On the sum of [formula omitted] largest distance eigenvalues of graphs

Abstract

For a connected graph G with order n and an integer k≥1, we denote by Sk(D(G))=λ1(D(G))+⋯+λk(D(G)) the sum of k largest distance eigenvalues of G. In this paper, we consider the sharp upper bound and lower bound of Sk(D(G)). We determine the sharp lower bounds of Sk(D(G)) when G is a connected graph and is a tree, respectively, and characterize both the extremal graphs. Moreover, we conjecture that the upper bound is attained when G is a path of order n and prove some partial result supporting the conjecture. To prove our result, we obtain a sharp upper bound of λ2(D(G)) in terms of the order and the diameter of G, where λ2(D(G)) is the second largest distance eigenvalue of G. As applications, we prove a general inequality involving λ2(D(G)), the independence number of G, and the number of triangles in G. An immediate corollary is a conjecture of Fajtlowicz, which was confirmed in Lin (2015) by a different argument. We conclude this paper with some open problems for further study.