Characteristics of Jaco Graphs, $J_\infty(a), a \in \Bbb N$
  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. 
Johan Kok, Paul Fisher, Bettina Wilkens, Mokhwetha Mabula, Vivian   Mukungunugwa
math.CO