By using the techniques of the transfer matrix, a formula for the number of Kekulé structures in ... more By using the techniques of the transfer matrix, a formula for the number of Kekulé structures in capped armchair nanotubes is established. In effective, according to the symmetric aspect of the tubule, the size of the transfer matrix could be decreased. The study shows that the Kekulé counts in the capped armchair nanotubes is no less than 2 (w(hK3)K1)a (as h/CN), where h and w are the length and circumference of the tubule, respectively, and az0.4661K(2.3887/w). In particular, the above lower bound can be improved to be (4/3) (n/2)Kw 2 (w(hK3)K1)a if the tubule, in terms of graph theory, is bipartite (e.g. the boron-nitride nanotubes), where n is the total number of the vertices on its two caps. As an application, the closed expression for a type is given out and the numerical results for three types with length up to 50 are listed. q Journal of Molecular Structure: THEOCHEM 725 (2005) 223-230 www.elsevier.com/locate/theochem 0166-1280/$ -see front matter q
The enumeration of perfect matchings of graphs is equivalent to the dimer problem which has appli... more The enumeration of perfect matchings of graphs is equivalent to the dimer problem which has applications in statistical physics. A graph $G$ is said to be $n$-rotation symmetric if the cyclic group of order $n$ is a subgroup of the automorphism group of $G$. Jockusch (Perfect matchings and perfect squares, J. Combin. Theory Ser. A, 67(1994), 100-115) and Kuperberg (An exploration of the permanent-determinant method, Electron. J. Combin., 5(1998), #46) proved independently that if $G$ is a plane bipartite graph of order $N$ with $2n$-rotation symmetry, then the number of perfect matchings of $G$ can be expressed as the product of $n$ determinants of order $N/2n$. In this paper we give this result a new presentation. We use this result to compute the entropy of a bulk plane bipartite lattice with $2n$-notation symmetry. We obtain explicit expressions for the numbers of perfect matchings and entropies for two types of cylinders. Using the results on the entropy of the torus obtained by...
An edge cut of a connected graph is called restricted if it separates this graph into components ... more An edge cut of a connected graph is called restricted if it separates this graph into components each having order at least 2; a graph G is super restricted edge connected if GÀS contains an isolated edge for every minimum restricted edge cut S of G. It is proved in this paper that k-regular connected graph G is super restricted edge connected if k > |V(G)|/2+1. The lower bound on k is exemplified to be sharp to some extent. With this observation, we determined the number of edge cuts of size at most 2kÀ2 of these graphs.
In this paper the authors generalize the classic random bipartite graph model, and define a model... more In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows: let m = m(n) be a positive integer-valued function on n and G(n, m; {p k }) the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A = {a1, a2, · · · , an} and B = {b1, b2, · · · , bm}, in which the numbers t a i ,b j of the edges between any two vertices ai ∈ A and bj ∈ B are identically distributed independent random variables with distribution
In this paper, the acyclic Kekulean molecules with greatest HOMO-LUMO separation are determined. ... more In this paper, the acyclic Kekulean molecules with greatest HOMO-LUMO separation are determined. The values of the HOMO-LUMO separations of these molecules are also determined. ? 0166-218X/99/$ -see front matter ? 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 -2 1 8 X ( 9 9 ) 0 0 1 1 9 -5
A connected graph is said to be k-cycle resonant if, for 1 6 t 6 k, any t disjoint cycles in G ar... more A connected graph is said to be k-cycle resonant if, for 1 6 t 6 k, any t disjoint cycles in G are mutually resonant, that is, there is a perfect matching M of G such that each of the t cycles is an M -alternating cycle. The concept of k-cycle resonant graphs was introduced by the present authors in 1994. Some necessary and su cient conditions for a graph to be k-cycle resonant were also given. In this paper, we improve the proof of the necessary and su cient conditions for a graph to be k-cycle resonant, and further investigate planar k-cycle resonant graphs with k = 1; 2. Some new necessary and su cient conditions for a planar graph to be 1-cycle resonant and 2-cycle resonant are established. ?
