On the distance spectral radius and the distance energy of graphs

Abstract

The D-eigenvalues {μ1, μ2, … , μ p } of a connected graph G are the eigenvalues of its distance matrix D. The D-energy of a graph G is the sum of the absolute values of its D-eigenvalues denoted by E D (G). In this article, we obtain a lower bound for the largest D-eigenvalue of G and an upper bound for E D (G) which improve Indulal's bounds [G. Indulal, Sharp bounds on the distance spectral radius and the distance energy of graphs, Linear Algebra Appl. 430 (2009), pp. 106–113]. In the final section of the article, we give an important remark on the distance regular graphs.