On ABC eigenvalues and ABC energy

Let G be a graph with n vertices, and let di be the degree of its i-th vertex. The ABC matrix of G is the square matrix of order n whose (i,j)-entry is equal to (di+dj−2)/(didj) if the i-th vertex and the j-th vertex of G are adjacent, and 0 otherwise. This matrix, related closely to the atom-bond connectivity (abbreviated as ABC) index, was recently introduced by Estrada as a matrix representation of the probability of visiting a nearest neighbor edge from one side or the other of a given edge in a graph, which in the context of molecular graphs can be related to the polarizing capacity of the bond considered. The ABC eigenvalues of G are the eigenvalues of its ABC matrix, and the ABC energy of G is the sum of the absolute values of its ABC eigenvalues. In this paper, some new results for the ABC eigenvalues and the ABC energy of a graph are presented.