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

TL;DR

This paper investigates the brush number of Mycielski Jaco graphs, introduces the concept of a brush centre, and explores optimal orientations to facilitate maintenance and cleaning strategies in graph-based models.

Contribution

It introduces the concept of a brush centre and applies the brush number to Mycielski Jaco graphs with respect to optimal orientations.

Findings

Determined the brush number for specific Mycielski Jaco graphs.

Proposed the concept of a brush centre for graph maintenance.

Analyzed the implications of optimal orientations on brush numbers.

Abstract

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.