Одељење за математику, 17. септембар 2021.

Наредни састанак Семинара биће одржан у петак, 17. септембра 2021. са почетком у 14.15 часова, онлајн и у сали 301ф
Математичког института САНУ.

Предавач: Драган Стевановић, Математички институт САНУ

Наслов предавања: ON CIRCULANT NUT GRAPH

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 entries. Motivated by a recent 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] initiated the study of circulant nut graphs. We continue this study by first showing that the generator set of a circulant nut graph contains t even and t odd integers for some t ≥ 1, while its number of vertices n is even and at least 4t+4. We further show that certain generator sets are universal in the sense that all circulant graphs with such generator set on an even number of n ≥ 4t+4 vertices, are nut graphs. Our main result shows that the generator set {1,...,2t+1} \ {t} is universal if and only if t is odd such that t ≢_10 1 and t ≢_18 15. This fully resolves one and partially resolves another conjecture of Bašić et al. [ibid.], and provides a positive answer to the question of Fowler et al. for a large share of the feasible pairs (n,d).

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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