Seminar za analizu, 24. maj 2021.

• 19. Maj,  2021
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Naredni sastanak Seminara biće održan u ponedeljak, 24. maja 2021. u sali 718, kao i onlajn, sa početkom u 11:15.

Predavač: Dragan Stevanović, Matematički institut SANU, Beograd

Naslov predavanja: ON CIRCULANT NUT GRAPHS

Apstrakt:
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. Here we take this study to the next level by:

- showing that the generator set of a circulant nut graph necessarily contains equally many even and odd integers;

- characterizing circulant nut graphs with the consecutive generator set $\{x,x+1,...,x+2t-1\}$ for $x,t\in N$, which generalizes the result of Bašić et al. for $x=1$;

- characterizing circulant nut graphs with the almost consecutive generator set $\{1,...,2t+1\}\setminus\{t\}$, which yields nut graphs of every even order $n\geq 4t+4$ whenever $t$ is odd such that $t\neq 1\pmod{10}$ and $t\neq 15\pmod{18}$, which resolves a conjecture of Bašić et al. and partially answers a question of Fowler.

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.

Detalji pristupa:

https://matf.webex.com/matf/j.php?MTID=m57b0fa2f2f0b0f7350fd77ee8e72cbe3
Meeting number: 137 278 8505
Password: 2D7gNg6kqwJ

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