Two families of circulant nut graphs

A circulant nut graph is a non-trivial simple graph whose adjacency matrix is a circulant matrix of nullity one such that its non-zero null space vectors have no zero elements. The study of circulant nut graphs was originally initiated by Basic et al. [Art Discrete Appl. Math. 5(2) (2021) #P2.01], where a conjecture was made regarding the existence of all the possible pairs (n, d) for which there exists a d-regular circulant nut graph of order n. Later on, it was proved by Damnjanovic and Stevanovic [Linear Algebra Appl. 633 (2022) 127-151] that for each odd t & GE; 3 such that t .10 1 and t .18 15, the 4t-regular circulant graph of order n with the generator set {1, 2, 3, ... , 2t + 1} \ {t}) must necessarily be a nut graph for each even n & GE; 4t + 4. In this paper, we extend these results by constructing two families of circulant nut graphs. The first family comprises the 4t-regular circulant graphs of order n which correspond to the generator sets {n } { n } {1, 2, ... , t -1} ? 4 , n 4 + 1 ? 2 - (t -1), ... , n 2 - 2, 2 n - 1 , for each odd t & ISIN; N and n & GE; 4t + 4 divisible by four. The second family consists of the 4t-regular circulant graphs of order n which correspond to the generator {n+2 } {n } sets {1, 2, ..., t-1}?4,n+6?2-(t-1), ...,2n-2,n , for each t & ISIN; N and n & GE; 4t + 6 such that n & EQUIV;42. 4 2 - 1 We prove that all of the graphs which belong to these families are indeed nut graphs, thereby fully resolving the 4t-regular circulant nut graph order-degree existence problem whenever t is odd and partially solving this problem for even values of t as well.