The spectral excess theorem for distance-biregular graphs

TL;DR

This paper extends the spectral excess theorem to bipartite distance-biregular graphs, providing a new characterization of their structure based on spectral and combinatorial properties.

Contribution

It introduces a novel version of the spectral excess theorem specifically for bipartite distance-biregular graphs, expanding the theoretical framework.

Findings

Derived a new spectral excess theorem for bipartite distance-biregular graphs

Established conditions under which these graphs are characterized by spectral and average excess

Enhanced understanding of the structural properties of distance-biregular graphs

Abstract

The spectral excess theorem for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess. A bipartite graph is distance-biregular when it is distance-regular around each vertex and the intersection array only depends on the stable set such a vertex belongs to. In this note we derive a new version of the spectral excess theorem for bipartite distance-biregular graphs.