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

The D-eigenvalues {μ1,μ2,…,…,μp} of a graph G are the eigenvalues of its distance matrix D and form the D-spectrum of G denoted by specD(G). The greatest D-eigenvalue is called the D-spectral radius of G denoted by μ1. The D-energy ED(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 μ1 and characterize those graphs for which these bounds are best possible. We also obtain an upperbound for ED(G) and determine those maximal D-energy graphs.