let  be a group, and let  be a set of group elements such that the identity element . The Cayley graph associated with  is then defined as the directed graph having one vertex associated with each group element and directed edges  whenever . The Cayley graph may depend on the choice of a generating set, and is connected iff  generates  (i.e., the set  are group generators of ).

Care is needed since the term "Cayley graph" is also used when  is implicitly understood to be a set of generators for the group, in which case the graph is always connected (but in general, still dependent on the choice of generators). This sort of Cayley graph of a group  may be computed in the Wolfram Language using CayleyGraph[G], where the generators used are those returned by GroupGenerators[G].

To complicate matters further, undirected versions of proper directed Cayley graphs are also known as Cayley graphs.

An undirected Cayley graph of a particular generating set of the alternating group  is sometimes known as a alternating group graph. The Cayley graph of the cyclic group  is the cycle graph , and of the dihedral group  is the prism graph . Other classes of graphs that are Cayley graphs are circulant graphs (connected if requiring a generating set; possibly disconnected if not), cube connected cycles, Hamming graphs, and hypercube graphs.

A directed graph Cayley graph has the same edge multiplicity for each node. A (directed or undirected) Cayley graph is always vertex-transitive, but the converse need not hold. However, a large fraction of small vertex-transitive graphs are Cayley graphs (McKay and Royle 1990).

Royle maintains a list of not-necessarily-connected vertex-transitive graphs with Cayley or non-Cayley designations up to 31 vertices and although the values for 27, 28 and 30 vertices have not been independently verified (though an error in the groups can only affect the graphs if somehow a minimal transitive group has been omitted, so errors are unlikely). The numbers of not-necessarily-connected Cayley graphs on , 2, ... nodes are 1, 2, 2, 4, 3, 8, 4, 14, 9, 20, 8, 74, ... (OEIS A185959; McKay and Royle 1990, McKay and Praeger 1994), and the numbers of nodes on which non-Cayley vertex-transitive graphs exist are 10, 15, 16, 18, 20, 24, 26, 28, 30, ....

The smallest vertex-transitive non-Cayley graph is the Petersen graph  (McKay and Praeger 1994) and the smallest disconnected vertex-transitive non-Cayley graph is two copies of .

Cayley graphs can be generated by starting with a set of generator permutations {P_i}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline21.svg" style="height:22px; width:29px" /> (which does not include the identity permutation) and mutually permuting elements until no new permutations are reached. This produces a set  that is closed under permutations of elements. Connecting each pair of permutations  with an edge if  for some  then gives a Cayley graph.

The only groups that can give planar Cayley graphs are exactly , , , , , and , as proved by Maschke (1896).

The following table lists some graphs that are undirected versions of Cayley graphs generated by small numbers of small permutations.

