395 KB40 KB172 KB4 KB200820252023Mednykh, AlexanderMednykh, Ilyaštevilka:1letnik:23P1.08 (16 str.)DOI:10.26493/1855-3974.2530.e7cCOBISSID_HOST:149277443ISSN:1855-3966URN:URN:NBN:SI:doc-IS4XZ1GYenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneaChebyshev polynomialcirculant graphcirkulantČebiševljev polinomLaplaceova matrikaLaplacian matrixMahler measureMahlerjeva meraspanning treevpeto drevoComplexity of circulant graphs with non-fixed jumps, its arithmetic properties and asymptotics|In the present paper, we investigate a family of circulant graphs with non-fixed jumps ?$$ \begin{gathered} G_n = C_{\beta n} (s_1, \dots, s_k, \alpha_1 n,\dots, \alpha_{\ell} n), \\ 1 \leq s_1 <\dots < s_k \leq \frac{\beta n}{2}, 1 \leq \alpha_1 < \dots < \alpha_{\ell} \leq \frac{\beta}{2}. \end{gathered}$$? Here ?$n$? is an arbitrary large natural number and integers ?$s_1,\ldots, s_k, \alpha_1, \ldots, \alpha_{\ell}, \beta$? are supposed to be fixed. First, we present an explicit formula for the number of spanning trees in the graph ?$G_n$?. This formula is a product of ?$\beta s_k -1$? factors, each given by the ?$n$?-th Chebyshev polynomial of the first kind evaluated at the roots of some prescribed polynomial of degree ?$s_k$?. Next, we provide some arithmetic properties of the complexity function. We show that the number of spanning trees in ?$G_n$? can be represented in the form ?$\tau (n) = pna(n)^2$?, where ?$a(n)$? is an integer sequence and ?$p$? is a prescribed natural number depending of parity of ?$\beta$? and ?$n$?. Finally, we find an asymptotic formula for ?$\tau (n)$? through the Mahler measure of the associated Laurent polynomials differing by a constant from ?$2k-\sum_{i = 1}^k (z^{s_i}+z^{-s_i})$?V članku preučujemo družino cirkulantov z nefiksiranimi skoki ?$$ \begin{gathered} G_n = C_{\beta n} (s_1, \dots, s_k, \alpha_1 n,\dots, \alpha_{\ell} n), \\ 1 \leq s_1 <\dots < s_k \leq \frac{\beta n}{2}, 1 \leq \alpha_1 < \dots < \alpha_{\ell} \leq \frac{\beta}{2}. \end{gathered} $$? Tukaj je ?$n$? poljubno veliko naravno število, cela števila ?$s_1,\ldots, s_k, \alpha_1, \ldots, \alpha_{\ell}, \beta$? pa so fiksirana. Najprej predstavimo eksplicitno formulo za število vpetih dreves v grafu ?$G_n$?. Ta formula je produkt ?$\beta s_k -1$? faktorjev, vsak od njih je podan z ?$n$?-tim Čebiševljevim polinomom prve vrste, evaluiranim pri ničlah nekega predpisanega polinoma stopnje ?$s_k$?. Nadalje predstavimo nekaj aritmetičnih lastnosti kompleksnostne funkcije. Dokažemo, da se da število vpetih dreves v grafu ?$G_n$? zapisati v obliki ?$\tau (n) = pna(n)^2$?, kjer je ?$a(n)$? celoštevilsko zaporedje, ?$p$? pa dano naravno število, odvisno od sodosti števil?$\beta$? in ?$n$?. Poiščemo še asimptotsko formulo za ?$\tau (n)$? preko Mahlerjeve mere Laurentovih polinomov, ki se od ?$2k-\sum_{i = 1}^k (z^{s_i}+z^{-s_i})$? razlikujejo samo za konstantoTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije