On the First Eigenvalue of Bipartite Graphs

TL;DR

This paper investigates the maximum largest eigenvalue of bipartite graphs with fixed edges and bipartition sizes, proposing a conjecture analogous to the Brualdi-Hoffman conjecture and proving it in specific cases.

Contribution

It introduces a conjecture for the maximum eigenvalue in bipartite graphs and proves it in certain special cases, extending spectral graph theory.

Findings

Conjecture on maximum eigenvalue for bipartite graphs proposed

Proved the conjecture in specific cases

Provides bounds and structural insights

Abstract

In this paper we study the maximum value of the largest eigenvalue for simple bipartite graphs, where the number of edges is given and the number of vertices on each side of the bipartition is given. We state a conjectured solution, which is an analog of the Brualdi- Hoffman conjecture for general graphs, and prove the conjecture in some special cases.