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 $$\cal{C}\cal{T}$$ 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 $$\cal{C}\cal{T}$$ , we prove its eccentricity matrix to be irreducible, and then determine its inertia showing that every graph in $$\cal{C}\cal{T}$$ 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 $$\cal{C}\cal{T}$$ 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 $$\cal{C}\cal{T}$$ 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.