A Study on Linear Jaco Graphs

We introduce the concept of a family of finite directed graphs (\emph{positive integer order,} $f(x)=mx+c;x,m\in \mathbb{N}$ and $c \in \Bbb N_0)$ 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; $V(J_{\mathrm{\infty }}(f(x)))=\{v_i:i\in \mathbb{N}\}$ 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 $k\in \mathbb{N}$ is a tail vertex then all vertices $v_\ell, k < \ell < j$aretailsofarcsto$v_j$andfinally,thedegreeofavertex$v_k$is$d(v_k) = mk + c$. The family of finite directed graphs are those limited to $n\in \mathbb{N}$ vertices by lobbing off all vertices (and corresponding arcs) $v_t,t>n\mathrm{.}$ Hence, trivially we have $d(v_i)\le mi+c$ for $i\in \mathbb{N}\mathrm{.}$ It is meant to be an \emph{introductory paper} to encourage further research.