On the eccentricity matrices of trees: Inertia and spectral symmetry

Iswar Mahato, M. Rajesh Kannan

Published 2022-03-30Version 1

The eccentricity matrix $\mathcal{E}(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by keeping the largest non-zero entries in each row and each column, and leaving zeros in the remaining ones. The eigenvalues of $\mathcal{E}(G)$ are called the $\mathcal{E}$-eigenvalues of $G$. In this article, we find the inertia of the eccentricity matrices trees. Interestingly, any tree on more than $5$ vertices with odd diameter has two positive and two negative $\mathcal{E}$-eigenvalues (irrespective of the structure of the tree). The number of positive and negative $\mathcal{E}$-eigenvalues of trees with even diameter are the equal and equals to the number of 'diametrically distinguished' vertices (see Definition 3.1). Besides we prove that the spectrum of the eccentricity matrix of a tree is symmetric with respect to the origin if and only if the tree has odd diameter. As an application, we characterize the trees with three distinct eccentricity $\mathcal{E}$-eigenvalues.