Inertia and Spectral Symmetry for the Eccentricity Matrices of Clique Trees

The eccentricity matrix 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 CT of clique trees whose blocks contain at most two cut-vertices of the clique tree. Along with studying the structural properties of a clique tree in CT , we prove its eccentricity matrix to be irreducible, and then determine its inertia showing that every graph in CT with more than four vertices and odd diameter has two positive and two negative E-eigenvalues. Positive E-eigenvalues and negative E-eigenvalues turn out to be equal in number even for graphs in CT with even diameter; that shared cardinality also counts the `diametrally distinguished' vertices. Finally, we prove that the spectrum of the eccentricity matrix of a clique tree G in CT is symmetric with respect to the origin if and only if G has an odd diameter and exactly two adjacent central vertices. Our results generalize those achieved on trees by I. Mahato and M. R. Kannan in 2022.