A Study on Linear Jaco Graphs

الأصل · EN

We introduce the concept of a family of finite directed graphs (positive integer order, f(x) = mx + c; x,m ∈ N and c ∈ N₀) which are directed graphs derived from an infinite directed graph called the f(x)-root digraph. The f(x)-root digraph has four fundamental properties which are; V(J∞(f(x))) = {vᵢ: i ∈ N} and, if vⱼ is the head of an 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 vℓ, k < ℓ < j are tails of arcs to vⱼ and finally, the degree of a vertex vₖ is d(vₖ) = mk + c. The family of finite directed graphs are those limited to n ∈ N vertices by lobbing off all vertices (and corresponding arcs) vₜ, t > n. Hence, trivially we have d(vᵢ) ≤ mi + c for i ∈ N. It is meant to be an introductory paper to encourage further research.