Asymptotics and Arithmetical Properties of Complexity for Circulant Graphs

Mednykh, Alexander; Mednykh, I. A.

doi:10.1134/s1064562418020138

Cited by 17 publications

(18 citation statements)

References 11 publications

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“…We mention that the number of spanning trees for circulant graphs is expressed is terms of the Chebyshev polynomials; it was found in [24], [23] and [22]. Similar results are also true for the I-graph [15]. The number of spanning trees in cobordism of two circulant graphs was investigated in [1].…”

Section: Introductionsupporting

confidence: 54%

Counting rooted spanning forests in cobordism of two circulant graphs

Абросимов¹,

Baigonakova²,

Grunwald³

et al. 2020

Sib. Elektron. Mat. Izv.

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We consider a family of graphs Hn(s1, . . . , s k ; t1, . . . , t ℓ ), which is a generalization of the family of I-graphs, which in turn, includes the generalized Petersen graphs and the prism graphs. We present an explicit formula for the number fH (n) of rooted spanning forests in these graphs in terms of Chebyshev polynomials and find its asymptotics. Also, we show that the number of rooted spanning forests can be represented in the form fH (n) = p a(n) 2 , where a(n) is an integer sequence and p is a prescribed integer depending on the number of odd elements in the sequence s1, . . . , s k , t1, . . . , t ℓ and the parity of n.

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Section: Introductionsupporting

confidence: 54%

Counting rooted spanning forests in cobordism of two circulant graphs

Абросимов¹,

Baigonakova²,

Grunwald³

et al. 2020

Sib. Elektron. Mat. Izv.

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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%

“…In the recent papers by the authors [14] and [15], explicit formulas for the number of spanning trees τ (n) in circulant graphs C n (s 1 , s 2 , . .…”

Section: Introductionmentioning

confidence: 99%

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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Asymptotics and Arithmetical Properties of Complexity for Circulant Graphs

Cited by 17 publications

References 11 publications

Counting rooted spanning forests in cobordism of two circulant graphs

Counting rooted spanning forests in cobordism of two circulant graphs

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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