A connected graph is called elementary if the union of all perfect matchings forms a connected su... more A connected graph is called elementary if the union of all perfect matchings forms a connected subgraph. In this paper we mainly study various properties of plane elementary bipartite graphs so that many important results previously obtained for hexagonal systems are treated in a uniÿed way. Firstly, we show that a plane bipartite graph G is elementary if and only if the boundary of each face (including the inÿnite face) is an alternating cycle with respect to some perfect matching of G. For a plane bipartite graph G all interior vertices of which are of the same degree, a stronger result is obtained; namely, G is elementary if and only if the boundary of the inÿnite face of G is an alternating cycle with respect to some perfect matching of G. Second, the concept of the Z-transformation graph Z(G) of a hexagonal system G (whose vertices represent the perfect matchings of G) is extended to a plane bipartite graph G and some results analogous to those for hexagonal systems are obtained. A peripheral face f of G is called reducible if the removal of the internal vertices and edges of the path that is the intersection of f and the exterior face of G results in a plane elementary bipartite graph. Thirdly, we obtain the reducible face decomposition for plane elementary bipartite graphs. Furthermore, sharp upper and lower bounds for the number of reducible faces are derived. Conversely, we can construct any plane elementary bipartite graphs by adding new peripheral faces one by one. As applications of this approach, we give simple construction methods for several types of plane elementary bipartite graphs G that contain a forcing edge (which belongs to exactly one perfect matching of G) and whose Z-transformation graphs Z(G) contain vertices of degree one. ?
The method of graphical vertex-condensation for enumerating perfect matchings of plane bipartite ... more The method of graphical vertex-condensation for enumerating perfect matchings of plane bipartite graph was found by Propp (Theoret. Comput. Sci. 303(2003), 267-301), and was generalized by Kuo (Theoret. Comput. Sci. 319 (2004), 29-57) and Yan and Zhang (J. Combin. Theory Ser. A, 110(2005), 113-125). In this paper, by a purely combinatorial method some explicit identities on graphical vertex-condensation for enumerating perfect matchings of plane graphs (which do not need to be bipartite) are obtained. As applications of our results, some results on graphical edge-condensation for enumerating perfect matchings are proved, and we count the sum of weights of perfect matchings of weighted Aztec diamond. Proposition 1.1 (Propp [13]) Let G = (U, V ) be a plane bipartite graph in which |U| = |V |. Let vertices a, b, c and d form a 4−cycle face in G, a, c ∈ U, and b, d ∈ V . Then M(G)M(G − {a, b, c, d}) = M(G − {a, b})M(G − {c, d}) + M(G − {a, d})M(G − {b, c}). By a combinatorial method, Kuo [12] generalized Propp's result above as follows. Proposition 1.2 (Kuo [12]) Let G = (U, V ) be a plane bipartite graph in which |U| = |V |. Let vertices a, b, c, and d appear in a cyclic order on a face of G. (1) If a, c ∈ U, and b, d ∈ V , then M(G)M(G − {a, b, c, d}) = M(G − {a, b})M(G − {c, d}) + M(G − {a, d})M(G − {b, c}).
Physica A: Statistical Mechanics and its Applications, 2004
In this paper, we define a family of links which are similar to but more complex than pretzel lin... more In this paper, we define a family of links which are similar to but more complex than pretzel links. We compute the exact expressions of the Jones polynomials for this family of links. Motivated by the connection with the Potts model in statistical mechanics, we investigate accumulation points of zeros of the Jones polynomials for some subfamilies.
Physica A: Statistical Mechanics and its Applications, 2008
Dimer problem 8.8.4 lattice 8.8.6 lattice Hexagonal lattice Cylindrical and toroidal boundary con... more Dimer problem 8.8.4 lattice 8.8.6 lattice Hexagonal lattice Cylindrical and toroidal boundary conditions a b s t r a c t 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 hydrogenbonded SnCl 2 ·2H 2 O 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?
Let Qn and Bn denote a quasi-polyomino chain with n squares and a quasi-hexagonal chain with n he... more Let Qn and Bn denote a quasi-polyomino chain with n squares and a quasi-hexagonal chain with n hexagons, respectively. In this paper, the authors establish a relation between the Wiener numbers of Qn and Bn: W (Qn) = 1 4 W (Bn) − 8 3 n 3 + 14 3 n + 3 . And the extremal quasi-polyomino chains with respect to the Wiener number are determined. Furthermore, several classes of polyomino chains with large Wiener numbers are ordered.
