Characteristics of Finite Jaco Graphs, $J_n(1), n \in \Bbb N$

TL;DR

This paper introduces finite Jaco graphs derived from an infinite 1-root digraph, explores their properties, and presents a Fibonacci-Zeckendorf result along with an algorithm for analyzing these graphs, aiming to stimulate further research.

Contribution

The paper defines finite Jaco graphs from an infinite digraph, establishes their fundamental properties, and introduces a novel Fibonacci-Zeckendorf result and an algorithm for their analysis.

Findings

Established properties of finite Jaco graphs.

Presented a Fibonacci-Zeckendorf relation within these graphs.

Developed the Fisher Algorithm for graph analysis.

Abstract

We introduce the concept of a family of finite directed graphs (order 1) which are directed graphs derived from an infinite directed graph (order 1), called the 1-root digraph. The 1-root digraph has four fundamental properties which are; $V(J_\infty(1)) = \{v_i|i \in \Bbb N\}$ and, if $v_j$ is the head of an edge (arc) then the tail is always a vertex $v_i, i<j$ and, if $v_k$, for smallest $k \in \Bbb N$ is a tail vertex then all vertices $v_\ell, k< \ell <j$ are tails of arcs to $v_j$ and finally, the degree of vertex $k$ is $d(v_k) = k.$ The family of finite directed graphs are those limited to $n \in \Bbb N$ vertices by lobbing off all vertices (and edges arcing to vertices) $v_t, t> n.$ Hence, trivially we have $d(v_i) \leq i$ for $i \in \Bbb N$. We present an interesting Fibonaccian-Zeckendorf result and present the Fisher Algorithm to table particular values of interest. It is meant to be an introductory paper to encourage exploratory research.