Introduction to the McPherson number, Υ(G)\Upsilon(G) of a simple connected graph

The concept of the \emph{McPherson number} of a simple connected graph $G$ on $n$ vertices denoted by $\Upsilon(G)$, is introduced. The recursive concept, called the \emph{McPherson recursion}, is a series of \emph{vertex explosions} such that on the first interation a vertex $v \in V(G)$ explodes to arc (directed edges) to all vertices $u \in V(G)$ for which the edge $vu \notin E(G)$, to obtain the mixed graph $G'_1.$ Now $G'_1$ is considered on the second iteration and a vertex $w \in V(G'_1) = V(G)$ may explode to arc to all vertices $z \in V(G'_1)$ if edge $wz \notin E(G)$ and arc $(w, z)$ or $(z, w) \notin E(G'_1).$ The \emph{McPherson number} of a simple connected graph $G$ is the minimum number of iterative vertex explosions say $\ell,$ to obtain the mixed graph $G'_\ell$ such that the underlying graph of $G'_\ell$ denoted $G^*_\ell$ has $G^*_\ell \simeq K_n.$ We determine the \emph{McPherson number} for paths, cycles and $n$-partite graphs. We also determine the \emph{McPherson number} of the finite Jaco Graph $J_n(1), n \in \Bbb N.$ It is hoped that this paper will encourage further exploratory research.Comment: 9 pages. To be submitted to the Pioneer Journal of Mathematics and Mathematical Science

A Note on the Gutman Index of Jaco Graphs

The concept of the \emph{Gutman index}, denoted $Gut(G)$ was introduced for a connected undirected graph $G$. In this note we apply the concept to the underlying graphs of the family of Jaco graphs, (\emph{directed graphs by definition}), and describe a recursive formula for the \emph{Gutman index} $Gut(J^*_{n+1}(x)).$ We also determine the \emph{Gutman index} for the trivial \emph{edge-joint} between Jaco graphs.Comment: 8 pages. The paper has been approved in terms of notation, and its alignment to the unifying definition of the family of Jaco graph

A note on the Brush Numbers of Mycielski Graphs, μ(G)\mu(G)

The concept of the brush number $b_r(G)$ was introduced for a simple connected undirected graph $G$. The concept will be applied to the Mycielskian graph $\mu(G)$ of a simple connected graph $G$ to find $b_r(\mu(G))$ in terms of an \emph{optimal orientation} of $G$. We prove a surprisingly simple general result for simple connected graphs on $n \geq 2$ vertices namely: $b_r(\mu(G))= b_r(\mu^{\rightarrow}(G)) = 2\sum\limits_{i=1}^{n}d^+_{G^{\rightarrow}_{b_r(G)}}(v_i).$Comment: 5 pages. arXiv admin note: substantial text overlap with arXiv:1501.0138

A Study on Linear Jaco Graphs

We introduce the concept of a family of finite directed graphs (\emph{positive integer order,} $f(x) = mx + c; x,m \in \Bbb N$ and $c \in \Bbb N_0)$ which are directed graphs derived from an infinite directed graph called the $f(x)$-root digraph. The $f(x)$-root digraph has four fundamental properties which are; $V(J_\infty(f(x))) = \{v_i: i \in \Bbb N\}$ and, if $v_j$ is the head of an arc then the tail is always a vertex $v_i, i < j$ and, if $v_k$ for smallest $k \in \Bbb N$ is a tail vertex then all vertices $v_\ell, k < \ell < j$ are tails of arcs to $v_j$ and finally, the degree of a vertex $v_k$ is $d(v_k) = mk + c$. The family of finite directed graphs are those limited to $n \in \Bbb N$ vertices by lobbing off all vertices (and corresponding arcs) $v_t, t > n.$ Hence, trivially we have $d(v_i) \leq mi + c$ for $i \in \Bbb N.$ It is meant to be an \emph{introductory paper} to encourage further research.Comment: 15 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1404.171

Competition Graphs of Jaco Graphs and the Introduction of the Grog Number of a Simple Connected Graph

