Sharp bounds on the distance spectral radius and the distance energy of graphs

Abstract

The D-eigenvalues { μ 1 , μ 2 , … , … , μ p } of a graph G are the eigenvalues of its distance matrix D and form the D-spectrum of G denoted by spec D ( G ) . The greatest D-eigenvalue is called the D-spectral radius of G denoted by μ 1 . The D-energy E D ( G ) of the graph G is the sum of the absolute values of its D-eigenvalues. In this paper we obtain some lower bounds for μ 1 and characterize those graphs for which these bounds are best possible. We also obtain an upperbound for E D ( G ) and determine those maximal D-energy graphs.