Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Contemplating some invariants of the Jaco Graph, $J_n(1), n \in \Bbb N$   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 $0 \leq m(J_n(1)) \leq 3.$

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The categories of this paper are: quant-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: chem-ph
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The categories of this paper are: nlin
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The categories of this paper are: hep-th
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ex
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The categories of this paper are: mtrl-th