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The eccentricity matrix ϵ (G) of a connected graph G is obtained from the distance matrix of G by retaining the largest distance in each row and each column and setting the remaining entries as 0. The eccentricity matrices of graphs are closely related to the distance matrices of graphs, nevertheless a number of properties of eccentricity matrices are substantially different from those of the distance matrices. The eccentricity matrices of trees need not be invertible though the distance matrices of trees are invertible. In this paper, we establish a necessary and sufficient condition for the eccentricity matrix of a tree to be invertible. The largest eigenvalue of ϵ (G) is called the ε-spectral radius, and the eccentricity energy (or the ε-energy) of G is the sum of the absolute values of the eigenvalues of ϵ (G). We obtain some bounds for the ε-spectral radius and characterize the extreme graphs. Two graphs are said to be ε-equienergetic if they have the same ε-energy. For any n ≥ 5 , we construct a pair of ε-equienergetic graphs on n vertices, which are not ε-cospectral. [ABSTRACT FROM AUTHOR]

The eccentricity matrix of a connected graph G is obtained from the distance matrix of G by retaining the largest distances in each row and each column, and setting the remaining entries as 0. In this article, a conjecture about the least eigenvalue of eccentricity matrices of trees, presented in the article (Wang et al., 2018), is solved affirmatively. In addition to this, the spectra and the inertia of eccentricity matrices of various classes of graphs are investigated. [ABSTRACT FROM AUTHOR]

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]