# 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* ∈ {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.