graph	generators
16-cell graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline32.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline33.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline34.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline35.svg" style="height:22px; width:6px" />2,1,3,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline36.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline37.svg" style="height:22px; width:6px" />2,1,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline38.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline39.svg" style="height:22px; width:6px" />3,4,1,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline40.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline41.svg" style="height:22px; width:6px" />3,4,2,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline42.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline43.svg" style="height:22px; width:6px" />
circulant graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline45.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline46.svg" style="height:22px; width:6px" />1,2,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline47.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline48.svg" style="height:22px; width:6px" />1,2,4,3,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline49.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline50.svg" style="height:22px; width:6px" />2,1,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline51.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline52.svg" style="height:22px; width:6px" />2,1,5,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline53.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline54.svg" style="height:22px; width:6px" />
complete bipartite graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline56.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline57.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline58.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline59.svg" style="height:22px; width:6px" />2,1,3,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline60.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline61.svg" style="height:22px; width:6px" />3,4,2,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline62.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline63.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline64.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline65.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline66.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline67.svg" style="height:22px; width:6px" />2,1,3,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline68.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline69.svg" style="height:22px; width:6px" />3,4,1,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline70.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline71.svg" style="height:22px; width:6px" />4,3,2,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline72.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline73.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline74.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline75.svg" style="height:22px; width:6px" />1,2,4,5,6,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline76.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline77.svg" style="height:22px; width:6px" />2,1,4,5,6,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline78.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline79.svg" style="height:22px; width:6px" />
complete graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline81.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline82.svg" style="height:22px; width:6px" />1,3,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline83.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline84.svg" style="height:22px; width:6px" />2,1,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline85.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline86.svg" style="height:22px; width:6px" />2,3,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline87.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline88.svg" style="height:22px; width:6px" />3,2,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline89.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline90.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline91.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline92.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline93.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline94.svg" style="height:22px; width:6px" />1,3,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline95.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline96.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline97.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline98.svg" style="height:22px; width:6px" />1,4,3,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline99.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline100.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline101.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline102.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline103.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline104.svg" style="height:22px; width:6px" />1,3,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline105.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline106.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline107.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline108.svg" style="height:22px; width:6px" />1,4,2,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline109.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline110.svg" style="height:22px; width:6px" />1,4,3,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline111.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline112.svg" style="height:22px; width:6px" />
cubical graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline113.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline114.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline115.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline116.svg" style="height:22px; width:6px" />3,4,2,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline117.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline118.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline119.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline120.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline121.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline122.svg" style="height:22px; width:6px" />2,1,3,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline123.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline124.svg" style="height:22px; width:6px" />3,4,1,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline125.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline126.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline127.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline128.svg" style="height:22px; width:6px" />1,2,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline129.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline130.svg" style="height:22px; width:6px" />1,4,5,3,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline131.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline132.svg" style="height:22px; width:6px" />
cubic symmetric graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline134.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline135.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline136.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline137.svg" style="height:22px; width:6px" />1,3,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline138.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline139.svg" style="height:22px; width:6px" />3,2,1,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline140.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline141.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline142.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline143.svg" style="height:22px; width:6px" />1,2,3,4,6,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline144.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline145.svg" style="height:22px; width:6px" />2,5,4,6,3,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline146.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline147.svg" style="height:22px; width:6px" />
cubic symmetric graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline149.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline150.svg" style="height:22px; width:6px" />1,3,2,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline151.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline152.svg" style="height:22px; width:6px" />2,1,4,3,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline153.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline154.svg" style="height:22px; width:6px" />4,5,3,1,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline155.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline156.svg" style="height:22px; width:6px" />
cubic vertex-transitive graph Ct19	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline157.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline158.svg" style="height:22px; width:6px" />1,2,3,4,6,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline159.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline160.svg" style="height:22px; width:6px" />1,2,4,3,6,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline161.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline162.svg" style="height:22px; width:6px" />5,6,3,4,1,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline163.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline164.svg" style="height:22px; width:6px" />
cubic vertex-transitive graph Ct23	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline165.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline166.svg" style="height:22px; width:6px" />1,2,3,4,6,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline167.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline168.svg" style="height:22px; width:6px" />2,3,1,5,6,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline169.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline170.svg" style="height:22px; width:6px" />
cubic vertex-transitive graph Ct28	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline171.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline172.svg" style="height:22px; width:6px" />1,3,2,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline173.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline174.svg" style="height:22px; width:6px" />2,3,4,1,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline175.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline176.svg" style="height:22px; width:6px" />
cubic vertex-transitive graph Ct37	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline177.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline178.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline179.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline180.svg" style="height:22px; width:6px" />1,3,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline181.svg" style="height:22px; width:6px" />,{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline182.svg" style="height:22px; width:6px" />3,4,1,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline183.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline184.svg" style="height:22px; width:6px" />
cubic vertex-transitive graph Ct38	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline185.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline186.svg" style="height:22px; width:6px" />1,2,3,4,5,7,6}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline187.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline188.svg" style="height:22px; width:6px" />2,3,1,6,7,4,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline189.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline190.svg" style="height:22px; width:6px" />
cubic vertex-transitive graph Ct41	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline191.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline192.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline193.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline194.svg" style="height:22px; width:6px" />1,3,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline195.svg" style="height:22px; width:6px" />,{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline196.svg" style="height:22px; width:6px" />2,1,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline197.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline198.svg" style="height:22px; width:6px" />
cubic vertex-transitive graph Ct42	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline199.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline200.svg" style="height:22px; width:6px" />1,2,3,4,5,7,6}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline201.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline202.svg" style="height:22px; width:6px" />2,3,4,1,6,5,7}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline203.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline204.svg" style="height:22px; width:6px" />
cuboctahedral graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline205.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline206.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline207.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline208.svg" style="height:22px; width:6px" />2,3,1,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline209.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline210.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline211.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline212.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline213.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline214.svg" style="height:22px; width:6px" />1,4,2,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline215.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline216.svg" style="height:22px; width:6px" />2,3,1,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline217.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline218.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline219.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline220.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline221.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline222.svg" style="height:22px; width:6px" />1,4,2,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline223.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline224.svg" style="height:22px; width:6px" />2,3,1,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline225.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline226.svg" style="height:22px; width:6px" />3,1,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline227.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline228.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline229.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline230.svg" style="height:22px; width:6px" />1,2,4,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline231.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline232.svg" style="height:22px; width:6px" />1,3,4,2,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline233.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline234.svg" style="height:22px; width:6px" />
5-cycle graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline235.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline236.svg" style="height:22px; width:6px" />2,3,4,5,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline237.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline238.svg" style="height:22px; width:6px" />5,1,2,3,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline239.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline240.svg" style="height:22px; width:6px" />
6-cycle graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline241.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline242.svg" style="height:22px; width:6px" />1,3,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline243.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline244.svg" style="height:22px; width:6px" />2,1,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline245.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline246.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline247.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline248.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline249.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline250.svg" style="height:22px; width:6px" />1,3,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline251.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline252.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline253.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline254.svg" style="height:22px; width:6px" />1,2,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline255.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline256.svg" style="height:22px; width:6px" />1,2,4,3,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline257.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline258.svg" style="height:22px; width:6px" />
