A Study on Linear Jaco Graphs

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