The anti-adjacency matrix of a graph: Eccentricity matrix

In this paper we introduce a new graph matrix, named the anti-adjacency matrix or eccentricity matrix, which is constructed from the distance matrix of a graph by keeping for each row and each column only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping for each row and each column only the distances equal to 1. We show that the eccentricity matrix of trees is irreducible, and we investigate the relations between the eigenvalues of the adjacency and eccentricity matrices. Finally, we give some applications of this new matrix in terms of molecular descriptors, and we conclude by proposing some further research problems.