A spectral version of the Moore problem for bipartite regular graphs

TL;DR

This paper establishes a new upper bound for the maximum size of bipartite $k$-regular graphs with constrained second largest eigenvalue, and characterizes the existence of certain bipartite distance-regular graphs based on diameter and girth.

Contribution

It provides a general upper bound for $b(k, heta)$ and characterizes the existence of bipartite distance-regular graphs with specific girth and diameter conditions.

Findings

Derived an upper bound for $b(k, heta)$ for all $0 \\leq \\theta < 2\\sqrt{k-1}$.

Exact values of $b(k, heta)$ when certain bipartite distance-regular graphs exist.

Proved non-existence of bipartite distance-regular graphs with $g \\geq 2d-2$ for $d=11$ or $d \\geq 15$.

Abstract

Let $b(k,\theta)$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $\theta$. In this paper, we obtain a general upper bound for $b(k,\theta)$ for any $0\leq \theta< 2\sqrt{k-1}$. Our bound gives the exact value of $b(k,\theta)$ whenever there exists a bipartite distance-regular graph of degree $k$, second largest eigenvalue $\theta$, diameter $d$ and girth $g$ such that $g\geq 2d-2$. For certain values of $d$, there are infinitely many such graphs of various valencies $k$. However, for $d=11$ or $d\geq 15$, we prove that there are no bipartite distance-regular graphs with $g\geq 2d-2$.