Some properties of generalized distance eigenvalues of graphs

Let G be a simple connected graph with vertex set V(G) = {v1, v2, …, vn} and edge set E(G), and let dvi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d_{{v_i}}}$$\end{document} be the degree of the vertex vi. Let D(G) be the distance matrix and let Tr(G) be the diagonal matrix of the vertex transmissions of G. The generalized distance matrix of G is defined as Dα(G) = αTr(G) + (1 − α)D(G), where 0 ⩽ α ⩽ 1. Let λ1(Dα(G)) ⩾ λ2(Dα(G)) ⩾ … ⩾ λn(Dα(G)) be the generalized distance eigenvalues of G, and let k be an integer with 1 ⩽ k ⩽ n. We denote by Sk(Dα(G)) = λ1(Dα(G)) + λ2(Dα(G)) +… + λk (Dα(G)) the sum of the k largest generalized distance eigenvalues. The generalized distance spread of a graph G is defined as DαS(G) = λ1(Dα(G)) − λn(Dα(G)). We obtain some bounds on Sk((Dα(G))) and DαS(G) of graph G, respectively.