Characteristics of Finite Jaco Graphs, Jn(1), n ∈ N
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∞(1)) = \vi|i ∈ N\ and, if vj is the head of an edge (arc) then the tail is always a vertex vi, i<j and, if vk, for smallest k ∈ N is a tail vertex then all vertices v, k< <j are tails of arcs to vj and finally, the degree of vertex k is d(vk) = k. The family of finite directed graphs are those limited to n ∈ N vertices by lobbing off all vertices (and edges arcing to vertices) vt, t> n. Hence, trivially we have d(vi) ≤ i for i ∈ 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.