Some Spectral and Quasi-Spectral Characterizations of Distance-Regular Graphs

TL;DR

This paper introduces preintersection numbers derived from a graph's spectrum to provide new spectral and quasi-spectral characterizations of distance-regular graphs, especially those with large girth or odd-girth.

Contribution

It generalizes intersection numbers using spectral data and offers novel characterizations of distance-regularity based on these preintersection numbers.

Findings

Preintersection numbers are determined by the spectrum of the adjacency matrix.

New spectral and quasi-spectral characterizations of distance-regular graphs are provided.

Results are particularly applicable to graphs with large girth or odd-girth.

Abstract

In this paper we consider the concept of preintersection numbers of a graph. These numbers are determined by the spectrum of the adjacency matrix of the graph, and generalize the intersection numbers of a distance-regular graph. By using the preintersection numbers we give some new spectral and quasi-spectral characterizations of distance-regularity, in particular for graphs with large girth or large odd-girth.