Locally s‐distance transitive graphs

Abstract

Abstract. We give a unified approach to analysing, for each positive integer s, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally s-arc transitive graphs. A graph is in the class if it is connected and if, for each vertex v, the subgroup of automorphisms fixing v acts transitively on the set of vertices at distance i from v, for each i from 1 to s. We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for s ≥ 2, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph, or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex orbits, or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups.

On imprimitive rank 3 permutation groups

“…Those admitting primitive rank 3 groups include symmetric designs [8][9][10], partial linear spaces [11,12] and transitive graph decompositions [1], while those admitting imprimitive rank 3 groups as an automorphism group include Latin square designs [14] (equivalent to partial linear spaces with three blocks of imprimitivity) and transitive graph decompositions [24] (for which Theorem 1.1 below is required). As well as contributing to the classification of imprimitive rank 3 permutation groups, this paper answers a question in [13] about complete multipartite graphs (see Remark 2), and identifies and fills a gap in the proof in [16] of the classification of antipodal distance transitive covers of complete graphs (see Remark 3).…”

Finite 2-Geodesic Transitive Abelian Cayley Graphs

“…A graph is said to be locally 2-distance transitive if, for each vertex v of the stabilizer A v of A := Aut( ) is transitive on both 1 (v) and 2 (v). Devillers, Giudici, Li and Praeger [2] gave a reduction of the family of locally 2-distance transitive graphs. Note that every 2-geodesic transitive graph is locally 2-distance transitive.…”

“…The octahedron K 3 [2] is a circulant of Z 6 and K 4 [2] is a circulant of Z 8 . Alspach et al [1], Kovács [10] and Li [13] gave a nice characterisation of arc transitive circulants.…”

“…We denote by K m [b] the complete multipartite graph with m parts, and each part has size b, where m ≥ 3, b ≥ 2, and K 3 [2] is the octahedron. (4) has been studied extensively, see [1,8,10,15,20].…”

Finite 2‐Distance Transitive Graphs

“…Devillers et al. studied the class of locally s ‐distance transitive graphs, using the normal quotient strategy developed for s ‐arc transitive graphs in . The condition of s ‐geodesic transitivity was investigated in several articles .…”