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

Abstract

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.