本文選題：圖譜理論 + 譜能量　； 參考：《蘭州理工大學(xué)》2017年碩士論文

【摘要】：圖譜理論是圖論研究的重要分支,其中對圖能量的研究是近年的熱點。圖能量是用圖譜表示圖中的量,其在計算機科學(xué)、物理、化學(xué)、生命科學(xué)、控制工程等領(lǐng)域均有重要應(yīng)用。圖的譜與圖結(jié)構(gòu)密切相關(guān)。設(shè)G = (V(G),E(G))是一個簡單無向圖,其中V(G)和E(G)分別對應(yīng)頂點集和邊集。圖G對應(yīng)的鄰接矩陣記為A(G)、拉普拉斯矩陣記為L(G)、無符號拉普拉斯矩陣記為Q(G),它們的特征根構(gòu)成圖G的鄰接譜、拉普拉斯譜、無符號拉普拉斯譜。圖能量的研究對象是圖的各類矩陣及其譜。本文研究的幾類圖:由圖G1和圖G2得到的剖分冠點圖G1◇G2、剖分冠邊圖G1%紾2,由圖G和圖H1,H2,…,Hn得到的廣義R-冠點圖R(G)(?)∧inHi,阿基米德晶格圖等。通過圖的廣義矩陣得到圖的廣義特征多項式,進而求得其A-譜、L-譜、Q-譜,解決了這幾類復(fù)雜圖難于求譜的問題。作為應(yīng)用,計算了圖的生成樹數(shù)目及Kirchhoff指數(shù),求得了幾類能量。提出了一種計算機求解能量的方法,得到了阿基米德晶格圖在有限點內(nèi)的能量指標(biāo)。本文的主要成果如下:(1)計算并證明了廣義R-冠點圖R(G)(?)∧inHi的廣義特征多項式及一些特定情況下的生成樹數(shù)目和Kirchhoff指數(shù),同時構(gòu)造了一類廣義同譜圖;(2)用計算機編程的方法給出了阿基米德晶格圖的三類譜能量的具體能量值;(3)給出了剖分冠點圖G1◇G2、剖分冠邊圖G1%紾2的廣義特征多項式并擴大了選圖范圍;(4)用計算機統(tǒng)計Estrada指數(shù)的數(shù)值范圍,找到Estrada幾乎同能圖,同時統(tǒng)計了HOMO-LUMO距離和指數(shù)。
[Abstract]:Map theory is an important branch of graph theory, among which the study of graph energy is a hot topic in recent years. The energy of graph is represented by graph, which has important applications in computer science, physics, chemistry, life science, control engineering and so on. The spectrum of a graph is closely related to its structure. Let G = V ~ (+) G ~ (+) be a simple undirected graph, where V _ (G) and E _ (G) correspond to vertex set and edge set, respectively. The adjacent matrix corresponding to graph G is denoted as Agna Gi, Laplace matrix is denoted as Ln Gn, and unsigned Laplace matrix is denoted as QG. Their characteristic roots form the adjacent spectrum, Laplace spectrum and unsigned Laplace spectrum of graph G. The research object of graph energy is the matrix of graph and its spectrum. In this paper, we study several kinds of graphs: G _ 1 G _ 2, G _ 1% G _ 2, G _ 1 and H _ 1H _ 2 obtained from G _ 1 and G _ 2. The generalized R- crown-point graph R _ G _ (G ~ (+) A _ (H _ n) in Hi, Archimedean lattice diagram and so on. The generalized characteristic polynomial of a graph is obtained by the generalized matrix of a graph, and its A-spectrum L- spectrum Q- spectrum is obtained, which solves the problem that it is difficult to obtain the spectrum of these complex graphs. As an application, the number of spanning trees and Kirchhoff exponents of graphs are calculated, and several kinds of energy are obtained. In this paper, a computer method for solving energy is proposed, and the energy index of Archimedes lattice graph at finite point is obtained. The main results of this paper are as follows: 1) in this paper, we calculate and prove the generalized characteristic polynomials of the generalized R- crown graph R ~ (1) and the number of spanning trees and the Kirchhoff exponent in some special cases. At the same time, a class of generalized isospectral graphs is constructed. The specific energy values of three kinds of spectral energy of Archimedean lattice graphs are given by computer programming.) the generalized characteristics of G _ 1 ~ G _ 2 and G _ 1% ~ (2) C _ 2 are given. The range of Estrada exponent is calculated by computer, and the numerical range of Estrada exponent is calculated by computer. The Estrada almost isomorphic graph is found and the HOMO-LUMO distance and exponent are also calculated.
【學(xué)位授予單位】：蘭州理工大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：TP391.41;O157.5

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