关于图谱理论与符号模式矩阵幂敛性质的研究

The research about the spectral graph theory and the property of a sign pattern's power sequence becomes a research hotspot in the graph theory and combinatoric matrix theory now, which not only have relations with many fields of mathematics, but also have many applications in communication science, biology chemestry, ecnomics, theoretical computer science. Recently, based on the work of the predecessors, we have studied the relations between the existence of some subgraphs and the graph spectrum, the relations between the existence of some graph minors and the graph spectrum, the relations between the strctural parameters and the graph spectrum as well; we have explored the applications of the property of a sign pattern's power sequece furtherly, determined the base set of some special classes of sign patterns and obtained some signifcant results on the characterizition of the base of the general primitive nopowerful sign patterns. We hope that the internal relations between the graph structure property and matrix property is studied from different viewpoints such as algebraic, combinatoric, toplogy, number theory, matrix theory, statistic method and technology,and so on. With graph structure and toplogy property, we also hope that the property of power sequence of the sign patterns is studied by combinatoric, number theory and statistic method and technology. As well, we hope that some new tools for studying graphs spectra and the property of a sign pattern's power sequece can be explored and enriched such that some new problems can be solved. For the new tools of studying the spectra of graphs and the property of a sign pattern's power sequece, we will furtherly explore their theoretical and realitic applications.

图谱理论和符号模式矩阵幂敛性质的研究是图论和组合矩阵论中的研究热点，其不仅与众多的数学领域有密切联系，而且在信息科学、生物学、化学、经济学和理论计算机科学等许多方面都有具体的应用背景。基于前人的基础，我们近来研究了一些子图存在性、图子式 (minor)、结构参数与图的谱之间的关系，进一步探讨了符号模式矩阵的幂敛性质在信息传播方面的应用，刻画了一些特殊矩阵类的基，并且在刻画一般本原非可幂符号模式矩阵的基方面取得了进展，得到一些有意义成果。项目组希望通过本项目的研究，利用代数、组合、拓扑、数论、矩阵论、统计的方法和技巧，从不同的视角研究图的矩阵性质与图的结构性质之间的内在联系；利用图的结构和拓扑性质、组合、数论、统计的方法技巧研究符号模式矩阵的幂敛性质。挖掘和丰富图谱理论和符号模式矩阵幂序列性质的研究工具，以期推动一些新问题的解决，并进一步探索它们的理论与实际应用。

图谱理论和符号模式矩阵幂敛性质不仅与众多的数学领域有密切联系，而且在信息科学、生物学、化学、经济学和理论计机科学等许多方面都有具体的应用背景。近些年来，在这两方面的研究得到迅速发展，已经成为组合矩阵论研究的热点和重要方向，其研究主要涉及到组合图论，随机，代数，拓扑、数论、统计等方法。本项目组主要考虑图矩阵与图结构之间的内在联系。主要研究内容有：1、基于图谱研究图的结构并且基于图结构研究其图谱性质；2、利用图的结构和拓扑性质、组合、数论、统计的方法技巧研究符号模式矩阵的幂敛性质，并基于幂敛指数研究图结构。通过本研究项目，我们展开了对有向图谱半径的研究；深入开展了对距离谱的研究，率先开展了对距离谱小根的研究；率先展开了对有向图距离谱的研究；展开了对无符号拉普拉斯谱的研究；深入研究了符号模式矩阵的基、局部基与Lewin指数。我们还将图谱的研究方法拓展应用到对匹配多项式的研究，并得到一些有意义的成果。在这些研究过程中，我们不断的挖掘和创新工具，使得一些极图得以刻画，解决了相关领域的一些公开问题，取得了一些原创性成果，积极的推动该领域更多问题的解决。