A Study on Linear Jaco Graphs

Abstract

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

15 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1404.1714