拉普拉斯谱与（类）广义拉普拉斯能量研究

Graph spectral theory is an important research branch in algebraic graph theory and combinatorial matrix theory, it has important applications in physics, chemistry, biology, electrical engineering and other disciplines, in which Laplacian spectrum of graphs and related indices are hot topics in resent years. General Laplacian energy and general Laplacian-energy like of graphs are both invariants based on Laplacian spectrum, which can be associated with some other indices, such as Laplacian energy, Laplacian-energy like and Kirhhoff index. Thus these two indices are of great meanings no matter in theory or in applications. In our project, we use algebraic graph theory, combinatorics, matrix theory, and group theory as tools to study on the spectral width of Laplacian spectrum of graphs, find the upper and lower bounds and the extremal graphs with general Laplacian energy and general Laplacian-energy like, consider the effects on the two indices by adding or deleting some edges or vertices. Then we investigate Coulson-type and Gutman-type integral formula of these indices to find the relationship between the two indices and Laplacian characteristic polynomials, so as to get new method to solve energy comparison and extremal problems. Furthermore, we analyze the relationship between the two indices and other indices such as graph energy and incidence energy. The project is expected to solve the above key problems, get some breakthrough in the thoughts and methods, so as to enrich and expand the spectra and related index theory of graphs.

图谱理论是代数图论和组合矩阵论的重要研究领域，在物理、化学、生物、电子工程等学科中有重要应用，图的拉普拉斯谱及相关拓扑指标的研究是近年来的热点问题。图的广义和类广义拉普拉斯能量都是基于拉普拉斯谱的不变量，可以与图的拉普拉斯能量、类拉普拉斯能量、边数、Kirhhoff指标等联系起来，具有重要的科学意义。本项目以代数图论、组合数学、矩阵理论、群论等为工具，研究图的拉普拉斯谱的谱宽及与其他不变量的关系；找出（类）广义拉普拉斯能量的上下界及对应的极图，考虑增减图的边或点对指标的影响；探究指标的各型积分公式，找出它们与拉普拉斯特征多项式及其系数的关系，寻找解决大小比较和极值问题的全新方法；进一步研究（类）广义拉普拉斯能量与图能量、关联能量等其他图指标之间的关系。本项目预计解决上述问题，取得创新性成果，发展丰富图谱和相关的指标理论。

图矩阵及其特征多项式是代数图论和组合矩阵论的重要研究内容，图的拉普拉斯矩阵、扩展邻接矩阵及相关拓扑指标的研究都是近年来的热点问题。图的广义和类广义拉普拉斯能量都是基于拉普拉斯特征值的不变量，可以与图的拉普拉斯能量、类拉普拉斯能量、边数、Kirhhoff指标等联系起来，具有重要的科学意义。本项目以代数图论、组合数学、矩阵理论、群论等为工具，研究了图的（类）广义拉普拉斯能量在指数是有理数、无理数时的积分表达式，给出了指数为正整数倒数时，类广义拉普拉斯能量与拉普拉斯特征多项式系数之间的关系；讨论了多项式的广义能量和图的类广义拉普拉斯能量之间的关系，得到了图和多项式广义能量的积分公式；给出了 (复) 多项式的(复) 能量以及图的偶数阶谱距的积分公式。另一方面，图的扩展邻接矩阵相对于一般的邻接矩阵，对化学结构的识别性更强，在对不同结构的物理化学性质以及生物活性等方面的研究中更有优势。本项目研究了基于扩展邻接矩阵特征值的扩展Estrada指标的上下界和相应极图，讨论了友谊图和完全二部图的扩展Estrada指标的值，以及合成图与原图的扩展Estrada指标之间的关系。