边冠图的若干研究

发布时间：2018-05-23 18:07

本文选题：边冠图 + Monomer-dimer模型　；参考：《集美大学》2017年硕士论文

【摘要】：给定两个简单图G_1和G_2,其中G1含有m条边.取m个G_2的拷贝,记为G_2~1,G_2~2,…,G_2~m,把G_1的每一条边e_i=(u,v)(i=1,2,…,m)的两个顶点与G_2~i的每个顶点相连得到的图称为G1和G2的边冠图,记为G_1?G_2.在G_1?G_2中删去G_1中的所有边得到的图称为G_1与G_2的修正边冠图,记为G_1[G_2].把简单图G进行一种特殊的边冠运算之后变成R(G),简称R变换,其中R(G)是把图G中的每一条边变成一个三角形,相当于把G和单点图K_1进行边冠图运算得到的图,即R(G)=G?K_1.对R(G)再进行一次R变换得到R~2(G),对R~2(G)再接着做R变换得到R~3(G),一直这样进行下去得到Rn(G)(其中R~n(G)=R(R~(n-1)(G)),n=1,2,3,…,特别,R~0(G)=G).张等人(Physica A,391(2012),828-833)研究了R~n(K_3)的Monomer-dimer问题,给出了求解公式,并得到了熵的表达式.本文第二章推广了前面的结果,对任意的图G,我们考虑了R~n(G)的Monomer-dimer问题,得到了求解公式,并证明了其熵与G的选择无关.对于边冠图G_1?G_2的邻接矩阵、拉普拉斯矩阵、无符号拉普拉斯矩阵以及标准化拉普拉斯矩阵的谱,理论已经比较完善,但对于修正的边冠图G_1[G_2]的相关谱还未被研究。本文第三章得到了G_1[G_2]的邻接矩阵、拉普拉斯矩阵、无符号拉普拉斯矩阵以及标准化拉普拉斯矩阵的谱,并给出了G_1[G_2]的生成树数目、基尔霍夫指标等的计算公式。
[Abstract]:Given two simple graphs G _ s _ 1 and G _ s _ 2, G _ 1 contains m edges. Take a copy of the G2s and write them as G2s. Take each side of the GSP / G / 1 / or / or A graph of two vertices connected to each of the vertices of GSP 2i is called the G 1 and G 2 edges, denoted as G _ 1 / G _ 2. The graph obtained by deleting all the edges of Gsta1 in G_1?G_2 is called the modified edgewise graph of GStus 1 and G2, which is noted as GSP 1 [GSP 2]. The simple graph G is transformed into a special edge crown operation after a special edge crown operation, which is called R transformation, in which R G is transformed into a triangle from each edge of the graph G, which is equivalent to the graph of G and single point graph K1 for edge graph operation, that is, RG / G / G / G / G / K1. If we do R transformation again, we can get RGN ~ (2) and R ~ (2 +) ~ (2) then R ~ (2 +) and R ~ (3) G ~ (1), and then we can get Rn ~ (3) G ~ (3), and we can get Rnn ~ (2) G ~ (1) ~ (1) ~ (1) ~ (1) ~ (1) ~ (1) ~ (1) ~ (1) ~ (1) ~ (1). In 1998, I was very special. Zhang et al. (Physica An 391 / 2012 / 828-833) studied the Monomer-dimer problem of RGN KS _ 3), gave the solution formula and obtained the expression of entropy. In the second chapter, we generalize the previous results. For any graph G, we consider the Monomer-dimer problem of RG, obtain the solution formula, and prove that its entropy is independent of the choice of G. For the spectrum of G_1?G_2 adjacent matrix, Laplace matrix, unsigned Laplace matrix and standardized Laplace matrix, the theory has been perfect, but the correlation spectrum of modified edge graph GSP 1 [G2] has not been studied. In the third chapter, we obtain the spectrum of the adjacent matrix, Laplacian matrix, unsigned Laplace matrix and standardized Laplacian matrix of G _ s _ 1 [G _ S _ 2], and give the calculation formulas of the number of spanning trees and the Kirchhoff index of G _ S _ 1 [G _ 2].
【学位授予单位】：集美大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.5

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