Polynomial Characterizations of Distance-Biregular Graphs

TL;DR

This paper extends polynomial characterizations from distance-regular graphs to distance-biregular graphs, providing new spectral tools for analyzing bipartite graphs with specific regularity properties.

Contribution

It generalizes existing characterizations and the spectral excess theorem to the broader class of distance-biregular graphs, enhancing spectral analysis methods.

Findings

Extended polynomial characterizations to distance-biregular graphs

Developed spectral tools for bipartite graphs with distance-regular halved graphs

Provided new insights into the spectrum of distance-biregular graphs

Abstract

Fiol, Garriga, and Yebra introduced the notion of pseudo-distance-regular vertices, which they used to develop a new characterization of distance-regular graphs. Building on that work, Fiol and Garriga developed the spectral excess theorem for distance-regular graphs. We extend both these characterizations to distancebiregular graphs and show how these characterizations can be used to study bipartite graphs with distance-regular halved graphs and graphs with the spectrum of a distance-biregular graph.