Let $G^\rightarrow$ be a simple connected directed graph on $n \geq 2$ vertices and let $V^*$ be a non-empty subset of $V(G^\rightarrow)$ and denote the undirected subgraph induced by $V^*$ by, $\langle V^* \rangle.$ We show that the \emph{competition graph} of the Jaco graph $J_n(1), n \in \Bbb N, n \geq 5,$ denoted by $C(J_n(1))$ is given by:\\ \\ $C(J_n(1)) = \langle V^* \rangle_{V^* = \{v_i|3 \leq i \leq n-1\}} - \{v_iv_{m_i}| m_i = i + d^+_{J_n(1)}(v_i), 3 \leq i \leq n-2\} \cup \{v_1, v_2, v_n\}.$\\ \\ Further to the above, the concept of the \emph{grog number} $g(G^\rightarrow)$ of a simple connected directed graph $G^\rightarrow$ on $n \geq 2$ vertices as well as the general \emph{grog number} of the underlying graph $G$, will be introduced. The \emph{grog number} measures the efficiency of an \emph{optimal predator-prey strategy} if the simple directed graph models an ecological predator-prey web.\\ \\ We also pose four open problems for exploratory research.Comment: title has been correcte

Contemplating on Brush Numbers of Mycielski Jaco Graphs, μ(Jn(1)),n∈N\mu(J_n(1)), n \in \Bbb N

The concept of the brush number $b_r(G)$ was introduced for a simple connected undirected graph $G$. The concept will be applied to the Mycielski Jaco graph $\mu(J_n(1)), n \in \Bbb N,$ in respect of an \emph{optimal orientation} of $J_n(1)$ associated with $b_r(J_n(1)).$ Further for the aforesaid, the concept of a \emph{brush centre} of a simple connected graph will be introduced. Because brushes themselves may be technology of kind, the technology in real world application will normally be the subject of maintenance or calibration or virus vetting or alike. Finding a \emph{brush centre} of a graph will allow for well located maintenance centres of the brushes prior to a next cycle of cleaning.Comment: 8 page

Contemplating some invariants of the Jaco Graph, Jn(1),n∈NJ_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 $0 \leq m(J_n(1)) \leq 3.$Comment: 10 pages. To be submitted to the Pioneer Journal of Mathematics and Mathematical Science

Some New Results on the Curling Number of Graphs

Let $S=S_1S_2S_3\ldots S_n$ be a finite string. Write $S$ in the form $XYY\ldots Y=XY^k$, consisting of a prefix $X$ (which may be empty), followed by $k$ copies of a non-empty string $Y$. Then, the greatest value of this integer $k$ is called the curling number of $S$ and is denoted by $cn(S)$. Let the degree sequence of the graph $G$ be written as a string of identity curling subsequences say, $X^{k_1}_1\circ X^{k_2}_2\circ X^{k_3}_3 \ldots \circ X^{k_l}_l$. The compound curling number of $G$, denoted $cn^c(G)$ is defined to be, $cn^n(G) = \prod\limits^{l}_{i=1}k_i$. In this paper, we discuss the curling number and compound curling number of certain products of graphs.Comment: 11 Pages in Journal of Combinatorial Mathematics and Combinatorial Computing, 201

Curling Numbers of Certain Graph Powers

Given a finite nonempty sequence $S$ of integers, write it as $XY^k$, where $Y^k$ is a power of greatest exponent that is a suffix of $S$: this $k$ is the curling number of $S$. The concept of curling number of sequences has already been extended to the degree sequences of graphs to define the curling number of a graph. In this paper we study the curling number of graph powers, graph products and certain other graph operations.Comment: 8 Pages, Submitte

A study on the curling number of graph classes

Given a finite nonempty sequence $S$ of integers, write it as $XY^k$, consisting of a prefix $X$ (which may possibly be empty), followed by $k$ copies of a non-empty string $Y$. Then, the greatest such integer $k$ is called the curling number of $S$ and is denoted by $cn(S)$. The concept of curling number of sequences has already been extended to the degree sequences of graphs to define the curling number of a graph. In this paper we study the curling number of graph powers, graph products and certain other graph operations.Comment: 8 Pages. arXiv admin note: substantial text overlap with arXiv:1509.0022