On $12$-regular nut graphs

Abstract

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 $d\in \{3,4,\dots ,11\}$ 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 $d\equiv 0(\mathrm{mod}4)$ and no circulant nut graph of degree $d\equiv 2(\mathrm{mod}4)$.

12 pages, 20 references