LAUSR

A nut graph is a simple graph whose adjacency matrix has eigenvalue 0 with multiplicity 1 such that its eigenvector has no zero entries. Motivated by a question of Fowler et al. [Disc. Math. Graph Theory 40 (2020), 533–557] to determine the pairs (n, d) for which a vertex-transitive nut graph of order n and degree d exists, Bašić et al. [arXiv:2102.04418, 2021] initiated the study of circulant nut graphs. Here we first show that the generator set of a circulant nut graph necessarily contains equally many even and odd integers. Then we characterize circulant nut graphs with the generator set {x, x+1, . . . , x+2t−1} for x, t ∈ N, which generalizes the result of Bašić et al. for the generator set {1, . . . , 2t}. We further study circulant nut graphs with the generator set {1, . . . , 2t + 1} \ {t}, which appears to yield nut graphs of every even order n > 4t + 4 whenever t is odd such that t 6≡ 1 (mod 10) and t 6≡ 15 (mod 18). This observation is tested here computationally for all t 6 1300, and it fully resolves Conjecture 9 from Bašić et al. [arXiv:2102.04418, 2021]. We also study existence of 4t-regular circulant nut graphs for small values of t, which partially resolves Conjecture 10 of Bašić et al. [ibid.]. Mathematics Subject Classifications: 05C50 The corresponding author. The second author is supported by the Serbian Ministry of education, science and technological development through the Mathematical Institute of SASA, and by the project F-159 of the Serbian Academy of Sciences and Arts. the electronic journal of combinatorics 27 (2020), #P00 https://doi.org/10.37236/abcd