Linear Jaco Graphs: A Critical Review (Journal of Informatics and Mathematical Sciences, 2016)
The concept of linear Jaco graphs was introduced by Kok et al. [19,20]. Linear Jaco graphs are a family of finite directed graphs which are derived from an infinite directed graph, called the \(f(x)\)-root digraph. The incidence function is a linear function \(f(x) = mx + c\), \(x \in \mathbb{N}\), \(m,c \in \mathbb{N}_0\). Much research has been done for the case \(f(x) = x\).  Many interesting open problems remain for the case \(f(x)=x\) and certainly for the general case \(f(x) = mx + c\), \(m,c > 0\). Despite an elegant, almost simple definition of these graphs it remains hard and predictably impossible in some cases to derive closed formula for a number of well-known invariants. Interesting to note, is the ever so often appearance of Fibonacci and Lucas numbers as well as the Golden Ratio in some results. These observations suggest that a sound number theoretic approach might resolve some of the mystery surrounding Jaco graphs.
Keywords: Linear Jaco graph; Hope graph; Jaconian vertex; Jaconian set; Fisher algorithm; Bettina's theorem.