Graphs possessing a high degree of symmetry have often been considered in topological graph theory. For instance, a number of constructions of genus embeddings by means of current or voltage graphs is based on the observation that a graph can be represented as a Cayley graph for some group. Another kind of embedding problems where symmetrical graphs are encountered is connected with regular map...

This paper is a contribution towards the determination of all finite distance-transitive graphs. We obtain a classification of all the antipodal distance-transitive graphs having as antipodal quotient a complete graph Kn . Such a graph necessarily has diameter 2 or 3 (see for example [2, Proposition 4.2.2 (ii)]). Those of diameter 2 are simply the complete multipartite graphs Kr,...,r with n pa...

An interesting fact is that almost all the connected 2-arc-transitive nonnormal Cayley graphs on nonabelian simple groups with small valency or prime (provided solvable vertex stabilizers) are alternating An. This naturally motivates study of for arbitrary valency. In this paper, we characterize automorphism such graphs. particular, show a non-complete (G,2)-arc-transitive graph G simple, socle...

A code in a graph 0 is a non-empty subset C of the vertex set V of 0. Given C , the partition of V according to the distance of the vertices away from C is called the distance partition of C . A completely regular code is a code whose distance partition has a certain regularity property. A special class of completely regular codes are the completely transitive codes. These are completely regula...

A digraph is 3-quasi-transitive (resp. 3-transitive), if for any path x0x1 x2x3 of length 3, x0 and x3 are adjacent (resp. x0 dominates x3). César Hernández-Cruz conjectured that if D is a 3-quasi-transitive digraph, then the underlying graph of D, UG(D), admits a 3-transitive orientation. In this paper, we shall prove that the conjecture is true.

For an integer m ≥ 3, a near m-gonal graph is a pair (Σ,E) consisting of a connected graph Σ and a set E of m-cycles of Σ such that each 2-arc of Σ is contained in exactly one member of E, where a 2-arc of Σ is an ordered triple (σ, τ, ε) of distinct vertices such that τ is adjacent to both σ and ε. The graph Σ is call (G, 2)-arc transitive, where G ≤ Aut(Σ), if G is transitive on the vertex se...

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Let p and q be primes. It is known that no tetravalent half-arc-transitive graphs of order 2p2 exist and a tetravalent half-arctransitive graph of order 4p must be non-Cayley; such a non-Cayley graph exists if and only if p − 1 is divisible by 8 and it is unique for a given o...

An in nite family of cubic edge transitive but not vertex transitive graph graphs with edge stabilizer isomorphic to Z is constructed

We introduce a construction called the fractional multiples of a graph. These graphs are used to settle a question raised by E. Welzl: We show that if G and H are vertex-transitive graphs such that there exists a homomorphism fromG toH but no homomorphism fromH to G, then there exists a vertex-transitive graph that is homomorphically “in between” G and H.