图的电阻矩阵谱性质研究

Spectral graph theory is an important branch in graph theory, which concern relations between graph structure, graph parameters and spectra of graph matrices. The resistance distance is a distance function on graphs, which has extensive applications in random walks of graphs, chemistry and complex networks. The resistance matrix is a “distance matrix”, whose entries are resistance distances. The spectrum of the resistance matrix has important theoretical significance and applications in the study of image processing and Kirchhoff-type indices of graphs. In this project, we investigate spectral properties of the resistance matrix, including resistance spectral characterizations of graph structure and graph parameters, characteristic polynomial of the resistance matrix, bounds for the spectral radius of the resistance matrix, and extremal graphs with respect to the spectral radius of the resistance matrix. This project is an intersection of spectral graph theory and resistance distance theory, will pride new methods and results for the study of spectral properties of the resistance matrix.

图谱理论主要研究图矩阵的谱与图结构和图参数的关系，是图论的重要分支。电阻距离是图的一种距离函数，在图的随机游动、化学和复杂网络等方面有广泛应用。图的电阻矩阵是以电阻距离为元素形成的“距离矩阵”，它的谱在图的基尔霍夫型指数与图象处理的研究中有重要的理论和应用价值。本项目研究图的电阻矩阵的谱性质，包括电阻矩阵特征值与图结构和图参数刻画、电阻矩阵的特征多项式、电阻矩阵谱半径的界以及电阻矩阵谱半径的极值图类等研究课题。本项目的研究是图谱理论与图的电阻距离理论的交叉结合，将为图的电阻矩阵谱性质研究带来新的技术方法和研究成果。

图的电阻距离的研究是图距离、图谱理论、广义逆理论等多个领域的交叉结合，已经成为图论中的一个活跃的研究领域。图的电阻距离在图的随机游走、图结构分析、化学图论和图的生成树计数等图论问题的研究中有重要作用，并且在复杂网络、控制系统和随机算法等领域中有广泛应用。本项目主要研究图的电阻距离、电阻矩阵与图结构的关系，给出了电阻距离等价图的刻画、生成树均衡图的电阻刻画、图的电阻中心性指标与特征值的关系、非一致超图谱性质等新结果。项目发表SCI检索论文6篇，支撑了3名研究生的培养。