Arc-Transitive Graph

An arc-transitive graph, sometimes also called a flag-transitive graph, is a finite simple graph with at least one edge and no isolated vertices whose graph automorphism group acts transitively on its graph arcs (Godsil and Royle 2001, p. 59).

More generally, a graph satisfying these conventions is called -arc-transitive (or simply "-transitive") with if it has an s-route and if there is always a graph automorphism of sending each s-route onto any other s-route (Harary 1994, p. 173). In other words, a graph is -transitive if its automorphism group acts transitively on all the s-routes (Holton and Sheehan 1993, p. 203). Note that various authors prefer symbols other than , for example (Harary 1994, p. 173) or .

Arc-transitivity is an even stronger property than edge-transitivity or vertex-transitivity, so arc-transitive graphs have a very high degree of symmetry.

A 0-transitive graph is vertex-transitive. A 1-transitive graph is simply called an "arc-transitive graph" or even a "transitive graph." In terminology common in algebraic graph theory, arc-transitive graphs are also called symmetric graphs (Godsil and Royle 2001, p. 59). In this work, symmetric graph denotes the broader class of graphs that are both edge-transitive and vertex-transitive. A graph in this broader class that is not arc-transitive is called a half-arc-transitive graph (Conder and Žitnik 2016). The Doyle graph is the unique smallest example (Alspach et al. 1994).

A connected graph with no endpoints (i.e., with minimum vertex degree ) is said to be strictly -transitive (with ) if is -transitive but not -transitive (Holton and Sheehan 1993, p. 206). An -transitive graph for which exactly one graph automorphism satisfies for any two s-routes and is called -regular or -unitransitive. For connected cubic graphs, strict -transitivity is equivalent to this condition (Tutte 1947; Coxeter 1950, pp. 416-417; Frucht 1952).

The cycle graph (for ) is -transitive for all , as is for any positive integer (Holton and Sheehan 1993, p. 204).

The numbers of arc-transitive graphs on , 2, ... vertices are 0, 1, 1, 3, 2, 6, 2, 8, 5, ... (OEIS A180240), as summarized in the table below, where denotes a path graph, a cycle graph, is a ladder rung graph, a complete graph, a complete bipartite graph, a complete tripartite graph, a hypercube graph, a circulant graph, and a graph union of copies of .

2
3
4	, ,
5	,
6	, , , octahedral graph , , utility graph
7	,
8	, , , cubical graph , , , 16-cell graph ,
9	, , , generalized quadrangle ,

The numbers of connected arc-transitive graphs on , 2, ... vertices are 0, 1, 1, 2, 2, 4, 2, 5, 4, 8, ... (OEIS A286280).

A tree may be -transitive yet not -transitive. For example, the star graph with is edge-transitive and 2-transitive, but not 1-transitive. However, an connected -transitive graph that is not a tree is also -transitive for all (Holton and Sheehan 1993, p. 204).

The path graph is -transitive (Holton and Sheehan 1993, p. 203), and a cycle graph () is -transitive (Holton and Sheehan 1993, pp. 204 and 209, Exercise 6).

If is an -transitive graph, then the graph union is also -transitive for any (Holton and Sheehan 1993, p. 204). For arc-transitivity, every disconnected graph without isolated vertices is arc-transitive iff its connected components are isomorphic arc-transitive graphs. This reduction does not extend to arbitrary -transitivity, since a connected component with no s-routes does not affect the action on those routes.

Tutte (1947) showed that for any strictly -transitive connected cubic graph, (Holton and Sheehan 1993, p. 207; Harary 1994, p. 175; Godsil and Royle 2001, p. 63). Weiss (1981) established that a finite connected regular graph of degree that is strictly -transitive has or (Holton and Sheehan 1993, p. 208; Godsil and Royle 2001, p. 63).

If is a connected vertex-transitive cubic graph on vertices and is its automorphism group, then if 3 divides the group order of the stabilizer of a vertex , then is arc-transitive (Godsil and Royle 2001, p. 75).

Strictly -transitive cubic graphs need not be cage graphs (Harary 1994, p. 175). Examples include the strictly 1-transitive graph of girth 12 on 432 vertices discovered by Frucht (1952) constructed as the Cayley graph of the permutations (2, 1, 5, 8, 3, 6, 7, 4, 9), (3, 6, 1, 4, 9, 2, 7, 8, 5), and (4, 3, 2, 1, 5, 7, 6, 8, 9) and now more commonly known as the cubic symmetric graph ; the strictly 2-transitive cubical, dodecahedral graphs, Möbius-Kantor graph , and Nauru graph; and the strictly 3-transitive Desargues graph (Coxeter 1950). Some strictly -transitive graphs are illustrated above and summarized in the table below (partially based on the tables given by Coxeter 1950 and Harary 1994, p. 175).

			graph
1	432	3	cubic symmetric graph
2	4	3	tetrahedral graph
2	8	3	cubical graph
2	16	3	Möbius-Kantor graph
2	16	4	tesseract graph
2	20	3	dodecahedral graph
2	24	3	Nauru graph
2	32	5	5-hypercube graph
2	32	6	Kummer graph
2	64	6	6-hypercube graph
2	128	7	7-hypercube graph
2	256	8	8-hypercube graph
3	6	3	utility graph
3	20	3	Desargues graph
4	14	3	Heawood graph
5	30	3	Tutte 8-cage