The Primitive Hole Number of Certain Graphs

TL;DR

This paper introduces the concept of primitive holes in graphs, defines the primitive hole number, and calculates this number for various standard graphs and Jaco graphs, also exploring the primitive degree of vertices.

Contribution

It defines primitive holes and primitive hole number, and provides formulas for these in standard and Jaco graphs, advancing graph theory understanding.

Findings

Primitive hole number determined for standard graphs

Primitive hole number recursively calculated for Jaco graphs

Primitive degree of vertices analyzed in certain graphs

Abstract

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.