A Spectral Moore Bound for Bipartite Semiregular Graphs

TL;DR

This paper establishes a new spectral upper bound on the number of vertices with a given valency in bipartite semiregular graphs, linking eigenvalues to graph structure and tightness conditions.

Contribution

It introduces a spectral Moore bound for bipartite semiregular graphs and characterizes when this bound is tight using distance-biregular graphs.

Findings

Derived an upper bound for the number of vertices with valency k.

Proved the bound is tight under specific distance-biregular graph conditions.

Developed properties of distance-biregular graphs relevant to the bound.

Abstract

Let $b(k,\ell,\theta)$ be the maximum number of vertices of valency $k$ in a $(k,\ell)$-semiregular bipartite graph with second largest eigenvalue $\theta$. We obtain an upper bound for $b(k,\ell,\theta)$ for $0 < \theta < \sqrt{k-1} + \sqrt{\ell-1}$. This bound is tight when there exists a distance-biregular graph with particular parameters, and we develop the necessary properties of distance-biregular graphs to prove this.