文献类型：期刊文献

英文题名：On the sum of k-th largest distance eigenvalues of graphs

作者：Lin, Huiqiu[1]

机构：[1] Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

年份：2018

外文期刊名：arXiv

收录：EI(收录号：20200123905)

语种：英文

外文关键词：Machine learning - Trees (mathematics)

摘要：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 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 [10] by a different argument. We conclude this paper with some open problems for further study. Copyright ? 2018, The Authors. All rights reserved.