On $s$-distance-transitive graphs

TL;DR

This paper explores conditions under which s-distance-transitive graphs exhibit distance-regularity, extending known properties of distance-regular and distance-transitive graphs to broader classes with specific restrictions.

Contribution

It establishes new results showing that for certain values of s and additional restrictions, s-distance-transitivity implies distance-regularity.

Findings

s-distance-transitivity implies distance-regularity under specific conditions

Extends properties of distance-regular graphs to broader classes

Provides new criteria linking symmetry and regularity in graphs

Abstract

Distance-regular graphs have many beautiful combinatorial properties. Distance-transitive graphs have very strong symmetries, and they are distance-regular, i.e. distance-transitivity implies distance-regularity. In this paper, we give similar results, i.e. for special $s$ and graphs with other restrictions we show that $s$-distance-transitivity implies distance-regularity.