图的谱唯一及相关问题研究

项目名称： 图的谱唯一及相关问题研究

项目编号： No.11261059

项目类型： 地区科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 黄琼湘

作者单位： 新疆大学

项目金额： 50万元

中文摘要： 本项目以研究同谱与同构的关系为主线，结合代数和组合方法，以图论、图谱理论和代数理论为工具，利用图的谱、图的角、图的特征向量、图的自同构群研究图的谱唯一（graphs determined by their spectra）问题。研究主要涉及以下4个相关内容：①图同谱条件；②研究图谱特征、刻画同谱图类；③研究谱唯一条件、刻画谱唯一图类；④研究图的谱参数与图的组合参数之间的关系。 研究表明，具有n个不同主特征值的n阶图类很可能是主要的谱唯一图类，而其自同构群为单位群。基于此，本项目特别针对自同构群较小的图类研究其同谱及同构问题：一是研究具有n个不同主特征值的n阶图的谱唯一性，为推进E.R. van Dam, W.H.Haemers猜想（几乎所有图都是谱唯一的）的研究作出贡献。二是研究F.Harary提出的问题：刻画具有不同特征值的图，为推进图谱特征的研究提供新的代数方法。

中文关键词： 谱唯一问题；谱半径；整谱图；主特征值；六角系统

英文摘要： In this project, by threading the relationship between cospectrality and isomorphism as a main research clue, using graph theory, spectra theory and algebraic theory as a tool, and taking advantage of the spectra, the angles, the eigenvectors and the automorphism of a graph, we mainly study the problem of graphs determined by their spectra. It involves the following four related content: ①find the conditions of graphs have the same spectra; ②study spectral characterization of graphs and characterize cospectral graphs; ③research the conditions of graphs determined by their spectra and characterize these graphs;④learn the relationship between the spectral parameters and other graph combinatorial parameters. Previous study shows that the graphs with n distinct main eigenvalues and order n may be a majority ofthe graphs which are determined by their spectra, moreover, the automorphism of these graphs is identity group. Based on this fact, the project specially focuses on the problem of cospectral and isomorphism of the graphs whose order of automorphism group is small. In order to further and contribute E. R. van Dam and W. H. Haemers' conjecture which states that almost all graphs are determined by their spectra, one goal of this project is studying whether graphs with n distinct main eigenvalue and order n

英文关键词： DS-problem；spectral radius；integral graph；main eigenvalue；hexagonal system