  A circulant nut graph is a non-trivial simple graph such that its adjacency
matrix is a circulant matrix whose null space is spanned by a single vector
without zero elements. Regarding these graphs, the order-degree existence
problem can be thought of as the mathematical problem of determining all the
possible pairs $(n, d)$ for which there exists a $d$-regular circulant nut
graph of order $n$. This problem was initiated by Ba\v{s}i\'c et al. and the
first major results were obtained by Damnjanovi\'c and Stevanovi\'c, who proved
that for each odd $t \ge 3$ such that $t\not\equiv_{10}1$ and
$t\not\equiv_{18}15$, there exists a $4t$-regular circulant nut graph of order
$n$ for each even $n \ge 4t + 4$. Afterwards, Damnjanovi\'c improved these
results by showing that there necessarily exists a $4t$-regular circulant nut
graph of order $n$ whenever $t$ is odd, $n$ is even, and $n \ge 4t + 4$ holds,
or whenever $t$ is even, $n$ is such that $n \equiv_4 2$, and $n \ge 4t + 6$
holds. In this paper, we extend the aforementioned results by completely
resolving the circulant nut graph order-degree existence problem. In other
words, we fully determine all the possible pairs $(n, d)$ for which there
exists a $d$-regular circulant nut graph of order $n$.
