Existence of Regular Nut Graphs and the Fowler Construction

In this paper the problem of the existence of regular nut graphs is addressed. A generalization of Fowler's Construction which is a local enlargement applied to a vertex in a graph is introduced to generate nut graphs of higher order. Let $N(\rho)$ denote the set of integers $n$ such that there exists a regular nut graph of degree $\rho$ and order $n$. It is proven that $N(3) = \{12\} \cup \{2k : k \geq 9\}$ and that $N(4) = \{8,10,12\} \cup \{n: n \geq 14\}$. The problem of determining $N(\rho)$ for $\rho > 4$ remains completely open.