# On a version of the spectral excess theorem

## TL;DR

This paper characterizes when a graph's distance matrix is a polynomial in its adjacency matrix using spectral properties and extends the spectral excess theorem to more general graphs.

## Contribution

It provides a new spectral characterization for the polynomial relation between distance and adjacency matrices, generalizing the spectral excess theorem.

## Key findings

- Characterization of when the distance matrix is a polynomial in the adjacency matrix.
- Extension of the spectral excess theorem to general graphs using Laplacian spectrum.
- Conditions involving the spectrum and mean distances for such polynomial relations.

## Abstract

Given a regular (connected) graph $\Gamma=(X,E)$ with adjacency matrix $A$, $d+1$ distinct eigenvalues, and diameter $D$, we give a characterization of when its distance matrix $A_D$ is a polynomial in $A$, in terms of the adjacency spectrum of $\Gamma$ and the arithmetic (or harmonic) mean of the numbers of vertices at distance $\le D-1$ of every vertex. The same results is proved for any graph by using its Laplacian matrix $L$ and corresponding spectrum. When $D=d$ we reobtain the spectral excess theorem characterizing distance-regular graphs.