【作者】 张百科；

【导师】 范益政；

【作者基本信息】 安徽大学 ， 基础数学， 2013， 硕士

【摘要】 谱图理论主要是研究图的结构性质和图的谱性质之间的联系,期望通过图的谱性质来描述图的结构性质.谱图理论在结构化学中的一个重要应用是：通过对化学物质结构建立图模型,应用图的特征值来刻画分析该化学物质的物理性质或化学性质.在这种背景下,许多研究者提出了一些基于图谱的分子拓扑指数.2000年,由E.Estrada引入衡量化学分子长链折叠情况的一个指标,这个指标后来被研究者称为Estrada指数.该指标提出以后引起了广泛地关注,成为国内外学者的研究热点.本文从两个方面综述了近年来Estrada指数的研究成果,研究了Estrada指数与一个重要图参数—染色数之间的关系,刻画了给定染色数的图类中Estrada指数取到极大的图,部分刻画了在该图类中Estrada(?)指数取到极小的图.本文的组织结构如下.第一章主要介绍了Estrada指数的研究背景和意义,以及本文所用到的一些基本概念和记号.第二章从两个方面综述了谱图理论领域Estrada指数最新研究成果.第三章首先介绍了一些基本引理,其次给出了给定染色数的Estrada(?)指数的极大图,最后探讨了顶点数与染色数相差1的Estrada(?)指数的极小图.更多还原

【Abstract】 Spectral graph theory mainly discuss relations between the structural property and the spectral property of graphs by using the spectra of graphs. An important application of spectral graph theory in structural chemistry is that one can characterize the chemical or physical properties of chemical molecule structure by using the spectra of graphs corre-sponding to chemical molecule structure. So, researchers propose some molecular topological index based on spectrum of graphs.In2000, an index was introduced by E.Estrada to measure the degree of folding of a protein, which was called the Estrada index by other researchers. In recent years, the Estrada index have became a hot topic of spectral graph theory, have received much attention. In this dissertation, we survey the recent development of Estrada index from two aspects, discuss the relationship between the Estrada index and the chromatic number which is an important graph parameter, characterize the graph whose Estrada index attains maximum among all graphs with fixed order and given chromatic number, and partially characterize the minimizing graph in this graph set.The dissertation is organized as follows. In Charter one, we briefly introduce the back-ground of the Estrada index and some concepts and notations which will be used in following Charter. In Charter two, the recent development of the Estrada index is surveyed from two aspects. In final Charter, we characterize the maximizing graph among all graphs with fixed order and given chromatic number, and characterize the minimizing graph among all graphs whose chromatic number is one less than its order.更多还原