一类图的本原洞数
The Primitive Hole Number of Certain Graphs
摘要：
一个简单连通图$G$的洞是一个无弦圈$C_n,$，其中图$G$中的$n \in \Bbb N, n \geq 4,$。一个简单连通图$G$的围长是$G$中的最小圈，如果存在这样的圈的话。可以观察到，所有这些最小的周期必然是无弦的。我们称给定图$G$中的圈$C_3$为该图的本原洞。我们引入了图的本原洞数的概念，作为该图中存在的本原洞的数量。本文确定了某些标准图的本原洞数。另外，我们还确定了一个Jaco图的底层图$J^*_{n+1}(1),$的本原洞数，其中$n \in \Bbb N, n \geq 4$递归地表示底层Jaco图$J_n(1)$，且其顶点为素Jaconian顶点$v_i$ .本文还引入了图的顶点本原度的概念，并确定了某些图的顶点本原度。
英文摘要：
A hole of a simple connected graph $G$ is a chordless cycle $C_n,$ where $n \in \Bbb N, n \geq 4,$ in the graph $G$. The girth of a simple connected graph $G$ is the smallest cycle in $G$, if any such cycle exists. It can be observed that all such smallest cycles are necessarily chordless. We call the cycle $C_3$ in a given graph $G$ a primitive hole of that graph. We introduce the notion of the primitive hole number of a graph as the number of primitive holes present in that graph. In this paper, we determine the primitive hole number of certain standard graphs. Also, we determine the primitive hole number of the underlying graph of a Jaco graph, $J^*_{n+1}(1),$ where $n \in \Bbb N, n \geq 4$ recursively in terms of the underlying Jaco graph $J_n(1)$, with prime Jaconian vertex $v_i$. The notion of primitive degree of the vertices of a graph is also introduced and the primitive degree of the vertices of certain graphs is also determined in this paper.
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