Inertia and spectral symmetry of the eccentricity matrices of a class of bi-block graphs

TL;DR

This paper investigates the spectral properties of eccentricity matrices of a specific class of bi-block graphs, revealing symmetry conditions and inertia characteristics related to graph diameter.

Contribution

It characterizes the inertia and spectral symmetry of eccentricity matrices for bi-block graphs, including a criterion based on graph diameter.

Findings

Eigenvalues are symmetric about zero if and only if the graph's diameter is odd.

Eccentricity matrices of these graphs are irreducible.

Graphs with diameter greater than three have eigenvalues symmetric about zero only when diameter is odd.

Abstract

The eccentricity matrix of a simple connected graph G is obtained from the distance matrix of G by retaining the largest non-zero distance in each row and column, and the remaining entries are defined to be zero. A bi-block graph is a simple connected graph whose blocks are all complete bipartite graphs with possibly different orders. In this paper, we study the eccentricity matrices of a subclass B (which includes trees) of bi-block graphs. We first find the inertia of the eccentricity matrices of graphs in B, and thereby we characterize graphs in B with odd diameters. Precisely, if G in B with diameter of G greater than three, then we show that the eigenvalues of the eccentricity matrix of G are symmetric with respect to the origin if and only if the diameter of G is odd. Further, we prove that the eccentricity matrices of graphs in B are irreducible.