8-cycle graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline259.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline260.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline261.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline262.svg" style="height:22px; width:6px" />3,4,1,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline263.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline264.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline265.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline266.svg" style="height:22px; width:6px" />1,2,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline267.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline268.svg" style="height:22px; width:6px" />1,4,5,2,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline269.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline270.svg" style="height:22px; width:6px" />
10-cycle graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline271.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline272.svg" style="height:22px; width:6px" />1,3,2,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline273.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline274.svg" style="height:22px; width:6px" />2,1,4,3,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline275.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline276.svg" style="height:22px; width:6px" />
12-cycle graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline277.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline278.svg" style="height:22px; width:6px" />1,2,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline279.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline280.svg" style="height:22px; width:6px" />2,1,4,3,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline281.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline282.svg" style="height:22px; width:6px" />
Franklin graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline283.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline284.svg" style="height:22px; width:6px" />1,2,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline285.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline286.svg" style="height:22px; width:6px" />1,2,4,3,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline287.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline288.svg" style="height:22px; width:6px" />2,1,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline289.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline290.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline291.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline292.svg" style="height:22px; width:6px" />1,2,3,4,5,7,6}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline293.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline294.svg" style="height:22px; width:6px" />1,2,3,4,6,5,7}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline295.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline296.svg" style="height:22px; width:6px" />1,2,4,3,5,7,6}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline297.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline298.svg" style="height:22px; width:6px" />
great rhombicuboctahedral graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline299.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline300.svg" style="height:22px; width:6px" />1,2,3,4,5,7,6}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline301.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline302.svg" style="height:22px; width:6px" />1,2,3,4,6,5,7}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline303.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline304.svg" style="height:22px; width:6px" />1,3,2,5,4,7,6}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline305.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline306.svg" style="height:22px; width:6px" />
icosahedral graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline307.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline308.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline309.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline310.svg" style="height:22px; width:6px" />2,1,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline311.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline312.svg" style="height:22px; width:6px" />2,3,1,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline313.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline314.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline315.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline316.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline317.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline318.svg" style="height:22px; width:6px" />1,4,2,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline319.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline320.svg" style="height:22px; width:6px" />2,1,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline321.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline322.svg" style="height:22px; width:6px" />2,3,1,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline323.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline324.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline325.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline326.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline327.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline328.svg" style="height:22px; width:6px" />1,4,2,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline329.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline330.svg" style="height:22px; width:6px" />2,1,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline331.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline332.svg" style="height:22px; width:6px" />2,3,1,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline333.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline334.svg" style="height:22px; width:6px" />3,1,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline335.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline336.svg" style="height:22px; width:6px" />
4-Möbius ladder	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline337.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline338.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline339.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline340.svg" style="height:22px; width:6px" />2,1,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline341.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline342.svg" style="height:22px; width:6px" />3,4,1,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline343.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline344.svg" style="height:22px; width:6px" />
octahedral graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline345.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline346.svg" style="height:22px; width:6px" />1,3,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline347.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline348.svg" style="height:22px; width:6px" />2,1,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline349.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline350.svg" style="height:22px; width:6px" />2,3,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline351.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline352.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline353.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline354.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline355.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline356.svg" style="height:22px; width:6px" />1,3,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline357.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline358.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline359.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline360.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline361.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline362.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline363.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline364.svg" style="height:22px; width:6px" />1,3,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline365.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline366.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline367.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline368.svg" style="height:22px; width:6px" />1,4,2,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline369.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline370.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline371.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline372.svg" style="height:22px; width:6px" />1,2,4,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline373.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline374.svg" style="height:22px; width:6px" />2,1,4,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline375.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline376.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline377.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline378.svg" style="height:22px; width:6px" />1,2,4,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline379.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline380.svg" style="height:22px; width:6px" />2,1,4,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline381.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline382.svg" style="height:22px; width:6px" />
Pappus graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline383.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline384.svg" style="height:22px; width:6px" />1,2,3,4,6,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline385.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline386.svg" style="height:22px; width:6px" />2,3,1,5,4,6}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline387.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline388.svg" style="height:22px; width:6px" />
2-path graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline389.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline390.svg" style="height:22px; width:6px" />2,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline391.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline392.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline393.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline394.svg" style="height:22px; width:6px" />1,3,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline395.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline396.svg" style="height:22px; width:6px" />
pentatope graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline397.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline398.svg" style="height:22px; width:6px" />2,3,4,5,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline399.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline400.svg" style="height:22px; width:6px" />3,4,5,1,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline401.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline402.svg" style="height:22px; width:6px" />
5-prism graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline403.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline404.svg" style="height:22px; width:6px" />1,3,2,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline405.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline406.svg" style="height:22px; width:6px" />2,4,1,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline407.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline408.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline409.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline410.svg" style="height:22px; width:6px" />1,3,2,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline411.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline412.svg" style="height:22px; width:6px" />2,4,1,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline413.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline414.svg" style="height:22px; width:6px" />
6-prism graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline415.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline416.svg" style="height:22px; width:6px" />1,2,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline417.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline418.svg" style="height:22px; width:6px" />2,1,4,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline419.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline420.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline421.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline422.svg" style="height:22px; width:6px" />1,2,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline423.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline424.svg" style="height:22px; width:6px" />2,1,4,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline425.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline426.svg" style="height:22px; width:6px" />
small rhombicosidodecahedral graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline427.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline428.svg" style="height:22px; width:6px" />1,2,4,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline429.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline430.svg" style="height:22px; width:6px" />3,4,2,5,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline431.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline432.svg" style="height:22px; width:6px" />
small rhombicuboctahedral graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline433.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline434.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline435.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline436.svg" style="height:22px; width:6px" />2,3,4,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline437.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline438.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline439.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline440.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline441.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline442.svg" style="height:22px; width:6px" />1,4,2,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline443.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline444.svg" style="height:22px; width:6px" />2,3,4,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline445.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline446.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline447.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline448.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline449.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline450.svg" style="height:22px; width:6px" />1,4,2,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline451.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline452.svg" style="height:22px; width:6px" />2,3,4,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline453.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline454.svg" style="height:22px; width:6px" />4,1,2,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline455.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline456.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline457.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline458.svg" style="height:22px; width:6px" />1,2,4,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline459.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline460.svg" style="height:22px; width:6px" />1,3,4,5,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline461.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline462.svg" style="height:22px; width:6px" />
