关于两个图的一类新连接图的谱  

On the spectrum of a new join of two graphs

作　　者：刘剑萍[1] 吴先章 陈锦松[1] LIU Jianping;WU Xianzhang;CHEN Jinsong(College of Mathematics and Computer Science,Fuzhou University,Fuzhou 350116,China)

机构地区：[1]福州大学数学与计算机科学学院,福建福州350116

出　　处：《浙江大学学报（理学版）》2021年第2期180-188,195,共10页Journal of Zhejiang University（Science Edition）

基　　金：The Project Supported by NSFC(11771362);the Natural Science Foundation of Fujian Province(2019J01643,2019J01645)。

摘　　要：给定2个图G_(1)和G_(2),设G_(1)的边集E(G_(1))={e_(1),e_(2),…,e_(m1)},则图G_(1)⊙G_(2)可由一个G_(1),m_(1)个G_(2)通过在G_(1)对应的每条边外加一个孤立点,新增加的点记为U={u_(1),u_(2),…,u_(m1)},将u_(i)分别与第i个G_(2)的所有点以及G_(1)中的边e_(i)的端点相连得到,其中i=1,2,…,m_(1)。得到:(i)当G_(1)是正则图,G_(2)是正则图或完全二部图时,确定了G_(1)⊙G_(2)的邻接谱(A-谱)。(ii)当G_(1)是正则图,G_(2)是任意图时,给出了G_(1)⊙G_(2)的拉普拉斯谱(L-谱)。(iii)当G_(1)和G_(2)都是正则图时,给出了G_(1)⊙G_(2)的无符号拉普拉斯谱(Q-谱)。作为以上结论的应用,构建了无限多对A-同谱图、L-同谱图和Q-同谱图;同时当G_(1)是正则图时,确定了G_(1)⊙G_(2)支撑树的数量和Kirchhoff指数。Given graphs G_(1)and G_(2),let E(G_(1))={e_(1),e_(2),…,e_(m1)}be the edge set of G_(1),the graph G_(1)⊙G_(2)can be obtained from one copy of G_(1)and m1 copies of G_(2)by adding a new vertex corresponding to each edge of G_(1),letting the resulting new vertex set be U={u_(1),u_(2),…,u_(m1)},and joining uiwith each vertex of i-th copy of G_(2)and with the endpoints of ei,for i=1,2,…,m1.We can determine:(i)the adjacency spectrum of G_(1)⊙G_(2)for G_(1),G_(2)are both regular graphs,or G_(1)is regular graph,but G_(2)is a complete bipartite graph;(ii)the Laplacian spectrum of G_(1)⊙G_(2)when G_(1)is a regular graph and G_(2)is an arbitrary graph;(iii)the signless Laplacian spectrum of G_(1)⊙G_(2)for both G_(1)and G_(2)are regular graphs.As applications,we construct infinitely many pairs of A-cospectral graphs,L-cospectral graphs and Q-cospectral graphs.and determine the number of spanning trees and the Kirchhoff index of G_(1)⊙G_(2),where G_(1)is a regular graph.

关 键 词：谱 同谱图 支撑树 基尔霍夫(Kirchhoff)指数 

分 类 号：O157[理学—数学]