Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ρ and σ , where ρ is (m,n)semiregular for some integers m ≥ 1, n ≥ 2, and where σ normalizes ρ, cyclically permuting the orbits of ρ in such a way that σ has at least one fixed vertex. A halfarc-transitive graph is a vertexand edgebut not arc-transitive graph. In this arti...

A graph X is k-arc-transitive if its automorphism group acts transitively on the set of it-arcs of X. A circulant is a Cayley graph of a cyclic group. A classification of 2-arc-transitive circulants is given.

A permutation group is said to be quasiprimitive if each of its nontrivial normal subgroups is transitive. A structure theorem for finite quasiprimitive permutation groups is proved, along the lines of the O'NanScott Theorem for finite primitive permutation groups. It is shown that every finite, non-bipartite, 2-arc transitive graph is a cover of a quasiprimitive 2-arc transitive graph. The str...

This paper forms part of a study of 2-arc transitivity for finite imprimitive symmetric graphs, namely for graphs admitting an automorphism groupG that is transitive on ordered pairs of adjacent vertices, and leaves invariant a nontrivial vertex partition B. Such a group G is also transitive on the ordered pairs of adjacent vertices of the quotient graph B corresponding toB. If in additionG is ...

The present paper is devoted classification of $\mathrm {AT4}(p,p+2,r)$-graphs. There a unique {AT4}(p,p+2,r)$-graph with $p=2$, namely, the distance-transitive Soicher graph intersection array $\{56, 45, 16, 1;1, 8, 56\}$, whose local graphs are isomorphic to Gewirtz graph. existence an ${p>2}$ remains open question. It known that each strongly regular parameters $\big ((p+2)(p^2+4p+2),p(p+3),...

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...

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...

We consider a code to be a subset of the vertex set of a Hamming graph. The set of s-neighbours of a code is the set of vertices, not in the code, at distance s from some codeword, but not distance less than s from any codeword. A 2-neighbour transitive code is a code which admits a group X of automorphisms which is transitive on the s-neighbours, for s = 1, 2, and transitive on the code itself...

We explore the relation between vertexand edge-transitivity and arc-transitivity of various graphs. We exhibit several families of graphs whose vertexand edgetransitivity imply arctransitivity. In particular, we show that any vertexand edgetransitive graph with twice a prime number of vertices is arctransitive by simplifying the proof of a theorem by Cheng and Oxley, in which they classify all ...