Abstract

Let G be a simple undirected connected graph. Let D(G) be the distance matrix of G and let Tr(G) be the diagonal matrix of the vertex transmissions in G. Let alpha is an element of[0, 1]. In S-Y. Cui et al. (2019) [7] the matrix D-alpha(G) = alpha Tr(G)+ (1 - alpha) D(G) is introduced and several properties are obtained. In this paper, new properties on the D-alpha-matrix are derived including inequalities that involve the largest vertex transmission and the spectral radii of the distance matrix, distance signless Laplacian matrix and D-alpha-matrix. The necessary and sufficient condition for the equality in each of the inequalities is given. Moreover, some results on the D-alpha-matrix of a graph with independent sets of vertices sharing the same set of neighbors, including the case of a complete multipartite graph, are obtained. Finally, the spectrum of D-alpha (G)is determined when G is the H-join of regular graphs. (C) 2019 Elsevier Inc. All rights reserved.