On the Signed Complete Graphs with Maximum Index

TL;DR

This paper investigates the maximum eigenvalue of signed complete graphs with negative edges forming a unicyclic graph, identifying the structure that achieves this maximum.

Contribution

It characterizes the structure of signed complete graphs with unicyclic negative edges that maximize the index for large n.

Findings

Maximum index achieved when negative edges form a triangle with pendant vertices

The structure with a triangle and pendant vertices maximizes the eigenvalue

Results apply to signed complete graphs with unicyclic negative subgraphs

Abstract

Let $\Gamma=(K_{n},H^-)$ be a signed complete graph whose negative edges induce a subgraph $H$. The index of $\Gamma$ is the largest eigenvalue of its adjacency matrix. In this paper we study the index of $\Gamma$ when $H$ is a unicyclic graph. We show that among all signed complete graphs of order $n>5$ whose negative edges induce a unicyclic graph of order $k$ and maximizes the index, the negative edges induce a triangle with all remaining vertices being pendant at the same vertex of the triangle.