Strongly walk-regular graphs

TL;DR

This paper explores the properties and classifications of strongly walk-regular graphs, a generalization of strongly regular graphs, characterizing their structure based on eigenvalues and walk-regularity for different lengths.

Contribution

It provides a comprehensive classification of strongly walk-regular graphs, especially focusing on regular graphs with four eigenvalues and their walk-regularity for various lengths.

Findings

Strongly walk-regular graphs are limited to specific families including complete, empty, and certain regular graphs.

Regular four-eigenvalue graphs are characterized in terms of eigenvalues for their walk-regularity.

Infinite families of graphs are identified that are strongly -walk-regular, with open questions remaining for other cases.

Abstract

We study a generalization of strongly regular graphs. We call a graph strongly walk-regular if there is an $\ell >1$ such that the number of walks of length $\ell$ from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint union of complete bipartite graphs of the same size and isolated vertices, or a regular graph with four eigenvalues. Graphs from the first three families in this list are indeed strongly $\ell$-walk-regular for all $\ell$, whereas the graphs from the fourth family are $\ell$-walk-regular for every odd $\ell$. The case of regular graphs with four eigenvalues is the most interesting (and complicated) one. Such graphs cannot be strongly $\ell$-walk-regular for even $\ell$. We will characterize the case that regular four-eigenvalue graphs are strongly $\ell$-walk-regular for every odd $\ell$, in terms of the eigenvalues. There are several examples of infinite families of such graphs. We will show that every other regular four-eigenvalue graph can be strongly $\ell$-walk-regular for at most one $\ell$. There are several examples of infinite families of such graphs that are strongly 3-walk-regular. It however remains open whether there are any graphs that are strongly $\ell$-walk-regular for only one particular $\ell$ different from 3.