On the Eccentricity Matrices of Certain Bi-Block Graphs

The eccentricity matrix of a simple connected graph G is obtained from the distance matrix of G by retaining the largest nonzero 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 \({\mathscr {B}}\) (which includes trees) of bi-block graphs. We first find the inertia of the eccentricity matrices of graphs in \({\mathscr {B}}\), and thereby, we characterize graphs in \({\mathscr {B}}\) with odd diameters. Precisely, if the diameter of \(G\in {\mathscr {B}}\) is more 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 \({\mathscr {B}}\) are irreducible.