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

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.