Eigenvalues of signed graphs

TL;DR

This paper investigates the spectral properties of signed graphs, characterizing extremal graphs with maximum eigenvalues and providing bounds on distance eigenvalues, advancing spectral extremal graph theory.

Contribution

It characterizes extremal signed graphs with maximum spectral radius and eigenvalues among graphs with a spanning tree, and establishes bounds on distance eigenvalues for graphs with diameter at least 2.

Findings

Characterization of extremal signed graphs with maximum spectral radius.

Bounds on the least distance eigenvalue for graphs with diameter ≥ 2.

Extension of a conjecture by Aouchiche and Hansen to signed graphs.

Abstract

Signed graphs have their edges labeled either as positive or negative. $\rho(M)$ denote the $M$-spectral radius of $\Sigma$, where $M=M(\Sigma)$ is a real symmetric graph matrix of $\Sigma$. Obviously, $\rho(M)=\mbox{max}\{\lambda_1(M),-\lambda_n(M)\}$. Let $A(\Sigma)$ be the adjacency matrix of $\Sigma$ and $(K_n,H^-)$ be a signed complete graph whose negative edges induce a subgraph $H$. In this paper, we first focus on a central problem in spectral extremal graph theory as follows: Which signed graph with maximum $\rho(A(\Sigma))$ among $(K_n,T^-)$ where $T$ is a spanning tree? To answer the problem, we characterize the extremal signed graph with maximum $\lambda_1(A(\Sigma))$ and minimum $\lambda_n(A(\Sigma))$ among $(K_n,T^-)$, respectively. Another interesting graph matrix of a signed graph is distance matrix, i.e. $D(\Sigma)$ which was defined by Hameed, Shijin, Soorya, Germina and Zaslavsky [8]. Note that $A(\Sigma)=D(\Sigma)$ when $\Sigma\in (K_n,T^-)$. In this paper, we give upper bounds on the least distance eigenvalue of a signed graph $\Sigma$ with diameter at least 2. This result implies a result proved by Lin [11] was originally conjectured by Aouchiche and Hansen [1].