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

TL;DR

This paper introduces the McPherson number, a new graph invariant based on iterative vertex explosions, and calculates it for various classes of graphs including paths, cycles, and Jaco graphs.

Contribution

The paper defines the McPherson number and provides formulas for it across different graph families, expanding understanding of graph transformation processes.

Findings

McPherson number computed for paths, cycles, and n-partite graphs.

Explicit McPherson number for finite Jaco graphs.

Introduction of a recursive vertex explosion process in graph theory.

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.