snub cubical graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline463.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline464.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline465.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline466.svg" style="height:22px; width:6px" />2,3,1,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline467.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline468.svg" style="height:22px; width:6px" />2,3,4,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline469.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline470.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline471.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline472.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline473.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline474.svg" style="height:22px; width:6px" />2,3,1,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline475.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline476.svg" style="height:22px; width:6px" />2,3,4,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline477.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline478.svg" style="height:22px; width:6px" />3,1,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline479.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline480.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline481.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline482.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline483.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline484.svg" style="height:22px; width:6px" />2,3,1,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline485.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline486.svg" style="height:22px; width:6px" />2,3,4,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline487.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline488.svg" style="height:22px; width:6px" />3,1,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline489.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline490.svg" style="height:22px; width:6px" />4,1,2,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline491.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline492.svg" style="height:22px; width:6px" />
square antiprism graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline493.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline494.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline495.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline496.svg" style="height:22px; width:6px" />3,4,1,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline497.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline498.svg" style="height:22px; width:6px" />3,4,2,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline499.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline500.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline501.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline502.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline503.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline504.svg" style="height:22px; width:6px" />3,4,1,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline505.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline506.svg" style="height:22px; width:6px" />3,4,2,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline507.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline508.svg" style="height:22px; width:6px" />4,3,1,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline509.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline510.svg" style="height:22px; width:6px" />
square graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline511.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline512.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline513.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline514.svg" style="height:22px; width:6px" />2,1,3,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline515.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline516.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline517.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline518.svg" style="height:22px; width:6px" />1,2,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline519.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline520.svg" style="height:22px; width:6px" />1,3,2,4,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline521.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline522.svg" style="height:22px; width:6px" />
tesseract graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline523.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline524.svg" style="height:22px; width:6px" />1,2,3,4,6,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline525.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline526.svg" style="height:22px; width:6px" />1,2,4,3,5,6}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline527.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline528.svg" style="height:22px; width:6px" />3,4,2,1,5,6}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline529.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline530.svg" style="height:22px; width:6px" />
tetrahedral graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline531.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline532.svg" style="height:22px; width:6px" />2,1,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline533.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline534.svg" style="height:22px; width:6px" />3,4,2,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline535.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline536.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline537.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline538.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline539.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline540.svg" style="height:22px; width:6px" />2,1,3,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline541.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline542.svg" style="height:22px; width:6px" />2,1,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline543.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline544.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline545.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline546.svg" style="height:22px; width:6px" />1,3,2,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline547.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline548.svg" style="height:22px; width:6px" />1,4,5,3,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline549.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline550.svg" style="height:22px; width:6px" />
triangle graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline551.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline552.svg" style="height:22px; width:6px" />2,3,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline553.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline554.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline555.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline556.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline557.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline558.svg" style="height:22px; width:6px" />1,4,2,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline559.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline560.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline561.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline562.svg" style="height:22px; width:6px" />1,2,4,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline563.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline564.svg" style="height:22px; width:6px" />1,2,5,3,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline565.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline566.svg" style="height:22px; width:6px" />
triangular prism graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline567.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline568.svg" style="height:22px; width:6px" />1,3,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline569.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline570.svg" style="height:22px; width:6px" />2,3,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline571.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline572.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline573.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline574.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline575.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline576.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline577.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline578.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline579.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline580.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline581.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline582.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline583.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline584.svg" style="height:22px; width:6px" />1,4,2,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline585.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline586.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline587.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline588.svg" style="height:22px; width:6px" />1,2,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline589.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline590.svg" style="height:22px; width:6px" />1,2,4,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline591.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline592.svg" style="height:22px; width:6px" />
truncated cubical graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline593.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline594.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline595.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline596.svg" style="height:22px; width:6px" />2,3,1,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline597.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline598.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline599.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline600.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline601.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline602.svg" style="height:22px; width:6px" />2,3,1,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline603.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline604.svg" style="height:22px; width:6px" />3,1,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline605.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline606.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline607.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline608.svg" style="height:22px; width:6px" />1,2,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline609.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline610.svg" style="height:22px; width:6px" />1,3,4,2,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline611.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline612.svg" style="height:22px; width:6px" />
truncated dodecahedral graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline613.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline614.svg" style="height:22px; width:6px" />1,2,4,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline615.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline616.svg" style="height:22px; width:6px" />3,4,1,2,5}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline617.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline618.svg" style="height:22px; width:6px" />
truncated icosahedral graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline619.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline620.svg" style="height:22px; width:6px" />1,3,2,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline621.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline622.svg" style="height:22px; width:6px" />2,3,4,5,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline623.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline624.svg" style="height:22px; width:6px" />
truncated octahedral graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline625.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline626.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline627.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline628.svg" style="height:22px; width:6px" />2,3,4,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline629.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline630.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline631.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline632.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline633.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline634.svg" style="height:22px; width:6px" />1,3,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline635.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline636.svg" style="height:22px; width:6px" />2,1,3,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline637.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline638.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline639.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline640.svg" style="height:22px; width:6px" />1,2,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline641.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline642.svg" style="height:22px; width:6px" />1,3,4,5,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline643.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline644.svg" style="height:22px; width:6px" />
truncated tetrahedral graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline645.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline646.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline647.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline648.svg" style="height:22px; width:6px" />2,1,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline649.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline650.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline651.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline652.svg" style="height:22px; width:6px" />1,3,4,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline653.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline654.svg" style="height:22px; width:6px" />1,4,2,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline655.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline656.svg" style="height:22px; width:6px" />2,1,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline657.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline658.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline659.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline660.svg" style="height:22px; width:6px" />1,2,4,5,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline661.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline662.svg" style="height:22px; width:6px" />1,3,2,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline663.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline664.svg" style="height:22px; width:6px" />
utility graph	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline666.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline667.svg" style="height:22px; width:6px" />1,3,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline668.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline669.svg" style="height:22px; width:6px" />2,1,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline670.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline671.svg" style="height:22px; width:6px" />3,2,1}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline672.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline673.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline674.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline675.svg" style="height:22px; width:6px" />1,2,4,3}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline676.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline677.svg" style="height:22px; width:6px" />1,3,2,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline678.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline679.svg" style="height:22px; width:6px" />1,4,3,2}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline680.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline681.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline682.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline683.svg" style="height:22px; width:6px" />1,2,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline684.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline685.svg" style="height:22px; width:6px" />2,3,1,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline686.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline687.svg" style="height:22px; width:6px" />
	{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline688.svg" style="height:22px; width:6px" />{" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline689.svg" style="height:22px; width:6px" />1,2,3,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline690.svg" style="height:22px; width:6px" />, {" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline691.svg" style="height:22px; width:6px" />2,3,1,5,4}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline692.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/CayleyGraph/Inline693.svg" style="height:22px; width:6px" />

