图的秩与能量之间关系的研究

In recent years, the rank and the energy of graphs have been two important research topics in algebraic graph theory, chemical graph theory, extremal graph theory and so on. They have attracted extensive research by many scholars at home and abroad. From a quantum chemical point of view, the smaller is rank of graph, the weaker should be the bonding in the molecule and, consequently, the smaller should the energy of graph be. So, there is a close relationship between the rank and the energy of graphs. In this project, we first separately study the rank and the energy of several families of graphs, and further investigate the relationship between them. This project mainly includes the following research contents. 1. We will study the relationship between the rank and some parameters for oriented graphs, T-gain graphs and mixed graphs, respectively. 2. We will study the relationship between the energy and some parameters for oriented graphs, T-gain graphs and mixed graphs, respectively. 3. We will give the relationship between the rank and the energy of graphs in terms of some graph parameters. This project, involving spectral graph theory, chemical graph theory and so on, will deepen the study of the relationship between the rank and the energy of graphs, broaden the scope of T-gain graphs, mixed graphs and the parameters of graphs, and provide new methods for the research of the rank and the energy of graphs.

近年来，图的秩与能量已成为代数图论、化学图论、极值图论等研究领域中的两个重要研究课题，并引起了国内外众多学者的广泛关注。从量子化学的角度来看，图的秩越小，分子中的键越弱，图的能量越小。因此图的秩和能量之间存在着密不可分的关系。本项目首先分别研究几类图的秩和能量，从而进一步研究两者之间的关系。主要包括：1.研究定向图、T-gain图以及混合图的秩与图参数之间关系。2.研究定向图、T-gain图以及混合图的能量与图参数之间关系。3.通过图的参数，研究秩与能量之间的关系。本项目研究内容涉及到图谱理论、化学图论等，将深化图的秩与能量之间关系的研究，拓宽T-gain图、混合图以及图参数的范围，为图的秩与能量等领域的研究提供新的方法。

近年来，图的秩与能量已成为代数图论、化学图论、极值图论等研究领域中的两个重要研究课题，并引起了国内外众多学者的广泛关注。本项目的主要研究内容为：符号图的秩与图参数之间关系、T-gain图的秩与图参数之间关系以及图的Hamilton性等。所得重要结果如下：.1. 给出了符号图的秩与围长之间的关系。该结果推广了简单图的秩与围长之间的关系。研究了符号图的零度与匹配数之间的缺数问题。该结果进一步优化和完善了符号图的秩与匹配数之间关系的结果，也为研究更广的图类如T-gain图的秩与匹配数之间关系提供了理论依据和方法支撑。.2. 给出了T-gain图的惯性指数与匹配数之间的关系。该结果推广了T-gain图的秩与匹配数之间关系的结果，并可应用到符号图、混合图等图类。.3. 利用hyper-Zagreb指标给出了图是可迹、hamiltonian 或 Hamilton-连通的充分条件。此外，利用补图的hyper-Zagreb指标给出了图是Hamilton-连通的充分条件。.4. 研究了图的k-Hamilton性、 k-边-Hamilton性和 k-路-可覆盖性。分别利用ECI、EDS和CEI等指标给出了k-Hamilton图、k-边-Hamilton图和k-路-可覆盖图的充分条件。该结果推广了关于Hamilton图、Hamilton连通图和可迹图的结论，同时丰富了图的Hamilton问题的研究。