Jaco图的竞争图及简单连通图Grog数的引入
Competition Graphs of Jaco Graphs and the Introduction of the Grog Number of a Simple Connected Graph
摘要：
设$G^\rightarrow$是$n \geq 2$顶点上的简单连通有向图，并且$V^*$是$V(G^\rightarrow)$的非空子集，并且用$\langle V^* \rangle.$表示由$V^*$诱导的无向子图我们证明了由$C(J_n(1))$表示的Jaco图$J_n(1), n \in \Bbb N, n \geq 5,$的\emph{competition graph}由下式给出：$C(J_n(1)) = \langle V^* \rangle_{V^* = \{v_i|3 \leq i \leq n-1\}} - \{v_iv_{m_i}| m_i = i + d^+_{J_n(1)}(v_i), 3 \leq i \leq n-2\} \cup \{v_1, v_2, v_n\}.$进一步地，将引入$n \geq 2$顶点上的简单连通有向图$G^\rightarrow$的\emph{grog number} $g(G^\rightarrow)$以及基础图$G$的一般\emph{grog number}的概念。如果简单有向图模拟生态捕食者-猎物网络，则PDMATHAMV测量\emph{optimal predator-prey strategy}的效率。\\
英文摘要：
Let $G^\rightarrow$ be a simple connected directed graph on $n \geq 2$ vertices and let $V^*$ be a non-empty subset of $V(G^\rightarrow)$ and denote the undirected subgraph induced by $V^*$ by, $\langle V^* \rangle.$ We show that the \emph{competition graph} of the Jaco graph $J_n(1), n \in \Bbb N, n \geq 5,$ denoted by $C(J_n(1))$ is given by:\\ \\ $C(J_n(1)) = \langle V^* \rangle_{V^* = \{v_i|3 \leq i \leq n-1\}} - \{v_iv_{m_i}| m_i = i + d^+_{J_n(1)}(v_i), 3 \leq i \leq n-2\} \cup \{v_1, v_2, v_n\}.$\\ \\ Further to the above, the concept of the \emph{grog number} $g(G^\rightarrow)$ of a simple connected directed graph $G^\rightarrow$ on $n \geq 2$ vertices as well as the general \emph{grog number} of the underlying graph $G$, will be introduced. The \emph{grog number} measures the efficiency of an \emph{optimal predator-prey strategy} if the simple directed graph models an ecological predator-prey web.\\ \\ We also pose four open problems for exploratory research.
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