On 2-arc-transitivity of Cayley graphs

AbstractThe classification of 2-arc-transitive Cayley graphs of cyclic groups, given in (J. Algebra. Combin. 5 (1996) 83–86) by Alspach, Conder, Xu and the author, motivates the main theme of this article: the study of 2-arc-transitive Cayley graphs of dihedral groups. First, a previously unknown infinite family of such graphs, arising as covers of certain complete graphs, is presented, leading to an interesting property of Singer cycles in the group PGL(2,q), q an odd prime power, among others. Second, a structural reduction theorem for 2-arc-transitive Cayley graphs of dihedral groups is proved, putting us—modulo a possible existence of such graphs among regular cyclic covers over a small family of certain bipartite graphs—a step away from a complete classification of such graphs. As a byproduct, a partial description of 2-arc-transitive Cayley graphs of abelian groups with at most three involutions is also obtained

On vertex symmetric digraphs

AbstractIt is proved that if p is a prime, k and m⩽p are positive integers, and I is a vertex symmetric diagraph of order pk or mp, then Γ has an automorphism all of whose orbits have cardinality p. Vertex symmetric graphs of order 2p such that 2p−1 is not the square of a composite integer and vertex symmetric digraphs of order pk are characterised

Recent trends and future directions in vertex-transitive graphs

A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade

Symmetries of Hexagonal Molecular Graphs on the Torus

Symmetric properties of some molecular graphs on the torus are studied. In particular we determine which cubic cyclic Haar graphs are 1-regular, which is equivalent to saying that their line graphs are ½-arc-transitive. Although these symmetries make all vertices and all edges indistinguishable, they imply intrinsic chirality of the corresponding molecular graph

The strongly distance-balanced property of the generalized Petersen graphs

A graph ▫$X$▫ is said to be strongly distance-balanced whenever for any edge ▫$uv$▫ of ▫$X$▫ and any positive integer ▫$i$▫, the number of vertices at distance ▫$i$▫ from ▫$u$▫ and at distance ▫$i + 1$▫ from ▫$v$▫ is equal to the number of vertices at distance ▫$i + 1$▫ from ▫$u$▫ and at distance ▫$i$▫ from ▫$v$▫. It is proven that for any integers ▫$k ge 2$▫ and ▫$n ge k^2 + 4k + 1$▫, the generalized Petersen graph GP▫$(n, k)$▫ is not strongly distance-balanced