AI 中文总结
本文提出了一种方法，用于在广义连接有向图中计数定向生成树，并给出了具有固定根的定向生成树的计数公式。
AI 中文摘要
设G是一个顶点集为{1,2,...,n}的有向图，H₁,H₂,...,Hₙ是n个有向图。广义连接有向图Ḡ=G[H₁,H₂,...,Hₙ]是通过将G中的每个顶点i替换为Hᵢ，并且对于任意u∈V(Hᵢ)和v∈V(Hⱼ)，(u,v)∈E(Ḡ)当且仅当(i,j)∈E(G)所得到的有向图。在本文中，我们将Ḡ中定向生成树的数量用H₁,H₂,...,Hₙ的拉普拉斯特征值和G的定向生成树的数量来表示。此外，我们考虑了Ḡ中具有固定根的定向生成树的数量。首先，我们引入了双聚类有向星形变换公式，用于计数有向图中具有固定根的定向生成树。利用该公式，我们给出了Ḡ中根在特定Hᵢ(1≤i≤n)中的定向生成树总数的公式，该公式基于H₁,H₂,...,Hₙ的拉普拉斯特征值和G的定向生成树的数量。作为应用，当每个Hᵢ是一个给定的有向图时，Ḡ中具有固定根的定向生成树的计数公式可以从我们的研究中推导出来。
英文摘要
Let $G$ be a digraph with vertex set $\{1,2,...,n\}$ and $H_{1},H_{2},...,H_{n}$ be $n$ digraphs. The generalized join digraph $\overrightarrow{G}=G[H_{1},H_{2},...,H_{n}]$ is a digraph obtained from $G$ by replacing each vertex $i$ with $H_{i}$ and for any $u\in V(H_{i})$ and $v\in V(H_{j})$, $(u,v)\in E(\overrightarrow{G})$ if and only if $(i,j)\in E(G)$. In this paper we express the number of oriented spanning trees in $\overrightarrow{G}$ in terms of Laplacian eigenvalues of $H_{1},H_{2},...,H_{n}$ and oriented spanning trees of $G$. Furthermore, we consider the number of oriented spanning trees with a fixed root in $\overrightarrow{G}$. First, we introduce the biclique-directed star transformation formula for counting oriented spanning trees with a fixed root in digraphs. Using it, we give the formula for the total number of oriented spanning trees with roots in a certain $H_{i}$ $(1\leq i \leq n)$ of $\overrightarrow{G}$ in terms of Laplacian eigenvalues of $H_{1},H_{2},...,H_{n}$ and oriented spanning trees of $G$. As applications, when each $H_{i}$ is a given digraph, the enumerative formulas for oriented spanning trees with a fixed root of $\overrightarrow{G}$ are derived from our work.
