On the spectral radius of Dα-matrix of a connected graph

In this paper, we further study the convex combinations Dα(G) of Tr(G) and D(G), defined as Dα(G) = αTr(G)+(1−α)D(G), 0 ≤ α ≤ 1, where D(G) and Tr(G) denote the distance matrix and diagonal matrix of the vertex transmissions of a simple connected graph G, respectively. We obtain some upper and lower bounds for the spectral radius of the generalized distance matrix, in terms of various graph parameters and characterize the extremal graphs. We also obtain a lower bound for the generalized distance spectral radius of a graph with given edge connectivity, in terms of the order n, the edge connectivity r and the parameter α. Further, we obtain a lower bound for the generalized distance spectral radius of a tree, in terms of the order n, the diameter d and the parameter α. We characterize the extremal graphs for some values of diameter d.