论文摘要
谱图理论主要研究图的谱性质和图的结构性质之间的关系，期望通过谱性质来刻画结构性质。谱图理论在结构化学中的一个应用就是：通过对有机分子建立图模型，应用图的特征值定量分析其能量级和稳定性，从而产生图的能量的概念。本文主要研究：(1)单圈图的能量在删边情形下的变化情况，以及图的Laplacian的能量的积分表达形式；(2)按照图的Laplacian谱半径，对含有相同顶点数的单圈混合图进行排序。Gutman和Hou等人对树和单圈图的极端能量已给出明确的刻画。本文从另一个角度考虑图的能量问题，即图在局部变化下，其能量的变化情况。针对于单圈图，我们发现：边的删除一般会导致能量的减少，但是对某些单圈图也会存在例外的情形，即删边导致能量的增加。2005年，Gutman和Zhou把图的Laplace特征值引入到图的能量中，给出了图的Laplacian能量的定义。对于正则图，图的能量与图的Laplacian能量是一致的。考虑到图的能量有一个非常优美的Coulson积分公式。图的Laplacian能量是否也有一个类似的表达形式呢?我们给出了图的Laplacian能量的若干积分形式。根据图的谱半径(或其它极端特征值)，对一些图类(如树，单圈图等)进行排序，可以了解这些图类的一些极端性质。近期，Fan确定了Laplacian谱半径达到最大的非奇异单圈混合图，并与Tam和Zhou刻画Laplacian谱半径达到最大的双圈混合图。对含相同顶点数的所有单圈混合图，我们按照Laplacian谱半径的大小，分别确定了谱半径达到最大，次大和第三大的单圈图。本文组织结构为：第一章介绍谱图理论及图的能量的简要背景，常用的概念和术语，以及研究问题和研究结果；第二章主要讨论单圈图在删除一条边后能量的变化问题；并给出Laplacian能量公式的积分形式；第三章则对单圈混合图按照Laplacian谱半径进行排序。
论文目录
相关论文文献
- [1].Gradient Estimates for p-Laplacian Lichnerowicz Equation on Noncompact Metric Measure Space[J]. Chinese Annals of Mathematics,Series B 2020(03)
- [2].Equivalence Relation between Initial Values and Solutions for Evolution p-Laplacian Equation in Unbounded Space[J]. Communications in Mathematical Research 2020(01)
- [3].Ordering Quasi-Tree Graphs by the Second Largest Signless Laplacian Eigenvalues[J]. Journal of Mathematical Research with Applications 2020(05)
- [4].Existence of Solutions to Nonlinear Schr?dinger Equations Involving N-Laplacian and Potentials Vanishing at Infinity[J]. Acta Mathematica Sinica 2020(10)
- [5].含偶圈图的Laplacian谱刻画[J]. 运筹学学报 2018(04)
- [6].四阶p-Laplacian边值问题正解的存在性（英文）[J]. 数学季刊(英文版) 2018(04)
- [7].The Existence of Semiclassical States for Some P-Laplacian Equation with Critical Exponent[J]. Acta Mathematicae Applicatae Sinica 2017(02)
- [8].Soft Decision Based Gaussian-Laplacian Combination Model for Noisy Speech Enhancement[J]. Chinese Journal of Electronics 2018(04)
- [9].具有奇性的Laplacian型方程周期正解的存在性[J]. 韩山师范学院学报 2016(06)
- [10].Blow-up of p-Laplacian evolution equations with variable source power[J]. Science China(Mathematics) 2017(03)
- [11].Solvability for a Coupled System of Fractional p-Laplacian Differential Equations at Resonance[J]. Communications in Mathematical Research 2017(01)
- [12].The Existence of Nodal Solutions for the Half-Quasilinear p-Laplacian Problems[J]. Journal of Mathematical Research with Applications 2017(02)
- [13].Coiflet solution of strongly nonlinear p-Laplacian equations[J]. Applied Mathematics and Mechanics(English Edition) 2017(07)
- [14].Laplacian Energies of Regular Graph Transformations[J]. Journal of Donghua University(English Edition) 2017(03)
- [15].Existence of Solutions to a p-Laplacian Equation with Integral Initial Condition[J]. Communications in Mathematical Research 2017(04)
- [16].The Signless Laplacian Spectral Characterization of Strongly Connected Bicyclic Digraphs[J]. Journal of Mathematical Research with Applications 2016(01)
- [17].Multiplicity for Nonlinear Elliptic Boundary Value Problems of p-Laplacian Type Without Ambrosetti-Rabinowitz Condition[J]. Acta Mathematicae Applicatae Sinica 2015(01)
- [18].Existence of Solutions for a Four-point Boundary Value Problem with a p(t)-Laplacian[J]. Communications in Mathematical Research 2015(01)
- [19].奇异Φ-Laplacian周期边值问题解的存在性[J]. 山东大学学报(理学版) 2015(08)
- [20].单圈图的最小无号Laplacian谱展[J]. 华南师范大学学报(自然科学版) 2013(04)
- [21].循环图的Laplacian谱展[J]. 数学杂志 2013(06)
- [22].The Normalized Laplacian Spectrum of Pentagonal Graphs and Its Applications[J]. Journal of Mathematical Research with Applications 2019(04)
- [23].Evolutionary p(x)-Laplacian Equation with a Convection Term[J]. Acta Mathematicae Applicatae Sinica 2019(03)
- [24].Existence of Nonnegative Solutions for a Class of Systems Involving Fractional(p,q)-Laplacian Operators[J]. Chinese Annals of Mathematics,Series B 2018(02)
- [25].一类p-Laplacian方程单侧全局区间分歧及应用[J]. 数学物理学报 2018(04)
- [26].Energy and Laplacian of fractal interpolation functions[J]. Applied Mathematics:A Journal of Chinese Universities 2017(02)
- [27].图的Normalized Laplacian多项式系数[J]. 集美大学学报(自然科学版) 2016(01)
- [28].一类具有奇性p-Laplacian-Rayleigh方程的周期正解[J]. 韩山师范学院学报 2016(03)
- [29].Homoclinic Solutions for a Prescribed Mean Curvature Lienard p-Laplacian Equation with a Deviating Argument[J]. Journal of Donghua University(English Edition) 2016(03)
- [30].一类奇性的p-Laplacian-Rayleigh方程的周期正解的存在性[J]. 井冈山大学学报(自然科学版) 2016(04)