| Abstract | 本文主要研究图的拟Laplace能量、图的关联能量、以及若干图的Kirchhoff指标.全文共分五章.
第一章首先介绍了本文用到的一些基本概念、术语和记号,其次,介绍了图的能量的研究背景和研究进展,最后介绍了本文的主要结果.
第二章首先讨论一般图的拟Laplace能量的估计,得到了一个新的下界.然后,对正则图的线图、细分图、全图以及半正则图的线图,给出了它们的拟Laplace能量的上、下界,并刻画了可达界的极值图.
第三章刻画了一些特殊图类LEL的渐近性质. 具体来讲,对正则图G,证明了其迭代线图拟Laplace能量的渐近值与G的结构无关;对具有环面(柱面、自由)边界的方格子图、具有环面边界的六角形（及三角形）格子图证明它们的LEL的增长率仅与顶点数相关.
第四章给出图的关联能量的估计.主要讨论了一般图的关联能量的估计,得到了一个新的下界.然后,对正则图的全图以及半正则图的线图,给出了它们的关联能量的上、下界,并刻画了可达界的极值图.
第五章主要研究了若干图的Kirchhoff指标.首先对正则图的细分图、全图以及半正则图的线图,给出了它们的Kirchhoff指标的上、下界,并刻画了可达界的极值图;同时给出了这些图的Kirchhoff指标的公式. 其次,对正则图G,得到了R(G)和Q(G)的Kirchhoff指标的公式. |
| Other Abstract | As an important branch of Algebraic Graph Theory, the spectral graph theory mainly concerns the relation between the combinatorial properties of a graph and the algebraic properties of matrices (such as adjacency matrix, Laplacian matrix, incidence matrix, etc.) associated with the graph. It has broad but important application inphysics, quantum chemistry, information science and communication networks and so on.
This thesis does some researches on Laplacian-energy-like invariant of a graph, incidence energy of agraph and Kirchhoff indices of some graphs, obtains some new meaningful results, and consists of the following five chapters.
In Chapter one, besides introducing the background and developments of the graph energy, some fundamental concepts, terminologies andnotations, we give a brief induction to main results of our work.
In Chapter 2, we first discuss the bounds for LEL of graphs, and obtain a new lower bound. Then, for line graph, subdivision graph,total graph of a regular graph and the line graph of a semiregular graph, we give the bounds for LEL of them in terms of vertices and regularity (or semi-regularity), and determine the graphs which the bounds are sharp.
In Chapter 3, we investigate the asymptotic behavior of LEL of some special graphs. In particular, we show that the asymptotic value of the LEL of iterated line graph of a regular graph $G$ is independent of the structure of $G$. For square lattices with toroidal (orcylindrical, free)~boundary conditions, the hexagonal and triangular
lattices with toroidal boundary condition, we prove that the growth rate of the LEL of them is only dependent on the number of vertices of them.
In Chapter 4, we mainly study the bounds for IE of graphs, and obtain a new lower bound. Then for total graph of a regular graph and the line graph of a semiregular graph, we establish the relation between the signless Laplacian polynomials of them and the original graph". Based on this, we give the bounds for IE of them in terms of vertices and regularity (or semi-regularity), and determine the graphs which the bounds are sharp.
In Chapter 5, we consider the Kirchhoff index of some graphs. Firstly, for line graph, subdivision graph, total graph of a regular graph and the line graph of a semiregular graph, we give the bounds for kirchhoff indices of them in terms of vertices and regularity
(or semi-regularity), determine the graphs which the bounds are sharp, and obtain the formulas for Kirchhoff ... |
