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 $N(3)=\{12\}\cup \{2k:k\ge 9\}$ and that $N(4) = {8,10,12} \cup {n: n \geq 14}$\mathrm{.}Theproblemofdetermining$N(\rho)$for$\rho > 4$ remains completely open.