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

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 $0 \leq m(J_n(1)) \leq 3.$