It is well known that there are three types of dimers belonging to the three different orientatio... more 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.
Journal of Statistical Mechanics: Theory and Experiment, 2011
F.Y. Wu, J. Wang, S.-C. Chang and R. Shrock initiated the study of zeros of the Jones polynomial ... more F.Y. Wu, J. Wang, S.-C. Chang and R. Shrock initiated the study of zeros of the Jones polynomial since it was the special case of partition functions of the Potts model in physics. The Homfly polynomial is the generalization of the Jones polynomial. Let L be an oriented link, and P L (v, z) be its Homfly polynomial. In this paper, we study zeros of P L (v, z) with z fixed. We prove the so-called unit circle theorem for a family of generalized Jaeger's links {D n (G)|n = 1, 2, · · ·} which states that |v| = 1 is the limits of zeros of Homfly polynomials of generalized Jaeger's links {D n (G)|n = 1, 2, · · ·} if G is bridgeless. Similar to the result of the Jones polynomial, we also prove that zeros of Homfly polynomials are dense in the whole complex plane.
In this paper, we study new configurations of benzenoid hydrocarbons, called benzenoid links. Rou... more In this paper, we study new configurations of benzenoid hydrocarbons, called benzenoid links. Roughly speaking, a primitive corofusene is a closed narrow hexagonal ribbon with curvature 0. A primitive corofusene or the union of disjoint primitive corofusenes in R 3 is called a benzenoid link. In this paper, we determine the minimum number of hexagons needed for a nontrivial benzenoid link in different senses. We also determine the structures of the smallest and the second smallest nontrivial benzenoid links of different types and their numbers of Kekule structures. We list all the benzenoid Hopf links of type III with 22−25 hexagons by their canonical codes in the appendix.
Based on Clar aromatic sextet theory [Clar, The Aromatic Serxtet (Wiley, New York, 1972)] and the... more Based on Clar aromatic sextet theory [Clar, The Aromatic Serxtet (Wiley, New York, 1972)] and the concept of sextet polynomial introduced by Hosoya and Yamaguchi [Mathematical Concepts in Organic Chemistry (Springer, Berlin, 1986)], we define a new ordering of benzenoid systems. For two isomeric benzenoid systems B 1 and B 2 , we say B 1 > B 2 if each coefficient of sextet polynomial of B 1 is no less than the corresponding coefficient of sextet polynomial of B 2 . In this paper, we consider the ordering of the benzenoid chains. The maximal and second maximal benzenoid chains as well as the minimal, the second minimal up to the fourth minimal benzenoid chains are determined. Furthermore, under this ordering, we determine the maximal and second maximal cyclo-polyphenacenes as well as the minimal, the second minimal, and up to the seventh minimal cyclo-polyphenacenes. * Corresponding address. 293 0259-9791/05/0800-0293/0
In this paper, we first recall some known architectures of polyhedral links . Motivated by these ... more In this paper, we first recall some known architectures of polyhedral links . Motivated by these architectures we introduce the notions of polyhedral links based on edge covering, vertex covering, and mixed edge and vertex covering, which include all polyhedral links in as special cases. The analysis of chirality of polyhedral links is very important in stereochemistry and the Jones polynomial is powerful in differentiating the chirality . Then we give a detailed account of a result on the computation of the Jones polynomial of polyhedral links based on edge covering developed by the present authors in and, at the same time, by using this method we obtain some new computational results on polyhedral links of rational type and uniform polyhedral links with small edge covering units. These new computational results are helpful to judge the chirality of of polyhedral links based on edge covering. Finally, we give some remarks and pose some problems for further study. We introduce the notions of polyhedral links based on edge covering, vertex covering, and mixed edge and vertex covering. These new architectures include polyhedral links in [10-16] as special cases. We hope that these new architectures will become the potential synthetical objects of chemists and biologists. Then we review a general method on the computation of the Jones polynomial of polyhedral links based on the edge covering developed by the present authors in , at the same time, by using this method to obtain some new computational results on polyhedral links of rational type and uniform polyhedral links with small edge covering units. These new computational results are helpful to judge the chirality of polyhedral links based on edge covering. Finally, we give some remarks and pose some open problems for further study.
