论文标题
M-Step竞争图的Digraphs是树
Digraphs whose m-step competition graphs are trees
论文作者
论文摘要
在本文中，我们完全表征了$ n $的digraphs，其$ m $ step竞争图是正整数的星形图，$ 2 \ leq m <n $。 This result in matrix version identifies the solution set to the matrix equation $X^m(X^T)^m= Λ_n+I_n$ for positive integers $2\leq m < n$ where $I_n$ is the identity matrix of order $n$ and $Λ_n$ is a $(0,1)$ Boolean matrix such that the first row and the first column consist of $1$'s except $（1,1）$ - 条目和其余条目为$ 0 $，这是订单$ n $的星形图的邻接矩阵。 我们还获得了有意义的digraphs的属性，其$ m $步骤的竞争图是树。在此过程中，我们扩展了Helleloid〜 [连接的三角形$ m $ m $ - 步骤竞争图，iNCETE APPL。\ MATH。\MATH。\ 145（2005）376--383]，通过向所有积极整数$ M \ geq 2 $和$ n $显示，可连接的Triangle triangle-free-m $ m $ -Step竞争图是$ n $ n $ n $ Vertices。
In this paper, we completely characterize the digraphs of order $n$ whose $m$-step competition graphs are star graphs for positive integers $2\leq m < n$. This result in matrix version identifies the solution set to the matrix equation $X^m(X^T)^m= Λ_n+I_n$ for positive integers $2\leq m < n$ where $I_n$ is the identity matrix of order $n$ and $Λ_n$ is a $(0,1)$ Boolean matrix such that the first row and the first column consist of $1$'s except $(1,1)$-entry and the remaining entries are $0$, which is the adjacency matrix of a star graph of order $n$. We also derive meaningful properties of the digraphs whose $m$-step competition graphs are trees. In the process, we extend a result of Helleloid~[Connected triangle-free $m$-step competition graphs, Discrete Appl.\ Math.\ 145 (2005) 376--383] by showing that for all positive integers $m \geq 2$ and $n$, the connected triangle-free $m$-step competition graph on $n$ vertices is a tree.
