本文選題：邊冠圖 + Monomer-dimer模型　； 參考：《集美大學(xué)》2017年碩士論文

【摘要】：給定兩個(gè)簡(jiǎn)單圖G_1和G_2,其中G1含有m條邊.取m個(gè)G_2的拷貝,記為G_2~1,G_2~2,…,G_2~m,把G_1的每一條邊e_i=(u,v)(i=1,2,…,m)的兩個(gè)頂點(diǎn)與G_2~i的每個(gè)頂點(diǎn)相連得到的圖稱為G1和G2的邊冠圖,記為G_1?G_2.在G_1?G_2中刪去G_1中的所有邊得到的圖稱為G_1與G_2的修正邊冠圖,記為G_1[G_2].把簡(jiǎn)單圖G進(jìn)行一種特殊的邊冠運(yùn)算之后變成R(G),簡(jiǎn)稱R變換,其中R(G)是把圖G中的每一條邊變成一個(gè)三角形,相當(dāng)于把G和單點(diǎn)圖K_1進(jìn)行邊冠圖運(yùn)算得到的圖,即R(G)=G?K_1.對(duì)R(G)再進(jìn)行一次R變換得到R~2(G),對(duì)R~2(G)再接著做R變換得到R~3(G),一直這樣進(jìn)行下去得到Rn(G)(其中R~n(G)=R(R~(n-1)(G)),n=1,2,3,…,特別,R~0(G)=G).張等人(Physica A,391(2012),828-833)研究了R~n(K_3)的Monomer-dimer問題,給出了求解公式,并得到了熵的表達(dá)式.本文第二章推廣了前面的結(jié)果,對(duì)任意的圖G,我們考慮了R~n(G)的Monomer-dimer問題,得到了求解公式,并證明了其熵與G的選擇無關(guān).對(duì)于邊冠圖G_1?G_2的鄰接矩陣、拉普拉斯矩陣、無符號(hào)拉普拉斯矩陣以及標(biāo)準(zhǔn)化拉普拉斯矩陣的譜,理論已經(jīng)比較完善,但對(duì)于修正的邊冠圖G_1[G_2]的相關(guān)譜還未被研究。本文第三章得到了G_1[G_2]的鄰接矩陣、拉普拉斯矩陣、無符號(hào)拉普拉斯矩陣以及標(biāo)準(zhǔn)化拉普拉斯矩陣的譜,并給出了G_1[G_2]的生成樹數(shù)目、基爾霍夫指標(biāo)等的計(jì)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].
【學(xué)位授予單位】：集美大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O157.5

【參考文獻(xiàn)】

相關(guān)期刊論文 前2條

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2 YEH Yeong-Nan;;On the number of matchings of graphs formed by a graph operation[J];Science in China(Series A:Mathematics);2006年10期

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本文編號(hào)：1925784