On the index of unbalanced signed bicyclic graphs

TL;DR

This paper investigates the largest eigenvalue of adjacency matrices in connected signed graphs, especially unbalanced bicyclic graphs, providing bounds, perturbation results, and identifying extremal graphs for large vertex counts.

Contribution

It introduces new bounds and characterizations for the index of unbalanced signed bicyclic graphs, including extremal graphs for large sizes.

Findings

Determined the first five largest indices among unbalanced bicyclic graphs for n >= 36.

Identified extremal signed graphs that attain these largest indices.

Provided general perturbation results for the index of signed graphs.

Abstract

In this paper, we focus on the index ( largest eigenvalue) of the adjacency matrix of connected signed graphs. We give some general results on the index when the corresponding signed graph is perturbed. As applications, we determine the first five largest index among all unbalanced bicyclic graphs on n >= 36 vertices together with the corresponding extremal signed graphs whose index attain these values.