Research Projects

A graph is strongly connected if its adjacency matrix is irreducible. However, this is not true for the eccentricity matrix of a connected graph. The problem of irreducibility of eccentricity matrices has been studied for trees and special graphs like cycles and windmill graphs. Further research is needed for general graphs. The spectral radius and least eigenvalue of eccentricity matrices of graphs is less known. The concept of energy of a graph is defined as the sum of the absolute values of its adjacency eigenvalues. Recently, the concept of energy has been extended to eccentricity matrices, and analogues of some basic results have been obtained. One direction in energy of graphs is studying the extremal energy problem, determining graphs with maximal and minimal energies in certain families. The rank and nullity of adjacency matrices of graphs are well studied, but the concept of eccentricity matrices remains a challenge. The distance is well defined for strongly connected digraphs, and the eccentricity matrix has been defined for digraphs and its basic spectral properties have been studied.