On circulant nut graphs (Q2666910)

A nut graph is a simple graph whose adjacency matrix has the eigenvalue 0 of multiplicity 1 such that its corresponding eigenvector has no zero entry. Motivated by a question of \textit{P. W. Fowler} et al. [Discuss. Math., Graph Theory 40, No. 2, 533--557 (2020; Zbl 1433.05153)] to determine the pair \((n,d)\) for which a vertex-transitive nut graph of order \(n\) and degree \(d\) exists, \textit{N. Bašić} et al. [``On 12-regular nut graphs'', Preprint, \url{arXiv:2102.04418}] initiated the study of circulant nut graphs and put some conjunctures. \par Here, along with some other results, the authors solve Conjuncture 9 given in [Bašić et al., loc.cit.] by proving that the circulant graphs with the generator set \(\{x,x+1,\dots,x+2t-1\} \) for \(x,t\in N\) and of an even order \(n\geq 2x+4t\) are nut graphs if and only if \(\gcd( \frac{n}{2},t) =\gcd( \frac{n}{2},2x+2t-1)\), and the circulant graphs with the generator set \(\{ 1,2,\dots,2t+1\} \backslash \{ t\} \) for each odd \(t\geq 3\) such that \(t\neq 1+10k\) and \(t\neq 15+18k\) for some \(k\in N\), are nut graphs for each even \(n\geq 4t+4\). While, Conjuncture 10 [Bašić et al., loc. cit.] is partially answered by providing \(4t-\)regular circulant nut graphs for any \(t \in N\) and of order \(n\geq 4t+4\) having \(\gcd( \frac{n}{2},t) =1\).