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

Mednykh, Alexander; Mednykh, I. A.

doi:10.1142/s1005386720000085

Cited by 5 publications

(3 citation statements)

References 28 publications

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“…The idea of generating functions in the present context is not new [25][26][27][28][29], being used to study the number of spanning trees in various situations, like for circular graphs [30]. Nonetheless, in [24] the framework has been put in very solid and general grounds, establishing T(z) as an important tool for calculating z G .…”

Section: Introductionmentioning

confidence: 98%

Spanning tree generating functions for infinite periodic graphs L and connections with simple closed random walks on L

Portillo¹,

Luz²

2021

J. Phys. A: Math. Theor.

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A spanning tree generating function T(z) for infinite periodic vertex-transitive (vt) lattices L vt has been proposed by Guttmann and Rogers (2012 J. Phys. A: Math. Theor. 45 494001). Their spanning tree constants are then given by T(z = 1). Here an extended T e (z) is constructed, relaxing the previous vt condition to q-regular lattices L and likewise leading to z L = T e (1). Further, in the vt case the method to derive T e yields a new integral formula for the lattice Green function. As examples, spanning tree generating functions for all the eleven vt Archimedean (surprisingly, easier to obtain from T e than from T) and two relevant non-vt, martini and the (4, 82) covering/medial, lattices are derived. The importance of T e (and T)—beyond just being a tool to calculate z L —is illustrated with the proof that the free energy of the random walk loop soup model (defined from closed random walks, the loops, over L) can be written in terms of T e (z). From such result, it is shown that the system critical point—existing only for d = 1 and d = 2—is directly related to z L .

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Section: Introductionmentioning

confidence: 98%

Spanning tree generating functions for infinite periodic graphs L and connections with simple closed random walks on L

Portillo¹,

Luz²

2021

J. Phys. A: Math. Theor.

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“…More precisely, t(G) is the product of nonzero eigenvalues of the Laplacian of G, divided by the number of vertices of G. For families of graphs whose Laplacian eigenvalues can be computed, this method is very useful in computing t(G), except that the results sometimes need to be simplified since eigenvalues may not be rational integers. Extensive work has been done to simplify the formula for t(G) for circulant graphs (see [1,2,8]). For example, the derivation of the number t(C 2 n ) of spanning trees of the square cycle with n vertices using the matrix-tree theorem was done first by Baron et al [3].…”

Section: Introductionmentioning

confidence: 99%

Convex subgraphs and spanning trees of the square cycles

Munemasa¹,

Tanaka²

2023

Preprint

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“…The following theorem was obtained in [65]; it gives an affirmative answer to the question of Lando of whether the generating function for the number of spanning trees in a circulant graph is rational.…”

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confidence: 99%

Cyclic coverings of graphs. Counting rooted spanning forests and trees, Kirchhoff index, and Jacobians

Mednykh,

Mednykh

2023

Russian Math. Surveys

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The purpose of this survey is to describe invariants of cyclic coverings of graphs. The covered graph is assumed to be fixed, and the cyclic covering group has an arbitrarily large order. A classical example of such a covering is a circulant graph. It covers a one-vertex graph with a prescribed number of loops. More sophisticated objects representing the family of cyclic coverings include $I$-, $Y$-, and $H$-graphs, generalized Petersen graphs, sandwich-graphs, discrete tori, and many others. We present analytic formulae for counting rooted spanning forests and trees in cyclic coverings, establish their asymptotics, and study the arithmetic properties of these quantities. Moreover, in the case of circulant graphs we give exact formulae for computing the Kirchhoff index and present structural theorems for the Jacobians of such graphs. Bibliography: 95 titles.

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

Cited by 5 publications

References 28 publications

Spanning tree generating functions for infinite periodic graphs L and connections with simple closed random walks on L

Spanning tree generating functions for infinite periodic graphs L and connections with simple closed random walks on L

Convex subgraphs and spanning trees of the square cycles

Cyclic coverings of graphs. Counting rooted spanning forests and trees, Kirchhoff index, and Jacobians

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