图积和多项式理论中的图结构与极值问题

项目名称： 图积和多项式理论中的图结构与极值问题

项目编号： No.11501448

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 李巍

作者单位： 西北工业大学

项目金额： 18万元

中文摘要： 近年来图积和多项式在数学、化学领域的研究日益广泛：围绕Pólya问题、Borowiecki问题产生的积和多项式和特征多项式的转化问题、积和多项式的根和系数与图结构关系问题是当前研究的重要方面。针对其突出问题，本项目深入研究三方面内容：积和多项式和特征多项式的转化与图结构关系、积和多项式根的实虚性与图结构关系、积和多项式系数和的极值问题。具体思路为：1）深化并创新已有研究，引入定向图斜特征多项式的线性组合，研究其与积和多项式的转化，利用耳朵分解，刻画可转化图类结构；2）突破已有方法，运用组合分析与复数域多项式根理论相结合的方法，研究积和多项式的根均为纯虚数新图类的存在性及结构，以促进Borowiecki问题研究；3）借助图结构及匹配计数，建立Cata-型六角系统及其推广图类积和多项式系数和的界值，并刻画极值图。上述问题的研究，将为图积和多项式研究提供新思路，丰富发展图积和多项式理论体系。

中文关键词： 积和多项式；匹配；Pfaffian；定向；特征多项式

英文摘要： In recent years, the permanental polynomials of graphs are more and more widely investigated in the areas of Mathematics and Chemistry. Arising with Pólya problem and Borowiecki problem, the conversions of permanental polynomials and characteristic polynomials and the relation between the permanental roots and permanental coefficients and the structures of graphs are important research subjects. Following these problems, this project will focus on three aspects: the relation between the conversions of permanental polynomials and characteristic polynomials and the structures of graphs, the relation between the properties on real and imaginary permanental roots and the structures of graphs, as well as the extremal problem on the sum of coefficients of permanental polynomial. The explicit research approaches are as follows: 1)deepen the existing research and establish new ideas. By introducing the linear combination of the skew-characteristic polynomials of orientation graphs, we will first study the conversions of such linear combinations and the permanental polynomials, and then determine the structures the those graphs with this convertible properties in terms of ear decomposition. 2) by breaking through the existing method, we plan to improve the existed results of Borowiecki problem. By combining the combinatorial analysis and the theory on the roots of polynomials in the complex domain, we will determine the existence of new graphs whose permanental roots are pure imaginary numbers, and characterize such graphs. 3) with the help of graph structures and matching numbers, we will first study the sums of permanental coefficients of hexagonal system and extended graphs, and then establish the bounds of this sums and characterize the structures of extremal graphs. The research of above problems is expected to not only provide a new idea to the study of permanental polynomials of graphs, but also enrich and improve the theory of permanental polynomials of graphs.

英文关键词： Permanental polynomial;Matching;Pfaffian orientation;Characteristic polynomial