树和单圈图的斜谱矩

给定图$G$,对图$G$的每条边确定一个方向,称为$G$的定向图$G^\sigma$, $G$称为$G^\sigma$的基础图. $G^\sigma$的斜邻接矩阵$S(G^\sigma)$是反对称矩阵,其特征值是0或纯虚数. $S(G^\sigma)$所有特征值的$k$次幂之和称为$G^\sigma$的$k$阶斜谱矩,其中$k$是非负整数.斜谱矩序列可用于对图进行排序.本文主要研究定向树和定向单圈图的斜谱矩,并对这两类图的斜谱矩序列依照字典序进行排序.首先确定了直径为$d$的树作为基础图的所有定向树中,斜谱矩序最大的$2\lfloor\frac{d}{4}\rfloor$个图; 然后确定以围长为$g$的单圈图作为基础图的所有定向单圈图中, 斜谱矩序最大的$2\lfloor\frac{g}{4}\rfloor+1$个图.

On the Skew Spectral Moments of Trees and Unicyclic Graphs

Given a simple graph $G$, the oriented graph $G^\sigma$ is obtained from $G$ by orienting each edge and $G$ is called the underlying graph of $G^\sigma$. The skew-symmetric adjacency matrix $S(G^\sigma)$ of $G^\sigma$, where the $(u,v)$-entry is $1$ if there is an arc from $u$ to $v$, and $-1$ if there is an arc from $v$ to $u$ (and 0 otherwise), has eigenvalues of 0 or pure imaginary. The $k$-th-skew spectral moment of $G^\sigma$ is the sum of power $k$ of all eigenvalues of $S(G^\sigma)$, where $k$ is a non-negative integer. The skew spectral moments can be used to produce graph catalogues. In this paper, we researched the skew spectral moments of some oriented trees and oriented unicyclic graphs and produced their catalogues in lexicographical order. We determined the last $2\lfloor\frac{d}{4}\rfloor$ oriented trees with underlying graph of diameter $d$ and the last $2\lfloor\frac{g}{4}\rfloor+1$ oriented unicyclic graphs with underlying graph of girth $g$, respectively.