The maximum index of signed complete graphs whose negative edges induce a bicyclic graph

TL;DR

This paper investigates the maximum eigenvalue (index) of signed complete graphs where the negative edges form a bicyclic graph, identifying the structure that maximizes this index.

Contribution

It determines the structure of the bicyclic graph that maximizes the index of the signed complete graph with negative edges inducing it.

Findings

Identifies the bicyclic graph structure that yields the maximum index.

Provides a characterization of the maximum index for these signed graphs.

Advances understanding of spectral properties of signed complete graphs with specific negative edge subgraphs.

Abstract

Let $\Gamma=(K_n,H)$ be a signed complete graph whose negative edges induce a subgraph $H$. Let $A(\Gamma)$ be the adjacency matrix of the signed graph $\Gamma$. The largest eigenvalue of $A(\Gamma)$ is called the index of $\Gamma$. In this paper, the index of all the signed complete graphs whose negative edges induce a bicyclic graph $B$ is investigated. Specifically, the structure of the bicyclic graph $B$ such that $\Gamma=(K_n,B)$ has the maximum index is determined.