The Laplacian spectral excess theorem for distance-regular graphs

TL;DR

This paper extends the spectral excess theorem to non-regular graphs using the Laplacian spectrum, providing a new characterization of distance-regular graphs without the regularity assumption.

Contribution

It introduces a Laplacian spectral excess theorem applicable to non-regular graphs, broadening the scope of spectral graph theory.

Findings

Proves the spectral excess theorem using Laplacian spectrum

Characterizes distance-regular graphs without regularity

Establishes equality conditions for the Laplacian spectral excess

Abstract

The spectral excess theorem states that, in a regular graph G, the average excess, which is the mean of the numbers of vertices at maximum distance from a vertex, is bounded above by the spectral excess (a number that is computed by using the adjacency spectrum of G), and G is distance-regular if and only if equality holds. In this note we prove the corresponding result by using the Laplacian spectrum without requiring regularity of G.