A note on the Brush Number of Jaco Graphs, $J_n(1), n \in \Bbb N

TL;DR

This paper extends the concept of the brush number to directed Jaco graphs, providing a formula for calculating the brush number based on vertex degrees and graph structure.

Contribution

It introduces a new formula for the brush number of directed Jaco graphs, expanding the concept to a specific family of directed graphs.

Findings

Derived an explicit formula for the brush number of Jaco graphs.

Showed the relationship between vertex degrees and brush number in directed graphs.

Extended the concept of brush number to a new class of directed graphs.

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)\}.