点传递对称图若干问题的研究

The symmetry of graph has been a hot research in algebraic graph theory, which is mainly described by some transitive action of it's automorphism group. Since the graph with good symmetry , such as vertices transitive symmetric graph, has an important application in internet,information security and so on. Then it has an important theoretical and practical significance to study such graphs. For the research of graph symmetry, a major work is to give them a classification and characterization, which has been closely watched. To determine the full automorphism group is a key to study the graph symmetry and it is also a hot and difficult problem in the research of algebraic graph. Since there is a lot of limitations by the traditional method, the project strive for innovation in research methods, we plan to study the problem of the full automorphism groups by the spectrum of graph, feature and probability theory. Specific, our studies focus on the following three topics: (1) To classify locally primitive graphs of order odd; (2) To give a classification of s-transitive Cayley graphs with small valency; (3) To study the full automorphism groups of some vertices transitive symmetric graphs.

图的对称性一直是代数图论研究的热点，它主要通过图自同构群的某种传递作用来描述。由于对称性较好的图，比如点传递对称图，在计算机网络、信息安全等领域有着重要的应用，因此对这类图的研究将具有重要的理论意义和现实意义。 图对称性研究的一个主要工作是对其分类和刻画，一直倍受同行关注。而确定图的全自同构群是研究图对称性的关键，也是代数图论研究的热点和难点。由于传统的研究方法有很大的局限性，所以本项目力求在研究方法上有所创新，拟利用图的谱、特征以及概率等相关知识来研究有关点传递对称图的全自同构群问题。具体的，本项目主要研究以下三个方面的内容： （1）奇数阶局部本原图的分类； （2）小度数s-传递Cayley图的分类； （3）点传递对称图全自同构群问题的研究。

图的对称性一直是代数图论研究的热点，它主要通过图自同构群的某种传递作用来描述。由于对称性较好的图，比如点传递对称图，在计算机网络、信息安全等领域有着重要的应用，因此对这类图的研究将具有重要的理论意义和现实意义。图对称性研究的一个主要工作是对其分类和刻画，一直倍受同行关注。而确定图的全自同构群是研究图对称性的关键，也是代数图论研究的热点和难点。由于传统的研究方法有很大的局限性，所以本项目首先在研究方法上有所创新：利用全局分析法、局部分析法、计算机辅助等方法相结合来研究有关点传递对称图的自同构群，从而完成对此类图的一些分类问题。具体的，我们以本项目设定的三个方面的研究内容为核心开展研究工作，并对核心问题的周边问题做了拓展思考，并得到了一些科研成果。截止目前，在本项目支持下共发表SCI论文16篇，具体得到的科研成果总结如下：.（1）Classification or description of symmetric graphs under some restrictive conditions；.（2）Classification of Some s-transitive Cayley Graphs with Small Degrees and Prime Degrees；.（3）Characterization of Spectrum, Energy and Hamiltonian Property of Graphs.