Signed graphs with maximal index

TL;DR

This paper determines the maximum eigenvalue of signed graphs with given size and negative edges, settling a conjecture and advancing spectral graph theory.

Contribution

It provides the exact maximum index for complete signed graphs with specified negative edges, resolving a conjecture by Koledin and Stanić.

Findings

Maximal index for complete signed graphs with given parameters identified

Conjecture by Koledin and Stanić confirmed and corrected

Advances understanding of spectral properties of signed graphs

Abstract

The index of a signed graph is the largest eigenvalue of its adjacency matrix. For positive integers $n$ and $m\le n^2/4$, we determine the maximal index of complete signed graphs with $n$ vertices and $m$ negative edges. This settles (the corrected version of) a conjecture by Koledin and Stani\'c (2017).