Inertia and spectral symmetry of eccentricity matrices of some clique trees

TL;DR

This paper investigates the spectral properties of eccentricity matrices of certain clique trees, revealing symmetry and eigenvalue distribution patterns related to graph diameter and structure.

Contribution

It establishes the irreducibility, inertia, and spectral symmetry of eccentricity matrices for clique trees with specific block structures, linking eigenvalues to graph diameter and central vertices.

Findings

Eccentricity matrices of clique trees in the class are irreducible.

Graphs with more than 4 vertices and odd diameter have four nonzero eigenvalues with specific signs.

Spectral symmetry occurs if and only if the clique tree has odd diameter and two adjacent central vertices.

Abstract

The eccentricity matrix $\mathcal E(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by leaving unchanged the largest nonzero entries in each row and each column, and replacing the remaining ones with zeros. In this paper, we consider the set $\mathcal C \mathcal T$ of clique trees whose blocks have at most two cut-vertices \textcolor{blue}{of the clique tree}. After proving the irreducibility of the eccentricity matrix of a clique tree in $\mathcal C \mathcal T$ and finding its inertia indices, we show that every graph in $\mathcal C \mathcal T$ with more than $4$ vertices and odd diameter has two positive and two negative $\mathcal E$-eigenvalues. Positive $\mathcal E$-eigenvalues and negative $\mathcal E$-eigenvalues turn out to be equal in number even for graphs in $\mathcal C \mathcal T$ with even diameter; that shared cardinality also counts the \textcolor{blue}{`diametrally distinguished'} vertices. Finally, we prove that the spectrum of the eccentricity matrix of a clique tree $G$ in $\mathcal C \mathcal T$ is symmetric with respect to the origin if and only if $G$ has an odd diameter and exactly two adjacent central vertices.