A note on the Brush Number of Jaco Graphs, $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 $G$. This note extends the concept to a special family of directed graphs and declares that the brush number $b_r(J_n(1))$ of a finite Jaco graph, $J_n(1), n \in \Bbb N$ with prime Jaconian vertex $v_i$ is given by:\\ \\ $b_r(J_n(1)) = \sum\limits_{j=1}^{I}(d^+(v_j) - d^-(v_j)) + \sum\limits_{j=I+1}^{n}max\{0, (n-j) - d^-(v_j)\}.