On the eccentricity matrices of trees: Inertia and spectral symmetry.

The eccentricity matrix E (G) of a connected graph G is obtained from the distance matrix of G by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. The eigenvalues of E (G) are the E -eigenvalues of G. In this article, we find the inertia of the eccentricity matrices of trees. Interestingly, any tree on more than 4 vertices with odd diameter has exactly two positive and two negative E -eigenvalues (irrespective of the structure of the tree). Also, we show that any tree with even diameter, except the star, has the same number of positive and negative E -eigenvalues. Besides, we prove that the E -eigenvalues of a tree are 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 E -eigenvalues. [ABSTRACT FROM AUTHOR]