文献类型：期刊文献

英文题名：Spectral determinations and eccentricity matrix of graphs

作者：Wang, Jianfeng[1];Lu, Mei[2];Brunetti, Maurizio[3];Lu, Lu[4];Huang, Xueyi[5]

机构：[1]Shandong Univ Technol, Sch Math & Stat, Zibo 255049, Peoples R China;[2]TsingHua Univ, Dept Math Sci, Beijing 100084, Peoples R China;[3]Univ Naples Federico II, Dept Math & Applicat, Naples, Italy;[4]Cent South Univ, Sch Math & Stat, Changsha 410083, Peoples R China;[5]East China Univ Sci & Technol, Sch Math, Shanghai 200237, Peoples R China

年份：2022

卷号：139

外文期刊名：ADVANCES IN APPLIED MATHEMATICS

收录：;EI(收录号：20221812056232);WOS:【SCI-EXPANDED(收录号:WOS:000796451400001)】；

基金：We are grateful to the referees for their many helpful comments and suggestions, which have considerably improved the presentation of the paper. Jianfeng Wang would like to express his thanks to Professors Andreas Blass and Boyindureng Wu for their discussions about the random graph theory, and to Professors Sebastian M. Cioab? and Jack H. Koolen for their suggestions about the spectral graph theory. Jianfeng Wang is supported by NSFC (No. 11971274) . Mei Lu is supported by NSFC (No. 12171272 and 11971158) . Lu Lu is supported by NSFC (No. 12001544) and NSF of Hunan (No. 2021JJ40707) . Xueyi Huang is supported by NSFC (No. 11901540) .

语种：英文

外文关键词：Distance; Spectral determination; Self-centered graph; Eccentricity matrix; Antipodal graph

摘要：Let G be a connected graph on n vertices. For a vertex u is an element of G, the eccentricity of u is defined as epsilon(u)=max?{d(u,v)|v is an element of V(G)}, where d(u,v) denotes the distance between u and v. The eccentricity matrix E(G)=(?uv), where ?(uv ):= {d(u,v)if d(u,v)=min?{epsilon(u),epsilon(v)},0 otherwise, has been firstly introduced in Chemical Graph Theory. In literature, it is also known as the DMAX-matrix. Graphs with the diameter equal to the radius are called self-centered graphs. Two non-isomorphic graphs are said to be M-cospectral with respect to a given matrix M if they have the same M-eigenvalues. In this paper, we show that, when n ->infinity, the fractions of non-isomorphic cospectral graphs with respect to the adjacency and the eccentricity matrix behave like those only concerning the self-centered graphs with diameter two. Secondly, we prove that a graph G has just two distinct epsilon-eigenvalues if and only if G is an r-antipodal graph. Thirdly, we obtain many pairs of epsilon-cospectral graphs by using strong and lexicographic products. Finally we formulate some problems waiting to be solved in order to build up a spectral theory based on the eccentricity matrix. (C) 2022 Elsevier Inc. All rights reserved.