The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic

Mednykh, Alexander; Mednykh, I. A.

doi:10.1016/j.disc.2018.08.030

Cited by 19 publications

(19 citation statements)

References 32 publications

(44 reference statements)

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“…The multigraphs arising in abelian -towers of bouquets are circulant multigraphs. We refer the reader to [3] and the references therein for formulas that have been derived to calculate the number of spanning trees for such graphs.…”

Section: Examplesmentioning

confidence: 99%

On abelian $$\ell $$-towers of multigraphs II

McGown

Vallières

2021

Ann. Math. Québec

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Let be a rational prime. Previously, abelian -towers of multigraphs were introduced which are analogous to Z -extensions of number fields. It was shown that for a certain class of towers of bouquets, the growth of the -part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for Z -extensions of number fields). In this paper, we give a generalization to a broader class of regular abelian -towers of bouquets than was originally considered. To carry this out, we observe that certain shifted Chebyshev polynomials are members of a continuously parametrized family of power series with coefficients in Z and then study the special value at u = 1 of the Artin-Ihara L-function -adically. RésuméSoit un nombre premier rationnel. Dans un article précédent, les -tours abéliennes de multigraphes furent introduites qui peuvent être vues comme étant analogues aux Z -extensions de corps de nombres. Il fut démontré que pour une certaine famille de tours de bouquets, la valuation -adique du nombre d'arbres couvrants varie de façon prévisible (de manière analogue à un théorème connu d'Iwasawa pour les Z -extensions de corps de nombres). Dans cet article, nous généralisons ce résultat à une famille plus large de -tours régulières et abéliennes de bouquets que celles considérées précédemment. Afin d'accomplir ce but, nous observons que certains polynômes de Tchebychev décalés sont membres d'une famille continûment paramétrée de séries entières à coefficients dans Z et étudions ensuite la valeur spéciale en u = 1 des fonctions L d'Artin-Ihara -adiquement.

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Section: Examplesmentioning

confidence: 99%

On abelian $$\ell $$-towers of multigraphs II

McGown

Vallières

2021

Ann. Math. Québec

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“…2T sp (w). Different aspects of complexity for circulant graphs were investigated in the papers [29,30,9,21,20]. The number of rooted spanning forests for circulant graphs is investigated in [11].…”

Section: Arithmetical Properties Of F (N) For the Graph H Nmentioning

confidence: 99%

Counting Rooted Spanning Forests For Circulant Foliation Over A Graph

Grunwald¹,

Kwon²,

Mednykh³

2021

Preprint

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In this paper, we present a new method to produce explicit formulas for the number of rooted spanning forests f (n) for the infinite family of graphsthis foliation is the circulant graph on n vertices with jumps s i,1 , s i,2 , . . . , s i,k i . This family includes the family of generalized Petersen graphs, I-graphs, sandwiches of circulant graphs, discrete torus graphs and others.The formulas are expressed through Chebyshev polynomials. We prove that the number of rooted spanning forests can be represented in the form f (n) = p f (H)a(n) 2 , where a(n) is an integer sequence and p is a prescribed natural number depending on the number of odd elements in the set of s i,j . Finally, we find an asymptotic formula for f (n) through the Mahler measure of the associated Laurent polynomial.

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“…, s k ). It will be based on our earlier results [14,15], where the numbers of spanning trees was given in terms of the Chebyshev polynomials. By Theorem 1, formula (4) from [15], we have the following result.…”

Section: Complexity Of Circulant Graphs Of Even Valencymentioning

confidence: 99%

“…Graph C n (1, 2). By [3] and [15] we have τ (n) = nF 2 n , where F n is the n-th Fibonacci number. Hence,…”

Section: Examplesmentioning

confidence: 99%

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On Rationality of Generating Function for the Number of Spanning Trees in Circulant Graphs

Mednykh

2020

Algebra Colloq.

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Let F (x) = ∞ n=1 τ (n)x n be the generating function for the number τ (n) of spanning trees in the circulant graphs C n (s 1 , s 2 , . . . , s k ). We show that F (x) is a rational function with integer coefficients satisfying the property F (x) = F (1/x). A similar result is also true for the circulant graphs of odd valency C 2n (s 1 , s 2 , . . . , s k , n). We illustrate the obtained results by a series of examples.

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The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic

Cited by 19 publications

References 32 publications

On abelian $$\ell $$-towers of multigraphs II

On abelian $$\ell $$-towers of multigraphs II

Counting Rooted Spanning Forests For Circulant Foliation Over A Graph

On Rationality of Generating Function for the Number of Spanning Trees in Circulant Graphs

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