A note on the number of edges of the Jaco Graph, $J_n(1), n \in \Bbb N

TL;DR

This paper discusses the challenge of finding a closed-form formula for the number of edges in finite Jaco Graphs and proposes three alternative formulas as potential solutions.

Contribution

It introduces three new formulas for calculating the number of edges in Jaco Graphs, addressing an open problem in graph theory.

Findings

Presented three alternative formulas for edge count

These formulas offer new approaches to the open problem

Contributes to understanding the structure of Jaco Graphs

Abstract

Kok et.al. [3] introduced Jaco Graphs \emph{(order 1)}. It is hoped that as a special case, a closed formula can be found for the number of edges of a finite Jaco Graph $J_n(1)$. However, the algorithms discussed in Ahlbach et.al. [1] suggest this might not be possible. Finding a closed formula for the number of edges of a Jaco Graph $J_n(1), n \in \Bbb N$ remains an interesting open problem. In this note we present three alternative, \emph{formula}.