图变换及其在图的最小特征值的应用  

The Graft Transformations and Their Applications on the Least Eigenvalues of Graphs

作　　者：王东宜 冯小芸 张维娟[1] 王国平[1] WANG Dong-yi;FENG Xiao-yun;ZHANG Wei-juan;WANG Guo-ping(School of Mathematical Sciences,Xinjiang Normal University,Urumqi,Xinjiang,830017,China)

机构地区：[1]新疆师范大学数学科学学院,新疆乌鲁木齐830017

出　　处：《新疆师范大学学报（自然科学版）》2024年第4期62-67,共6页Journal of Xinjiang Normal University(Natural Sciences Edition)

基　　金：新疆维吾尔自治区自然科学基金项目(2023D01A38)。

摘　　要：假设G是一个简单连通图,其顶点集V(G)={v_(1),v_(2),⋯,vn}.图G的邻接矩阵表示为A(G)=(a_(ij))n×n,其中如果两个顶点vi和vj在图G中相邻,则a_(ij)=1;否则a_(ij)=0.由于A(G)是一个实对称矩阵,所以其特征值可以排列为λ_(1)(G)≥λ_(2)(G)≥⋯≥λn(G),A(G)的特征值也是图G的特征值。文章首先给出图的三个图变换,然后应用其确定存在两个有n≥12个顶点的连通图,其最小特征值可以达到所有单圈图的补图中最小,这修改了文献[9]中的主要结果。Suppose G is a connected simple graph with the vertex set V(G)={v_(1),v_(2),⋯,vn}.Then the adjacency matrix of G is A(G)=(a)ij n×n,where a_(ij)=1 if vi is adjacent to vj,and otherwise a_(ij)=0.Since A(G)is real and symmetric,its eigenvalues can be arranged asλ_(1)(G)≥λ_(2)(G)≥⋯≥λn(G),and the eigenvalues of A(G)are also called the eigenvalues of G.In this paper we first give three graft transformations on the least eigenvalues of graphs are given and then as their applications two connected graphs on n≥12 vertices whose least eigenvalues can be minimum among the complements of all unicyclic graphs are given,which modifies the main result in literature[9].

关 键 词：图变换 最小特征值 单圈图 补图 

分 类 号：O157.5[理学—数学]