Locally $s$-distance transitive graphs

TL;DR

This paper introduces a unified framework for analyzing a class of finite connected graphs that generalize distance transitive and locally s-arc transitive graphs, revealing their structural properties and symmetries.

Contribution

It characterizes the class of graphs, proves closure under normal quotients, and classifies nondegenerate graphs as either complete multipartite or normal covers of basic graphs.

Findings

The class is closed under forming normal quotients.

Nondegenerate, nonbasic graphs are either complete multipartite or covers of basic graphs.

Basic graphs admit quasiprimitive or biquasiprimitive actions on their vertices.

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 of diameter at least $s$. 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 $s\geq 2$, 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.