Denote by T n the set of polyomino chains with n squares. For any T n ∈ T n , let m k (T n ) and ... more Denote by T n the set of polyomino chains with n squares. For any T n ∈ T n , let m k (T n ) and i k (T n ) be the number of k-matchings and k-independent sets of T n , respectively. In this paper, we show that for any polyomino chain T n ∈ T n and any k 0, m k (L n ) m k (T n ) m k (Z n ) and i k (L n ) i k (T n ) i k (Z n ), with the left equalities holding for all k only if T n = L n , and the right equalities holding for all k only if T n = Z n , where L n and Z n are the linear chain and the zig-zag chain, respectively.
As the extension of the previous work by Ciucu and the present authors (J. Combin. Theory Ser. A ... more As the extension of the previous work by Ciucu and the present authors (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.
A "perfect matching" of a graph G with n vertices is a set of ⌊ n 2 ⌋ independent edges of G. In ... more A "perfect matching" of a graph G with n vertices is a set of ⌊ n 2 ⌋ independent edges of G. In this paper we succeeded in determining the trees whose complements have the extremal number of "perfect matchings" for two different group of trees, and some further problems are posed.
Suppose that G is a simple graph. We prove that if G contains small number of cycles of even leng... more Suppose that G is a simple graph. We prove that if G contains small number of cycles of even length then the matching polynomial of G can be expressed in terms of the characteristic polynomials of the skew adjacency matrix A(G e ) of an arbitrary orientation G e of G and the minors of A(G e ). In addition to a formula previously discovered by Godsil and Gutman, we obtain a different formula for the matching polynomial of a general graph.
By using the techniques of the transfer matrix, a formula for the number of Kekulé structures in ... more By using the techniques of the transfer matrix, a formula for the number of Kekulé structures in capped armchair nanotubes is established. In effective, according to the symmetric aspect of the tubule, the size of the transfer matrix could be decreased. The study shows that the Kekulé counts in the capped armchair nanotubes is no less than 2 (w(hK3)K1)a (as h/CN), where h and w are the length and circumference of the tubule, respectively, and az0.4661K(2.3887/w). In particular, the above lower bound can be improved to be (4/3) (n/2)Kw 2 (w(hK3)K1)a if the tubule, in terms of graph theory, is bipartite (e.g. the boron-nitride nanotubes), where n is the total number of the vertices on its two caps. As an application, the closed expression for a type is given out and the numerical results for three types with length up to 50 are listed. q Journal of Molecular Structure: THEOCHEM 725 (2005) 223-230 www.elsevier.com/locate/theochem 0166-1280/$ -see front matter q
The enumeration of perfect matchings of graphs is equivalent to the dimer problem which has appli... more The enumeration of perfect matchings of graphs is equivalent to the dimer problem which has applications in statistical physics. A graph $G$ is said to be $n$-rotation symmetric if the cyclic group of order $n$ is a subgroup of the automorphism group of $G$. Jockusch (Perfect matchings and perfect squares, J. Combin. Theory Ser. A, 67(1994), 100-115) and Kuperberg (An exploration of the permanent-determinant method, Electron. J. Combin., 5(1998), #46) proved independently that if $G$ is a plane bipartite graph of order $N$ with $2n$-rotation symmetry, then the number of perfect matchings of $G$ can be expressed as the product of $n$ determinants of order $N/2n$. In this paper we give this result a new presentation. We use this result to compute the entropy of a bulk plane bipartite lattice with $2n$-notation symmetry. We obtain explicit expressions for the numbers of perfect matchings and entropies for two types of cylinders. Using the results on the entropy of the torus obtained by...
An edge cut of a connected graph is called restricted if it separates this graph into components ... more An edge cut of a connected graph is called restricted if it separates this graph into components each having order at least 2; a graph G is super restricted edge connected if GÀS contains an isolated edge for every minimum restricted edge cut S of G. It is proved in this paper that k-regular connected graph G is super restricted edge connected if k > |V(G)|/2+1. The lower bound on k is exemplified to be sharp to some extent. With this observation, we determined the number of edge cuts of size at most 2kÀ2 of these graphs.
