A Study on Linear Jaco Graphs - Kok, Johan; C, Susanth; Kalayathankal, Sunny Joseph (Journal of Informatics and Mathematical Sciences, 2015)
We introduce the concept of a family of nite directed graphs (positive integer order, f(x) =mx+c; x, m N and c N_0) which are directed graphs derived from an innite directed graph called the f(x)-root digraph. The f(x)-root digraph has four fundamental properties which are; V (J_∞(f(x))) = {v_i : i N} and, if v_j is the head of an arc then the tail is always a vertex v_i, i < j and, if v_k for smallest k N is a tail vertex then all vertices v_, k < < j are tails of arcs to v_j and finally, the degree of vertex v_k is d(v_k) = mk + c. The family of nite directed graphs are those limited to n N vertices by lobbing o all vertices (and corresponding arcs) v_t, t > n. Hence, trivially we have d(v_i) mi + c for i N. It is meant to be an introductory paper to encourage further research.