On perturbations of almost distance-regular graphs

TL;DR

This paper characterizes certain almost distance-regular graphs, called $h$-punctually walk-regular graphs, through the cospectrality of their perturbed graphs, providing new insights into their structure and properties.

Contribution

It introduces a novel characterization of $h$-punctually walk-regular graphs via graph perturbations and cospectrality, extending the theory of distance-regular graphs.

Findings

Certain perturbations produce cospectral graphs if and only if others do in walk-regular graphs.

New characterizations of distance-regular graphs are obtained.

The study connects graph perturbation theory with properties of almost distance-regular graphs.

Abstract

In this paper we show that certain almost distance-regular graphs, the so-called $h$-punctually walk-regular graphs, can be characterized through the cospectrality of their perturbed graphs. A graph $G$ with diameter $D$ is called $h$-punctually walk-regular, for a given $h\le D$, if the number of paths of length $\ell$ between a pair of vertices $u,v$ at distance $h$ depends only on $\ell$. The graph perturbations considered here are deleting a vertex, adding a loop, adding a pendant edge, adding/removing an edge, amalgamating vertices, and adding a bridging vertex. We show that for walk-regular graphs some of these operations are equivalent, in the sense that one perturbation produces cospectral graphs if and only if the others do. Our study is based on the theory of graph perturbations developed by Cvetkovi\'c, Godsil, McKay, Rowlinson, Schwenk, and others. As a consequence, some new characterizations of distance-regular graphs are obtained.