有向图能量的极值问题研究

The concept of graph energy arose in the context of the study of conjugated Hydrocarbons. The corresponding research has been one of the very active directions in chemical graph theory. In recent years, many international mathematical chemists including Gutman and Rada studied the digraphs analogue of graph energy, which attracted considerable interest from theoretical chemists and mathematicians. Due to the complicated cases for the structures and eigenvalues of digraphs, and the non-trivial relationships between digraph energies and digraph spectra, it is often difficult to deal with the problems on digraph energies. The aim of this project is to consider the problems of characterizing the extremal digraphs on digraph energies of the directed hexacyclic systems, and the problems of the relationships between digraph energies and digraph spectra. These are all very meaningful topics in the study of graph theory.. In this project, what we mainly study are as follows. By proving that digraph energies of directed lattices for regular polygons with even length monotonically increases with respect to the coefficients of the characteristic polynomials, we characterize the extremal digraphs on digraph energies of a type of directed pericondensed hexacyclic systems, that is, directed Hollow k-polygons. And we research the relationships between digraph energies and spectral radii of digraphs and show some tight upper bounds of digraph energies.

图能量的概念起源于共轭碳氢化合物的研究，其相关问题的研究一直是化学图论研究最活跃的方向之一。近年来，国际数学化学家Gutman和Rada等人把图能量的研究推广到有向图，引起理论化学家和数学家的共同关注。由于有向图结构和特征根情况复杂，并且有向图能量与有向图谱的深层次关系研究尚未成熟，因此研究有向图能量方面的问题往往比较困难。本项目主要研究有向六角系统的有向图能量的极值图刻画问题，以及有向图能量与有向图的谱半径关系问题，这些问题都是图论问题研究中非常有意义的课题。. 本项目主要研究的内容：通过证明长度为偶数的正多边形有向格子图的有向图能量关于其特征多项式的系数是单调递增的，对有向Peri-型六角系统（即有向Hollow k-边形）的有向图能量的极值图进行刻画；探索有向图能量和有向图的谱半径的关系，并给出有向图能量紧的上界。

图能量的概念起源于共轭碳氢化合物的研究，其相关问题的研究一直是化学图论研究最活跃的方向之一。近年来，国际数学化学家Gutman和Rada等人把图能量的研究推广到有向图，引起理论化学家和数学家的共同关注。由于有向图结构和特征根情况复杂，并且有向图能量与有向图谱的深层次关系研究尚未成熟，因此研究有向图能量方面的问题往往比较困难。本项目主要研究有向六角系统的有向图能量的极值图刻画问题，以及有向图能量与有向图的谱半径关系问题，这些问题都是图论问题研究中非常有意义的课题。本项目主要研究的内容：通过证明长度为偶数的正多边形有向格子图的有向图能量关于其特征多项式的系数是单调递增的，对有向Peri-型六角系统（即有向Hollow k-边形）的有向图能量的极值图进行刻画；探索有向图能量和有向图的谱半径的关系，并给出有向图能量紧的上界。