{"?xml":{"@version":"1.0"},"edm:RDF":{"@xmlns:dc":"http://purl.org/dc/elements/1.1/","@xmlns:edm":"http://www.europeana.eu/schemas/edm/","@xmlns:wgs84_pos":"http://www.w3.org/2003/01/geo/wgs84_pos","@xmlns:foaf":"http://xmlns.com/foaf/0.1/","@xmlns:rdaGr2":"http://rdvocab.info/ElementsGr2","@xmlns:oai":"http://www.openarchives.org/OAI/2.0/","@xmlns:owl":"http://www.w3.org/2002/07/owl#","@xmlns:rdf":"http://www.w3.org/1999/02/22-rdf-syntax-ns#","@xmlns:ore":"http://www.openarchives.org/ore/terms/","@xmlns:skos":"http://www.w3.org/2004/02/skos/core#","@xmlns:dcterms":"http://purl.org/dc/terms/","edm:WebResource":[{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-IS4XZ1GY/58610384-8032-4225-b2f2-a33e8f6f352d/PDF","dcterms:extent":"395 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-IS4XZ1GY/8596a130-0ba4-4250-b98c-6ec11124b886/TEXT","dcterms:extent":"40 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-IS4XZ1GY/35d774cf-e05f-458f-8ed8-5e363c539475/PDF","dcterms:extent":"172 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-IS4XZ1GY/ef703200-e441-4684-827a-bdf700aa7c9a/TEXT","dcterms:extent":"4 KB"}],"edm:TimeSpan":{"@rdf:about":"2008-2025","edm:begin":{"@xml:lang":"en","#text":"2008"},"edm:end":{"@xml:lang":"en","#text":"2025"}},"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:doc-IS4XZ1GY","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2023","dc:creator":["Mednykh, Alexander","Mednykh, Ilya"],"dc:format":[{"@xml:lang":"sl","#text":"številka:1"},{"@xml:lang":"sl","#text":"letnik:23"},{"@xml:lang":"sl","#text":"P1.08 (16 str.)"}],"dc:identifier":["DOI:10.26493/1855-3974.2530.e7c","COBISSID_HOST:149277443","ISSN:1855-3966","URN:URN:NBN:SI:doc-IS4XZ1GY"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije"},"dc:subject":[{"@xml:lang":"en","#text":"Chebyshev polynomial"},{"@xml:lang":"en","#text":"circulant graph"},{"@xml:lang":"sl","#text":"cirkulant"},{"@xml:lang":"sl","#text":"Čebiševljev polinom"},{"@xml:lang":"sl","#text":"Laplaceova matrika"},{"@xml:lang":"en","#text":"Laplacian matrix"},{"@xml:lang":"en","#text":"Mahler measure"},{"@xml:lang":"sl","#text":"Mahlerjeva mera"},{"@xml:lang":"en","#text":"spanning tree"},{"@xml:lang":"sl","#text":"vpeto drevo"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"Complexity of circulant graphs with non-fixed jumps, its arithmetic properties and asymptotics|"},"dc:description":[{"@xml:lang":"sl","#text":"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 ?$2k-\\sum_{i = 1}^k (z^{s_i}+z^{-s_i})$?"},{"@xml:lang":"sl","#text":"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 konstanto"}],"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"znanstveno časopisje"},{"@xml:lang":"en","#text":"journals"},{"@rdf:resource":"http://www.wikidata.org/entity/Q361785"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:doc-IS4XZ1GY","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-IS4XZ1GY"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-IS4XZ1GY/58610384-8032-4225-b2f2-a33e8f6f352d/PDF"},"edm:rights":{"@rdf:resource":"http://creativecommons.org/licenses/by/4.0/"},"edm:provider":"Slovenian National E-content Aggregator","edm:intermediateProvider":{"@xml:lang":"en","#text":"National and University Library of Slovenia"},"edm:dataProvider":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:doc-IS4XZ1GY/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-IS4XZ1GY"}}}}