TY - JOUR

T1 - Complexity of circulant graphs with non-fixed jumps, its arithmetic properties and asymptotics

AU - Mednykh, Alexander

AU - Mednykh, Ilya

N1 - Публикация для корректирровки.

PY - 2022

Y1 - 2022

N2 - 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> . . . > α1≥ [β2].Here n is an arbitrary large natural number and integers s1, . . . , sk, α1, . . . , αl, β 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 given natural number depending on parity of β and n. Finally, we find an asymptotic formula for τ (n) through the Mahler measure of the Laurent polynomials differing by a constant from 2k - Pk i=1(zst+ z-st). Keywords: Spanning tree, circulant graph, Laplacian matrix, Chebyshev polynomial, Mahler measure.

AB - 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> . . . > α1≥ [β2].Here n is an arbitrary large natural number and integers s1, . . . , sk, α1, . . . , αl, β 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 given natural number depending on parity of β and n. Finally, we find an asymptotic formula for τ (n) through the Mahler measure of the Laurent polynomials differing by a constant from 2k - Pk i=1(zst+ z-st). Keywords: Spanning tree, circulant graph, Laplacian matrix, Chebyshev polynomial, Mahler measure.

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85149695642&origin=inward&txGid=56b695cebdb1e65a91582bc5686b0152

UR - https://www.mendeley.com/catalogue/4b78f3a6-59ec-3c4a-990b-8d4571da21de/

U2 - 10.26493/1855-3974.2530.e7c

DO - 10.26493/1855-3974.2530.e7c

M3 - Article

VL - 23

JO - Ars Mathematica Contemporanea

JF - Ars Mathematica Contemporanea

SN - 1855-3966

IS - 1

M1 - 2530

ER -