Signed spectral Tura\'{n} type theorems

TL;DR

This paper extends spectral bounds and Turán-type inequalities to signed graphs, solves an open problem related to eigenvalues and clique numbers, and explores conjectures and walk-eigenvalue relationships in the context of signed graphs.

Contribution

It provides new bounds and proofs for eigenvalues of signed graphs, addresses an open problem, and investigates conjectures and spectral properties specific to signed graphs.

Findings

Extended Turán's inequality for signed graphs

Solved a strengthened open problem on eigenvalues and clique number

Showed the Bollobás-Nikiforov conjecture does not always hold for signed graphs

Abstract

A signed graph $\Sigma = (G, \sigma)$ is a graph where the function $\sigma$ assigns either $1$ or $-1$ to each edge of the simple graph $G$. The adjacency matrix of $\Sigma$, denoted by $A(\Sigma)$, is defined canonically. In a recent paper, Wang et al. extended the eigenvalue bounds of Hoffman and Cvetkovi\'{c} for the signed graphs. They proposed an open problem related to the balanced clique number and the largest eigenvalue of a signed graph. We solve a strengthened version of this open problem. As a byproduct, we give alternate proofs for some of the known classical bounds for the least eigenvalues of the unsigned graphs. We extend the Tur\'{a}n's inequality for the signed graphs. Besides, we study the Bollob\'{a}s and Nikiforov conjecture for the signed graphs and show that the conjecture need not be true for the signed graphs. Nevertheless, the conjecture holds for signed graphs under some assumptions. Finally, we study some of the relationships between the number of signed walks and the largest eigenvalue of a signed graph.