Characteristics of Jaco Graphs, $J_{\infty }(a),a\in \mathbb{N}$

Abstract

We introduce the concept of a family of finite directed graphs (order a) which are directed graphs derived from an infinite directed graph (order a), called the a-root digraph. The a-root digraph has four fundamental properties which are; $V(J_{\infty }(a))=\{v_i|i\in \mathbb{N}\}$ 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 $k\in \mathbb{N}$ 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 $d(v_k)=ak.$ The family of finite directed graphs are those limited to $n\in \mathbb{N}$ vertices by lobbing off all vertices (and edges arcing to vertices)$v_t,t>n.$ Hence, trivially we have $d(v_i)\le ai$ for $i\in \mathbb{N}.$ We present an interesting Lucassian-Zeckendorf result and other general results of interest. It is meant to be an introductory paper to encourage exploratory research.

13 pages. arXiv admin note: text overlap with arXiv:1404.0484