New results on the Dα− matrix of connected graphs

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 α ∈ [ 0 , 1 ] . In S-Y. Cui et al. (2019) [7] the matrix D α ( G ) = α Tr ( G ) + ( 1 − α ) D ( G ) is introduced and several properties are obtained. In this paper, new properties on the D α -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 α -matrix. The necessary and sufficient condition for the equality in each of the inequalities is given. Moreover, some results on the D α -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 α ( G ) is determined when G is the H -join of regular graphs.