The spectral excess theorem for graphs with few eigenvalues whose distance-$2$ or distance-$1$-or-$2$ graph is strongly regular

TL;DR

This paper extends the spectral excess theorem to characterize regular graphs with few eigenvalues whose distance-2 or combined distance-1-or-2 graph is strongly regular, using spectral and combinatorial properties.

Contribution

It provides a new spectral characterization of such graphs, linking eigenvalues, mean distances, and strong regularity of derived graphs.

Findings

Characterization of graphs via spectrum and mean distance

Extension of spectral excess theorem to new graph classes

Identification of conditions for strong regularity in derived graphs

Abstract

We study regular graphs whose distance-$2$ graph or distance-$1$-or-$2$ graph is strongly regular. We provide a characterization of such graphs $\Gamma$ (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance $d$ from every vertex, where $d+1$ is the number of different eigenvalues of $\Gamma$. This can be seen as a another version of the so-called spectral excess theorem, which characterizes in a similar way those regular graphs that are distance-regular.