A nut graph is a simple graph whose adjacency matrix has the eigenvalue 0 with multiplicity 1 such that its corresponding eigenvector has no zero entries. Motivated by a question of Fowler et al. (2020) [5] to determine the pairs (n, d) for which a vertex-transitive nut graph of order n and degree d exists, Basic et al. (2021) [1] 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 + 2,..., x + 2t - 1} for x, t is an element of N, which generalizes the result of Basic et al. for the generator set {1, 2, 3,..., 2t}. We further study circulant nut graphs with the generator set {1, 2, 3,..., 2t + 1} \ {t}, which yields nut graphs of every even order n >= 4t + 4 whenever t is odd such that t not equivalent to(10) 1 and t not equivalent to(18) 15. This fully resolves Conjecture 9 from Basic et al. (2021) [1]. We also study the existence of 4t-regular circulant nut graphs for small values of t, which partially resolves Conjecture 10 of Basic et al. (2021) [1]. (C) 2021 Elsevier Inc. All rights reserved.
NSTL
Elsevier
