Spectral Radius of Bipartite Graphs

TL;DR

This paper determines the maximum spectral radius of bipartite graphs derived from complete bipartite graphs by edge deletion, identifying the optimal edge removal strategy and establishing new bounds based on degree sequences.

Contribution

It introduces sharp upper bounds on the spectral radius of bipartite graphs and characterizes the extremal graphs after edge deletion.

Findings

Maximum spectral radius achieved by deleting edges incident to a single vertex

Optimal edge deletion occurs at a vertex in the larger partite set

New bounds relate spectral radius to degree sequences

Abstract

Let k, p, q be positive integers with k < p < q+1. We prove that the maximum spectral radius of a simple bipartite graph obtained from the complete bipartite graph Kp,q of bipartition orders p and q by deleting k edges is attained when the deleting edges are all incident on a common vertex which is located in the partite set of order q. Our method is based on new sharp upper bounds on the spectral radius of bipartite graphs in terms of their degree sequences.