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

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} $\alpha (J_n(1))=|\mathbb{I}|$ with, $\mathbb{I}=\{v_{i,j}|v_1=v_{1,1}\in \mathbb{I}$ and $v_i=v_{i,j}=v_{(d^{+}(v_{m,(j-1)})+m+1)}\}.$ We also prove that for the Jaco Graph, $J_n(1),n\in \mathbb{N}$ with the prime Jaconian vertex $v_i$ the chromatic number, $\chi (J_n(1))$ is given by: \begin{equation*} χ(J_n(1)) \begin{cases} = (n-i) + 1, &\text{if and only if the edge $v_i{v}_n$ 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 $J_n(1),n\in \mathbb{N}$ we have that $0\le m(J_n(1))\le 3.$

10 pages. To be submitted to the Pioneer Journal of Mathematics and Mathematical Sciences