Geometric aspects of 2-walk-regular graphs

TL;DR

This paper explores the properties of 1- and 2-walk-regular graphs, extending key bounds and analyzing their structure, revealing that 2-walk-regular graphs have a richer combinatorial structure than 1-walk-regular graphs.

Contribution

It generalizes important bounds and structural results from distance-regular graphs to 1- and 2-walk-regular graphs, highlighting the complexity of 2-walk-regular graphs.

Findings

Generalized Delsarte's clique bound to 1-walk-regular graphs.

Extended Godsil's multiplicity bound and Terwilliger's local structure analysis to 2-walk-regular graphs.

Proved finiteness of non-geometric 2-walk-regular graphs with fixed eigenvalue and diameter.

Abstract

A $t$-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most $t$. Such graphs generalize distance-regular graphs and $t$-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular graphs, and study analogues of certain results that are important for distance regular graphs. We will generalize Delsarte's clique bound to 1-walk-regular graphs, Godsil's multiplicity bound and Terwilliger's analysis of the local structure to 2-walk-regular graphs. We will show that 2-walk-regular graphs have a much richer combinatorial structure than 1-walk-regular graphs, for example by proving that there are finitely many non-geometric 2-walk-regular graphs with given smallest eigenvalue and given diameter (a geometric graph is the point graph of a special partial linear space); a result that is analogous to a result on distance-regular graphs. Such a result does not hold for 1-walk-regular graphs, as our construction methods will show.