On the degrees of regular nut graphs and Cayley nut graphs

A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry. It is known that infinitely many d-regular nut graphs exist for 3 ≤ d ≤ 12 and for d ≥ 4 such that d ≡ 0 (mod 4). Here it is shown that infinitely many d-regular nut graphs exist for each degree d ≥ 3. Moreover, we prove that there are infinitely many d-regular Cayley nut graphs for each even d ≥ 4. This implies that we have identified all feasible degrees d for which a d-regular Cayley nut graph exists.