Vertex-transitive nut graph order–degree existence problem

A nut graph is a nontrivial simple graph whose adjacency matrix has a simple eigenvalue zero such that the corresponding eigenvector has no zero entries. It was recently shown that the order n and degree d of a vertex-transitive nut graph satisfy 4 | d, d ≥ 4, 2 | n and n ≥ d + 4; or d ≡ 2 (mod 4), d ≥ 6, 4 | n and n ≥ d + 6. Here, we prove that for each such n and d, there is a d-regular Cayley nut graph of order n. As a direct consequence, we find all the pairs (n, d) for which there exists a d-regular vertex-transitive (resp. Cayley) nut graph of order n.