On 12-regular nut graphs

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 ∈ {3, 4, …, 11} all values n such that there exists a d-regular nut graph of order n. In the present paper, we resolve the first open case d = 12, i.e. we show that there exists a 12-regular nut graph of order n if and only if n ≥ 16. We also present a result by which there are infinitely many circulant nut graphs of degree d ≡ 0 (mod  4) and no circulant nut graphs of degree d ≡ 2 (mod  4). The former result partially resolves a question by Fowler et al. on existence of vertex-transitive nut graphs of order n and degree d. We conclude the paper with problems, conjectures and ideas for further work.