For example, the dihedral group  is a permutation group on 14 elements that can be generated by two elements corresponding to a reversal and a rotation, respectively. Therefore, any two such elements give a connected Cayley graph that has 14 nodes and 28 edges. The left figure above shows the Cayley graph for the choice of generators given by (7, 1, 2, 3, 4, 5, 6) and (7, 6, 5, 4, 3, 2, 1), with reversals shown in red and rotations in blue. Any two elements corresponding to a rotation only with give a disconnected graph, and there are exactly 15 pairs of such elements since there are  ways to pick two elements from a six possible rotations. (Here, the number 6 appears instead of 7 since the unit element may not be a member of the subset giving the Cayley graph.) The right figure shows the Cayley graph for  generated by the elements (7, 1, 2, 3, 4, 5, 6) and (6, 7, 1, 2, 3, 4, 5), which is disconnected since these elements do not generate the group (in particular, without a flip, there is no way for permutations with positive order to switch to a negative order; hence two independent rings are obtained).

The figure above shows the Cayley graph for the alternating group  using the elements (2, 1, 4, 3) and (2, 3, 1, 4) as generators, which is a directed form of the truncated tetrahedral graph.

If three vertices of the complete graph  are covered with differently colored stones and any stone may be moved to the empty vertex, then the graph of all positions forms a Cayley graph with edges indicating neighboring positions (left figure). This corresponds to the Cayley graph of the symmetric group  using the elements (2, 1, 3, 4), (3, 2, 1, 4), and (4, 2, 3, 1) as generators. It turns out that this graph is a directed version of the unique cubic symmetric graph on 24 vertices (right figure).

