Seminar za biomatematiko in matematično kemijo - Arhiv

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. [Discuss. 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 yields nut graphs of every even order n ≥ 4t + 4 whenever t is odd such that t ≠ 1 (mod 10) and t ≠ 15 (mod 18). This fully resolves Conjecture 9 from Bašić et al. [ibid.]. We also study the existence of 4t-regular circulant nut graphs for small values of t, which partially resolves Conjecture 10 of Bašić et al. [ibid.].

While the original question is stated in terms of (spectral) graph theory, the talk will very quickly move from the setting of graph eigenvalues and eigenvectors to polynomial algebra, with most of the obtained results based on the properties of cyclotomic polynomials.

This is a joint work with Ivan Damnjanović.

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