论文标题
通过它们的光谱瞬间订购类似星光的树木
Ordering starlike trees by the totality of their spectral moments
论文作者
论文摘要
$ k $ -th的光谱矩$ m_k（g）的邻接矩阵〜$ g $表示封闭的长度〜$ k $ in〜 $ g $的封闭步行数量。我们在这里研究图形的部分订单$ \ preceq $，由$ g \ preceq h $定义，如果$ m_k（g）（g）\ leq m_k（h）$ for All $ k \ geq 0 $，并且对$ \ \ \\ preceq $何时在指定的图表中$ \ preceq $ a？我们的主要结果是，$ \ prepeq $是每组具有恒定顶点数量的星形树上的线性顺序。回想一下，如果有一个顶点〜$ u $，则连接的图形$ g $是一棵类似八哥的树，使得$ g-u $的组件是路径，称为〜$ g $的分支。事实证明，$ \ preceq $ starlike树的订购量不断，顶点数量与其分支长度的排序顺序相吻合。
The $k$-th spectral moment $M_k(G)$ of the adjacency matrix of a graph~$G$ represents the number of closed walks of length~$k$ in~$G$. We study here the partial order $\preceq$ of graphs, defined by $G\preceq H$ if $M_k(G)\leq M_k(H)$ for all $k\geq 0$, and are interested in the question when is $\preceq$ a linear order within a specified set of graphs? Our main result is that $\preceq$ is a linear order on each set of starlike trees with constant number of vertices. Recall that a connected graph $G$ is a starlike tree if it has a vertex~$u$ such that the components of $G-u$ are paths, called the branches of~$G$. It turns out that the $\preceq$ ordering of starlike trees with constant number of vertices coincides with the shortlex order of sorted sequence of their branch lengths.
