河北师范大学学报—

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刊名：河北师范大学学报（自然科学版）Journal of Hebei Normal University (Natural Science)
主办：河北师范大学
ISSN： 1000-5854
CN： 13-1061/N
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仙人掌图关于奇圈数的特征值重数的上界

(安徽理工大学数学与大数据学院,安徽淮南 232001)
DOI： 10.13763/j.cnki.jhebnu.nse.202601010

The upper bound of the multiplicity of eigenvalues by the number of odd cycles in a cactus

Wang Mengtao,Wang Long

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摘要/Abstract

摘要：

本文研究了仙人掌图的特征值重数上界问题.对于一个简单无向仙人掌图G,其邻接矩阵的特征值μ的重数用m(G,μ)表示,G中n阶奇圈的特征值组成的集合用U表示,即U={2cos2kπn∣k=0,1,…,n－1,n是一个奇数}.本文主要运用数学归μ纳法对多种仙人掌图进行分类讨论,证明了当μ ∉ U时,都有m(G,μ)≤2c_e(G)+c_o(G)+p(G),并给出了等号成立的充要条件,其中p(G)、c_o(G)与c_e(G)分别表示图G的悬挂点数、奇圈数与偶圈数.

Abstract：

This paper studies the problem of the upper bound on the multiplicity of eigenvalues of cactus.For a simple undirected cactus G,the multiplicity of an eigenvalue μ of its adjacency matrix is denoted by m(G,μ).The set consisting of eigenvalues of odd cycles of order n in G is denoted by U,that is U={2cos2kπn|k=0,1,…,n－1,where n is an odd number}.In this paper,we mainly use mathematical induction to classify and discuss various cactus graphs,and find that when μ ∉ U,it always holds that m(G,μ)≤2ce(G)+co(G)+p(G).We also give the necessary and sufficient conditions for the equality to hold,where p(G),c_o(G) and c_e(G) denote the number of pendant vertices,the number of odd cycles and the number of even cycles in graph G respectively.

关键词

关键词： 仙人掌图 ; 图的特征值 ; 邻接矩阵 ; 奇圈数

Key words： cactus ; eigenvalues of graph ; adjacency matrix ; number of odd cycles

参考文献 11

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