Royle has constructed all cubic Cayley graphs up to 1000 vertices, excluding those on 512 and 768 vertices.

The Cayley graphs of infinite groups provide interesting geometries. For example, the Cayley graphs of the free group on two generators are illustrated above (drawn out to successive levels), representing horizontal and vertical displacement respectively. Each new edge is drawn at half the size to give fractal images.

---

REFERENCES

Holton, D. A. and Sheehan, J. The Petersen Graph. Cambridge, England: Cambridge University Press, pp. 292-293, 1993.

Maschke, H. "The Representation of Finite Groups." Amer. J. Math. 16, 156-194, 1896.

McKay, B. D. and Praeger, C. E. "Vertex-Transitive Graphs Which Are Not Cayley Graphs. I." J. Austral. Math. Soc. Ser. A 56, 53-63, 1994.

McKay, B. D. and Royle, G. "Construction of All Simple Graphs with at Most 26 Vertices and Transitive Automorphism Group." Ars. Combin. 30, 161-176, 1990.

Royle, G. "Cayley Graphs." http://school.maths.uwa.edu.au/~gordon/remote/cayley/.Royle, G. "Transitive Graphs." http://school.maths.uwa.edu.au/~gordon/trans/.Sloane, N. J. A. Sequence A185959 in "The On-Line Encyclopedia of Integer Sequences."Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 938, 2002.

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Lobster Graph

Labeled Tree

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Prüfer Code

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Planted Planar Tree

Graph Strength