Enumeration of spanning trees of graphs with rotational symmetry

Methods of enumeration of spanning trees in a finite graph, a problem related to various areas of mathematics and physics, have been investigated by many mathematicians and physicists. A graph G is said to be n-rotational symmetric if the cyclic group of order n is a subgroup of the automorphism group of G. Some recent studies on the enumeration of spanning trees and the calculation of their asymptotic growth constants on regular lattices with toroidal boundary condition were carried out by physicists. A natural question is to consider the problem of enumeration of spanning trees of lattices with cylindrical boundary condition, which are the so-called rotational symmetric graphs. Suppose G is a graph of order N with n-rotational symmetry and all orbits have size n, which has n isomorphic induced subgraphs G"0,G"1,...,G"n"-"1. In this paper, we prove that if there exists no edge between G"i and G"j for j i-1,i+1(modn), then the number of spanning trees of G can be expressed in terms of the product of the weighted enumerations of spanning trees of n graphs D"i's for i=0,1,...,n-1, where D"i has N/n vertices if i=0 and N/n+1 vertices otherwise. As applications we obtain explicit expressions for the numbers of spanning trees and asymptotic tree number entropies for five lattices with cylindrical boundary condition in the context of physics.

[1]  D. Welsh.  Complexity: Knots, Colourings and Counting: Link polynomials and the Tait conjectures , 1993 .

[2]  G. Kirchhoff.  Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird , 1847 .

[3]  Fuji Zhang,et al.  The number of spanning trees of plane graphs with reflective symmetry , 2005, J. Comb. Theory, Ser. A.

[4]  F. Y. Wu.  The Potts model , 1982 .

[5]  On the Entropy of Spanning Trees on a Large Triangular Lattice , 2003, cond-mat/0309198.

[6]  H. Stanley,et al.  Phase Transitions and Critical Phenomena , 2008 .

[7]  F. Y. Wu,et al.  Spanning trees on graphs and lattices in d dimensions , 2000, cond-mat/0004341.

[8]  G. R. Allen.  Dimer models for the antiferroelectric transition in copper formate tetrahydrate , 1974 .

[9]  Fuji Zhang,et al.  Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians , 2004, Adv. Appl. Math..

[10]  Geoffrey Grimmett.  The Random-Cluster Model , 2002, math/0205237.

[11]  S. Salinas,et al.  Theory of the phase transition in the layered hydrogen-bonded SnCl2· 2H2O crystal , 1974 .

[12]  F. Y. Wu,et al.  Spanning trees on hypercubic lattices and nonorientable surfaces , 2000, Appl. Math. Lett..

[13]  Shu-Chiuan Chang,et al.  A pr 2 00 6 Spanning Trees on Lattices and Integration Identities , 2006 .

[14]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[15]  F. Y. Wu.  Number of spanning trees on a lattice , 1977 .

[16]  C. Fortuin,et al.  On the random-cluster model: I. Introduction and relation to other models , 1972 .

[17]  D. Cvetkovic,et al.  Spectra of graphs : theory and application , 1995 .

[18]  Russell Lyons.  Asymptotic Enumeration of Spanning Trees , 2005, Comb. Probab. Comput..

[19]  E. Dirksen.  Über die Auflösung der numerischen Gleichungen mit Einer Unbekannten. , 1835 .

[20]  R. Pemantle,et al.  PR ] 2 A pr 2 00 4 LOCAL CHARACTERISTICS , ENTROPY AND LIMIT THEOREMS FOR SPANNING TREES AND DOMINO TILINGS VIA TRANSFER-IMPEDANCES Running Head : LOCAL BEHAVIOR OF SPANNING TREES , 1993 .

[21]  S. M. Bhattacharjee,et al.  Dimer models on anisotropic lattices , 1989 .

[22]  David Bruce Wilson,et al.  Trees and Matchings , 2000, Electron. J. Comb..

[23]  Greg Kuperberg.  An Exploration of the Permanent-Determinant Method , 1998, Electron. J. Comb..

[24]  Fuji Zhang,et al.  Enumerating spanning trees of graphs with an involution , 2009, J. Comb. Theory, Ser. A.

[25]  M. Fisher.  On the Dimer Solution of Planar Ising Models , 1966 .

[26]  Wei-Shih Yang,et al.  Spanning Trees on the Sierpinski Gasket , 2006, cond-mat/0609453.

[27]  R. Shrock,et al.  Some exact results for spanning trees on lattices , 2006, cond-mat/0602574.

[28]  R. Pemantle,et al.  Local Characteristics, Entropy and Limit Theorems for Spanning Trees and Domino Tilings Via Transfer-Impedances , 1993, math/0404048.

[29]  Jin Ho Kwak,et al.  Characteristic polynomials of graph coverings , 2004 .

[30]  Dimers on two-dimensional lattices , 2003, cond-mat/0303251.

[31]  William Jockusch.  Perfect Matchings and Perfect Squares , 1994, J. Comb. Theory, Ser. A.

[32]  Norman Biggs.  Algebraic Graph Theory: Index , 1974 .