A Study on Linear Jaco Graphs

Abstract

We introduce the concept of a family of nite directed graphs (positive integer order, $f(x) =mx+c$; $x, m \in \mathbb{N}$ and $c \in \mathbb{N}_0$) 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; $V (J_\infty(f(x))) = \{v_i : i \in \mathbb{N}\}$ 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 $k \mathbb{N}$ 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 $v_k$ is $d(v_k) = mk + c$. The family of nite directed graphs are those limited to $n \in \mathbb{N}$ vertices by lobbing o all vertices (and corresponding arcs) $v_t$, $t > n$. Hence, trivially we have $d(v_i) mi + c$ for $i \in \mathbb{N}$. It is meant to be an introductory paper to encourage further research.