On the spectral radius of unbalanced signed bipartite graphs

TL;DR

This paper investigates the spectral radius of unbalanced signed bipartite graphs, identifying the unique maximum case and analyzing bounds when negative edges form a tree, contributing to spectral graph theory.

Contribution

It characterizes the unbalanced signed bipartite graph with maximum spectral radius and explores bounds for graphs with negative edges forming a tree.

Findings

Unique unbalanced signed bipartite graph with maximum spectral radius

Maximum spectral radius for graphs with negative edges forming a tree

Spectral properties depend on switching operations and edge configurations

Abstract

A signed graph is one that features two types of edges: positive and negative. Balanced signed graphs are those in which all cycles contain an even number of positive edges. In the adjacency matrix of a signed graph, entries can be $0$, $-1$, or $1$, depending on whether $ij$ represents no edge, a negative edge, or a positive edge, respectively. The index of the adjacency matrix of a signed graph $\dot{G}$ is less or equal to the index of the adjacency matrix of its underlying graph $G$, i.e., $\lambda_1(\dot{G}) \le \lambda_1(G)$. Indeed, if $\dot{G}$ is balanced, then $\lambda_1(\dot{G})=\lambda_1(G)$. This inequality becomes strict when $\dot{G}$ is an unbalanced signed graph. Recently, Brunetti and Stani\'c found the whole list of unbalanced signed graphs on $n$ vertices with maximum (resp. minimum) spectral radius. To our knowledge, there has been little research on this problem when unbalanced signed graphs are confined to specific graph classes. In this article, we demonstrate that there is only one unbalanced signed bipartite graph on $n$ vertices with maximum spectral radius, up to an operation on the signed edges known as switching. Additionally, we investigate unbalanced signed complete bipartite graphs on $n$ vertices with a bounded number of edges and maximum spectral radius, where the negative edges induce a tree.