In this paper the authors generalize the classic random bipartite graph model, and define a model... more In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows: let m = m(n) be a positive integer-valued function on n and G(n, m; {p k }) the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A = {a1, a2, · · · , an} and B = {b1, b2, · · · , bm}, in which the numbers t a i ,b j of the edges between any two vertices ai ∈ A and bj ∈ B are identically distributed independent random variables with distribution
In this paper, the acyclic Kekulean molecules with greatest HOMO-LUMO separation are determined. ... more In this paper, the acyclic Kekulean molecules with greatest HOMO-LUMO separation are determined. The values of the HOMO-LUMO separations of these molecules are also determined. ? 0166-218X/99/$ -see front matter ? 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 -2 1 8 X ( 9 9 ) 0 0 1 1 9 -5
A connected graph is said to be k-cycle resonant if, for 1 6 t 6 k, any t disjoint cycles in G ar... more A connected graph is said to be k-cycle resonant if, for 1 6 t 6 k, any t disjoint cycles in G are mutually resonant, that is, there is a perfect matching M of G such that each of the t cycles is an M -alternating cycle. The concept of k-cycle resonant graphs was introduced by the present authors in 1994. Some necessary and su cient conditions for a graph to be k-cycle resonant were also given. In this paper, we improve the proof of the necessary and su cient conditions for a graph to be k-cycle resonant, and further investigate planar k-cycle resonant graphs with k = 1; 2. Some new necessary and su cient conditions for a planar graph to be 1-cycle resonant and 2-cycle resonant are established. ?
A connected graph is called elementary if the union of all perfect matchings forms a connected su... more A connected graph is called elementary if the union of all perfect matchings forms a connected subgraph. In this paper we mainly study various properties of plane elementary bipartite graphs so that many important results previously obtained for hexagonal systems are treated in a uniÿed way. Firstly, we show that a plane bipartite graph G is elementary if and only if the boundary of each face (including the inÿnite face) is an alternating cycle with respect to some perfect matching of G. For a plane bipartite graph G all interior vertices of which are of the same degree, a stronger result is obtained; namely, G is elementary if and only if the boundary of the inÿnite face of G is an alternating cycle with respect to some perfect matching of G. Second, the concept of the Z-transformation graph Z(G) of a hexagonal system G (whose vertices represent the perfect matchings of G) is extended to a plane bipartite graph G and some results analogous to those for hexagonal systems are obtained. A peripheral face f of G is called reducible if the removal of the internal vertices and edges of the path that is the intersection of f and the exterior face of G results in a plane elementary bipartite graph. Thirdly, we obtain the reducible face decomposition for plane elementary bipartite graphs. Furthermore, sharp upper and lower bounds for the number of reducible faces are derived. Conversely, we can construct any plane elementary bipartite graphs by adding new peripheral faces one by one. As applications of this approach, we give simple construction methods for several types of plane elementary bipartite graphs G that contain a forcing edge (which belongs to exactly one perfect matching of G) and whose Z-transformation graphs Z(G) contain vertices of degree one. ?
The method of graphical vertex-condensation for enumerating perfect matchings of plane bipartite ... more The method of graphical vertex-condensation for enumerating perfect matchings of plane bipartite graph was found by Propp (Theoret. Comput. Sci. 303(2003), 267-301), and was generalized by Kuo (Theoret. Comput. Sci. 319 (2004), 29-57) and Yan and Zhang (J. Combin. Theory Ser. A, 110(2005), 113-125). In this paper, by a purely combinatorial method some explicit identities on graphical vertex-condensation for enumerating perfect matchings of plane graphs (which do not need to be bipartite) are obtained. As applications of our results, some results on graphical edge-condensation for enumerating perfect matchings are proved, and we count the sum of weights of perfect matchings of weighted Aztec diamond. Proposition 1.1 (Propp [13]) Let G = (U, V ) be a plane bipartite graph in which |U| = |V |. Let vertices a, b, c and d form a 4−cycle face in G, a, c ∈ U, and b, d ∈ V . Then M(G)M(G − {a, b, c, d}) = M(G − {a, b})M(G − {c, d}) + M(G − {a, d})M(G − {b, c}). By a combinatorial method, Kuo [12] generalized Propp's result above as follows. Proposition 1.2 (Kuo [12]) Let G = (U, V ) be a plane bipartite graph in which |U| = |V |. Let vertices a, b, c, and d appear in a cyclic order on a face of G. (1) If a, c ∈ U, and b, d ∈ V , then M(G)M(G − {a, b, c, d}) = M(G − {a, b})M(G − {c, d}) + M(G − {a, d})M(G − {b, c}).
