Connected graphs with large multiplicity of $-1$ in the spectrum of the eccentricity matrix

TL;DR

This paper characterizes connected graphs with a large multiplicity of the eigenvalue -1 in the spectrum of their eccentricity matrix, expanding understanding of spectral properties related to graph structure.

Contribution

It provides a complete characterization of connected graphs with high multiplicity of -1 as an eigenvalue in the eccentricity matrix spectrum for multiplicities up to five.

Findings

Identifies specific graph classes with eigenvalue -1 multiplicity n-i for i ≤ 5.

Connects spectral properties to the median eigenvalue problem.

Enhances understanding of eccentricity matrix spectra in graph theory.

Abstract

The eccentricity matrix of a simple connected graph is obtained from the distance matrix by only keeping the largest distances for each row and each column, whereas the remaining entries become zero. This matrix is also called the anti-adjacency matrix, since the adjacency matrix can also be obtained from the distance matrix but this time by keeping only the entries equal to $1$. It is known that, for $\lambda \not\in \{-1,0\}$ and a fixed $i\in \mathbb{N}$, there is only a finite number of graphs with $n$ vertices having $\lambda$ as an eigenvalue of multiplicity $n-i$ on the spectrum of the adjacency matrix. This phenomenon motivates researchers to consider the graphs has a large multiplicity of an eigenvalue in the spectrum of the eccentricity matrix, for example, the eigenvalue $-2$ [X. Gao, Z. Stani\'{c}, J.F. Wang, Grahps with large multiplicity of $-2$ in the spectrum of the eccentricity matrix, Discrete Mathematics, 347 (2024) 114038]. In this paper, we characterize the connected graphs with $n$ vertices having $-1$ as an eigenvalue of multiplicity $n-i$ $(i\leq5)$ in the spectrum of the eccentricity matrix. Our results also become meaningful in the framework of the median eigenvalue problem [B. Mohar, Median eigenvalues and the HOMO-LUMO index of graphs, Journal of Combinatorial Theory Series B, 112 (2015) 78-92].