Bipartite biregular Moore graphs

TL;DR

This paper improves the Moore bound for bipartite biregular graphs with odd diameter and presents new constructions of such graphs that attain this bound, including spectral properties and uniqueness results.

Contribution

It introduces an improved Moore bound for bipartite biregular graphs with odd diameter and provides new constructions that achieve this bound, some uniquely up to isomorphism.

Findings

New Moore bound for bipartite biregular graphs with odd diameter

Explicit constructions of graphs attaining the Moore bound

Uniqueness of certain extremal graphs for diameters 3, 4, and 5

Abstract

A bipartite graph $G=(V,E)$ with $V=V_1\cup V_2$ is biregular if all the vertices of a stable set $V_i$ have the same degree $r_i$ for $i=1,2$. In this paper, we give an improved new Moore bound for an infinite family of such graphs with odd diameter. This problem was introduced in 1983 by Yebra, Fiol, and F\`abrega.\\ Besides, we propose some constructions of bipartite biregular graphs with diameter $d$ and large number of vertices $N(r_1,r_2;d)$, together with their spectra. In some cases of diameters $d=3$, $4$, and $5$, the new graphs attaining the Moore bound are unique up to isomorphism.