The main result of this paper is that, if Γ is a connected 4-valent G-arc-transitive graph and v is a vertex of Γ, then either Γ is part of a well-understood infinite family of graphs, or |Gv| ≤ 2^4 x 3^6 or 2 x |Gv| x log2(|Gv|/2) ≤ |VΓ| and that this last bound is tight. As a corollary, we get a similar result for 3-valent vertex-transitive graphs.
A bicirculant is a graph admitting an automorphism with two cycles of equal length in its cycle decomposition. A graph is said to be arc-transitive if its automorphism group acts transitively on the set of its arcs. This paper gives a complete classification of connected pentavalent arc-transitive bicirculants.
An automorphism α of a Cayley graph Cay(G,S) of a group G is color-preserving if α(g,gs) = (h,hs) or (h, hs^−1) for every edge (g,gs)∈E(Cay(G,S)). If every color-preserving automorphism of Cay(G,S) is also affine, then Cay(G,S) is a CCA (Cayley color automorphism) graph. If every Cayley graph Cay(G,S) is a CCA graph, then G is a CCA group. In this paper it is shown that there is a unique non-CCA Cayley graph X of the non-abelian group F21 of order 21. It is also shown that if Cay(G,S) is a non-CCA graph of a group G of odd square-free order, then G = H×F21 for some CCA group H, and Cay(G,S) is a Cartesian product of Cay(H,T) and X.
The following problem is considered: if H is a semiregular abelian subgroup of a transitive permutation group G acting on a finite set X, find conditions for (non)existence of G -invariant partitions of X. Conditions presented in this paper are derived by studying spectral properties of associated G -invariant digraphs. As an essential tool, irreducible complex characters of H are used. Questions of this kind arise naturally when classifying combinatorial objects which enjoy a certain degree of symmetry. As an illustration, a new and short proof of an old result of Frucht et al. (Proc Camb Philos Soc 70:211–218, 1971) classifying edge- transitive generalized Petersen graphs, is given.
A positive integer n is a Cayley number if every vertex-transitive graph of order n is a Cayley graph. In 1983, Dragan Marušič posed the problem of determining the Cayley numbers. In this paper we give an infinite set S of primes such that every finite product of distinct elements from S is a Cayley number. This answers a 1996 outstanding question of Brendan McKay and Cheryl Praeger, which they “believe to be the key unresolved question” on Cayley numbers. We also show that, for every finite product n of distinct elements from S, every transitive group of degree n contains a semiregular element.