The generalized distance matrix

Let D(G) and Diag(Tr) denote the distance matrix and diagonal matrix of the vertex transmissions of a simple connected graph G, respectively. The distance signless Laplacian matrix of G is defined as DQ(G) = Diag(Tr)+ D(G). Heretofore, the spectral properties of D(G) and DQ(G) have attracted much more attention. In the present paper, we propose to study the convex combinations Da(G) of Diag(Tr) and D(G), defined as Da(G) = aDiag(Tr) + (1 a)D(G), 0 < a < 1. This study sheds new light on D(G) and DQ(G). Some spectral properties of Da,(G) are given and a few open problems are discussed. Furthermore, we take effort to obtain some upper and lower bounds of spectral radius of Da(G). Finally, the generalized distance spectra of some graphs obtained by operations are also studied. (C) 2018 Elsevier Inc. All rights reserved.