Complexity of circulant graphs with non-fixed jumps, its arithmetic properties and asymptotics Full article
| Journal |
Ars Mathematica Contemporanea
ISSN: 1855-3966 , E-ISSN: 1855-3974 |
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| Output data | Year: 2023, Volume: 23, Number: 1, Article number : #P1.08, Pages count : 16 DOI: 10.26493/1855-3974.2530.e7c | ||||
| Tags | Spanning tree, circulant graph, Laplacian matrix, Chebyshev polynomial, Mahler measure. | ||||
| Authors |
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| Affiliations |
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Funding (1)
| 1 |
Министерство науки и высшего образования РФ Mathematical Center in Akademgorodok |
075-15-2019-1613, 075-15-2022-281 |
Abstract:
In the present paper, we investigate a family of circulant graphs with non-fixed jumps Gn= Cβn(s1, ... , sk, α1n, ... , αln), 1 ≤ s1< ... < sk≤ [βn/2], 1 ≤ α1< ... < αl ≤ [β/2]. Here n is an arbitrary large natural number and integers s1, ... , sk, α1, ... , αl are supposed to be fixed. First, we present an explicit formula for the number of spanning trees in the graph Gn. This formula is a product of βsk-1 factors, each given by the n-th Chebyshev polynomial of the first kind evaluated at the roots of some prescribed polynomial of degree sk. Next, we provide some arithmetic properties of the complexity function. We show that the number of spanning trees in Gn can be represented in the form τ(n) = p n a(n)2, where a(n) is an integer sequence and p is a prescribed natural number depending of parity of β and n. Finally, we find an asymptotic formula for τ(n) through the Mahler measure of the associated Laurent polynomials differing by a constant from 2k - ∑i = 1k(zsi+z−si).
Cite:
Mednykh A.
, Mednykh I.
Complexity of circulant graphs with non-fixed jumps, its arithmetic properties and asymptotics
Ars Mathematica Contemporanea. 2023. V.23. N1. #P1.08 :1-16. DOI: 10.26493/1855-3974.2530.e7c WOS Scopus РИНЦ OpenAlex
Complexity of circulant graphs with non-fixed jumps, its arithmetic properties and asymptotics
Ars Mathematica Contemporanea. 2023. V.23. N1. #P1.08 :1-16. DOI: 10.26493/1855-3974.2530.e7c WOS Scopus РИНЦ OpenAlex
Dates:
| Submitted: | Jan 11, 2021 |
| Accepted: | Mar 23, 2022 |
| Published online: | Oct 21, 2022 |
| Published print: | Feb 8, 2023 |
Identifiers:
| Web of science: | WOS:000892434600005 |
| Scopus: | 2-s2.0-85149695642 |
| Elibrary: | 57161030 |
| OpenAlex: | W2905558881 |