  A nut graph is a simple graph whose adjacency matrix is singular with
$1$-dimensional kernel such that the corresponding eigenvector has no zero
entries. In 2020, Fowler et al. characterised for each $d \in
\{3,4,\ldots,11\}$ all values $n$ such that there exists a $d$-regular nut
graph of order $n$. In the present paper, we determine all values $n$ for which
a $12$-regular nut graph of order $n$ exists. We also present a result by which
there are infinitely many circulant nut graphs of degree $d \equiv 0 \pmod 4$
and no circulant nut graph of degree $d \equiv 2 \pmod 4$.
