Characteristics of Jaco Graphs, $J_\infty(a), a \in \Bbb N$

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; $V(J_\infty(a)) = {v_i|i \in \Bbb N}$ 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 $d(v_k) = ak.$ 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 $d(v_i) \leq ai$ for $i \in \Bbb N.$ 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.