Contemplating some invariants of the Jaco Graph, $J_n(1), n \in \Bbb N$

TL;DR

This paper investigates various invariants of Jaco Graphs, including recursive formulas for independence numbers, formulas for chromatic numbers based on prime Jaconian vertices, and bounds on the murtage number, expanding understanding of their structural properties.

Contribution

It introduces recursive formulas for independence numbers, characterizes chromatic numbers based on prime Jaconian vertices, and establishes bounds on the murtage number for Jaco Graphs.

Findings

Recursive formula for independence number of Jaco Graphs.

Chromatic number depends on the existence of specific edges.

Murtage number of Jaco Graphs is bounded between 0 and 3.

Abstract

Kok et.al. [7] introduced Jaco Graphs (\emph{order 1}). In this essay we present a recursive formula to determine the \emph{independence number} $\alpha(J_n(1)) = |\Bbb I|$ with, $\Bbb I = \{v_{i,j}| v_1 = v_{1,1} \in \Bbb I$ and $v_i = v_{i,j} =v_{(d^+(v_{m, (j-1)}) + m +1)}\}.$ We also prove that for the Jaco Graph, $J_n(1), n \in \Bbb N$ with the prime Jaconian vertex $v_i$ the chromatic number, $\chi(J_n(1))$ is given by: \begin{equation*} \chi(J_n(1)) \begin{cases} = (n-i) + 1, &\text{if and only if the edge $v_iv_n$ exists,}\\ \\ = n-i &\text{otherwise.} \end{cases} \end{equation*} We further our exploration in respect of \emph{domination numbers, bondage numbers} and declare the concept of the \emph{murtage number} of a simple connected graph $G$, denoted $m(G)$. We conclude by proving that for any Jaco Graph $J_n(1), n \in \Bbb N$ we have that $0 \leq m(J_n(1)) \leq 3.$