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Weigen Yan 《Physica A》2008,387(24):6069-6078
We obtain explicit expressions of the number of close-packed dimers and entropy for three types of lattices (the so-called 8.8.6, 8.8.4, and hexagonal lattices) with cylindrical boundary condition and the entropy of the 8.8.6 lattice with toroidal boundary condition. Our results and the one on 8.8.4 and hexagonal lattices with toroidal boundary condition by Salinas and Nagle [S.R. Salinas, J.F. Nagle, Theory of the phase transition in the layered hydrogen-bonded SnCl2⋅2H2O crystal, Phys. Rev. B 9 (1974) 4920-4931] and Wu [F.Y. Wu, Dimers on two-dimensional lattices, Inter. J. Modern Phys. B 20 (2006) 5357-5371] imply that the 8.8.6 (or 8.8.4) lattices with cylindrical and toroidal boundary conditions have the same entropy whereas the hexagonal lattices have not. Based on these facts we propose the following problem: under which conditions do the lattices with cylindrical and toroidal boundary conditions have the same entropy? 相似文献
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The Pfaffian method enumerating perfect matchings of plane graphs was discovered by Kasteleyn. We use this method to enumerate perfect matchings in a type of graphs with reflective symmetry which is different from the symmetric graphs considered in [J. Combin. Theory Ser. A 77 (1997) 67, MATCH—Commun. Math. Comput. Chem. 48 (2003) 117]. Here are some of our results: (1) If G is a reflective symmetric plane graph without vertices on the symmetry axis, then the number of perfect matchings of G can be expressed by a determinant of order |G|/2, where |G| denotes the number of vertices of G. (2) If G contains no subgraph which is, after the contraction of at most one cycle of odd length, an even subdivision of K2,3, then the number of perfect matchings of G×K2 can be expressed by a determinant of order |G|. (3) Let G be a bipartite graph without cycles of length 4s, s{1,2,…}. Then the number of perfect matchings of G×K2 equals ∏(1+θ2)mθ, where the product ranges over all non-negative eigenvalues θ of G and mθ is the multiplicity of eigenvalue θ. Particularly, if T is a tree then the number of perfect matchings of T×K2 equals ∏(1+θ2)mθ, where the product ranges over all non-negative eigenvalues θ of T and mθ is the multiplicity of eigenvalue θ. 相似文献
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Let G be a graph with n vertices and ν(G) be the matching number of G. Let η(G) denote the nullity of G (the multiplicity of the eigenvalue zero of G). It is well known that if G is a tree, then η(G)=n-2ν(G). Tan and Liu [X. Tan, B. Liu, On the nullity of unicyclic graphs, Linear Alg. Appl. 408 (2005) 212-220] proved that the nullity set of all unicyclic graphs with n vertices is {0,1,…,n-4} and characterized the unicyclic graphs with η(G)=n-4. In this paper, we characterize the unicyclic graphs with η(G)=n-5, and we prove that if G is a unicyclic graph, then η(G) equals , or n-2ν(G)+2. We also give a characterization of these three types of graphs. Furthermore, we determine the unicyclic graphs G with η(G)=0, which answers affirmatively an open problem by Tan and Liu. 相似文献
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As the extension of the previous work by Ciucu and the present authors [M. Ciucu, W.G. Yan, F.J. Zhang, The number of spanning trees of plane graphs with reflective symmetry, J. Combin. Theory Ser. A 112 (2005) 105-116], this paper considers the problem of enumeration of spanning trees of weighted graphs with an involution which allows fixed points. We show that if G is a weighted graph with an involution, then the sum of weights of spanning trees of G can be expressed in terms of the product of the sums of weights of spanning trees of two weighted graphs with a smaller size determined by the involution of G. As applications, we enumerate spanning trees of the almost-complete bipartite graph, the almost-complete graph, the Möbius ladder, and the almost-join of two copies of a graph. 相似文献
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一类广义系统的渐近稳定性 总被引:5,自引:0,他引:5
QiuWeigen LiuYongqing 《高校应用数学学报(英文版)》2000,15(2):123-127
Abstract. This paper is mainly used to determine the asymptotic stability of a class of nonlineargeneralized systems through its slow subsystems under the assumption of regularity. One exam-ple is given. 相似文献
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It is well known that there are three types of dimers belonging to the three different orientations in a honeycomb lattice,
and in each type all dimers are mutually parallel. Based on a previous result, we can compute the partition function of the
dimer problem of the plane (free boundary) honeycomb lattices with three different activities by using the number of its pure
dimer coverings (perfect matchings). The explicit expression of the partition function and free energy per dimer for many
types of plane honeycomb lattices with fixed shape of boundaries is obtained in this way (for a shape of plane honeycomb lattices,
the procedure that the size goes to infinite, corresponds to a way that the honeycomb lattice goes to infinite). From these
results, an interesting phenomena is observed. In the case of the regions of the plane honeycomb lattice has zero entropy
per dimer—when its size goes to infinite—though in the thermodynamic limit, there is no freedom in placing a dimer at all,
but if we distinguish three types of dimers with nonzero activities, then its free energy per dimer is nonzero. Furthermore,
a sufficient condition for the plane honeycomb lattice with zero entropy per dimer (when the three activities are equal to
1) is obtained. Finally, the difference between the plane honeycomb lattices and the plane quadratic lattices is discussed
and a related problem is proposed. 相似文献
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