An extending result on spectral radius of bipartite graphs

TL;DR

This paper investigates the maximum spectral radius of bipartite graphs with a given number of edges, extending known bounds and linking spectral properties to twin primes.

Contribution

It determines the maximal spectral radius for bipartite graphs that are not complete and connects spectral graph theory with number theory related to twin primes.

Findings

Maximal spectral radius for non-complete bipartite graphs with e edges.

Spectral characterization of twin prime pairs.

Extension of spectral radius bounds beyond complete bipartite graphs.

Abstract

Let $G$ denote a bipartite graph with $e$ edges without isolated vertices. It was known that the spectral radius of $G$ is at most the square root of $e$, and the upper bound is attained if and only if $G$ is a complete bipartite graph. Suppose that $G$ is not a complete bipartite graph, and $e-1$ and $e+1$ are not twin primes. We determine the maximal spectral radius of $G$. As a byproduct of our study, we obtain a spectral characterization of a pair $(e-1, e+1)$ of integers to be a pair of twin primes.