Physica A: Statistical Mechanics and its Applications, 2004
In this paper, we define a family of links which are similar to but more complex than pretzel lin... more In this paper, we define a family of links which are similar to but more complex than pretzel links. We compute the exact expressions of the Jones polynomials for this family of links. Motivated by the connection with the Potts model in statistical mechanics, we investigate accumulation points of zeros of the Jones polynomials for some subfamilies.
Physica A: Statistical Mechanics and its Applications, 2008
Dimer problem 8.8.4 lattice 8.8.6 lattice Hexagonal lattice Cylindrical and toroidal boundary con... more Dimer problem 8.8.4 lattice 8.8.6 lattice Hexagonal lattice Cylindrical and toroidal boundary conditions a b s t r a c t 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 hydrogenbonded SnCl 2 ·2H 2 O 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?
Let Qn and Bn denote a quasi-polyomino chain with n squares and a quasi-hexagonal chain with n he... more Let Qn and Bn denote a quasi-polyomino chain with n squares and a quasi-hexagonal chain with n hexagons, respectively. In this paper, the authors establish a relation between the Wiener numbers of Qn and Bn: W (Qn) = 1 4 W (Bn) − 8 3 n 3 + 14 3 n + 3 . And the extremal quasi-polyomino chains with respect to the Wiener number are determined. Furthermore, several classes of polyomino chains with large Wiener numbers are ordered.
It is well known that there are three types of dimers belonging to the three different orientatio... more 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.
Journal of Statistical Mechanics: Theory and Experiment, 2011
F.Y. Wu, J. Wang, S.-C. Chang and R. Shrock initiated the study of zeros of the Jones polynomial ... more F.Y. Wu, J. Wang, S.-C. Chang and R. Shrock initiated the study of zeros of the Jones polynomial since it was the special case of partition functions of the Potts model in physics. The Homfly polynomial is the generalization of the Jones polynomial. Let L be an oriented link, and P L (v, z) be its Homfly polynomial. In this paper, we study zeros of P L (v, z) with z fixed. We prove the so-called unit circle theorem for a family of generalized Jaeger's links {D n (G)|n = 1, 2, · · ·} which states that |v| = 1 is the limits of zeros of Homfly polynomials of generalized Jaeger's links {D n (G)|n = 1, 2, · · ·} if G is bridgeless. Similar to the result of the Jones polynomial, we also prove that zeros of Homfly polynomials are dense in the whole complex plane.
In this paper, we study new configurations of benzenoid hydrocarbons, called benzenoid links. Rou... more In this paper, we study new configurations of benzenoid hydrocarbons, called benzenoid links. Roughly speaking, a primitive corofusene is a closed narrow hexagonal ribbon with curvature 0. A primitive corofusene or the union of disjoint primitive corofusenes in R 3 is called a benzenoid link. In this paper, we determine the minimum number of hexagons needed for a nontrivial benzenoid link in different senses. We also determine the structures of the smallest and the second smallest nontrivial benzenoid links of different types and their numbers of Kekule structures. We list all the benzenoid Hopf links of type III with 22−25 hexagons by their canonical codes in the appendix.
Based on Clar aromatic sextet theory [Clar, The Aromatic Serxtet (Wiley, New York, 1972)] and the... more Based on Clar aromatic sextet theory [Clar, The Aromatic Serxtet (Wiley, New York, 1972)] and the concept of sextet polynomial introduced by Hosoya and Yamaguchi [Mathematical Concepts in Organic Chemistry (Springer, Berlin, 1986)], we define a new ordering of benzenoid systems. For two isomeric benzenoid systems B 1 and B 2 , we say B 1 > B 2 if each coefficient of sextet polynomial of B 1 is no less than the corresponding coefficient of sextet polynomial of B 2 . In this paper, we consider the ordering of the benzenoid chains. The maximal and second maximal benzenoid chains as well as the minimal, the second minimal up to the fourth minimal benzenoid chains are determined. Furthermore, under this ordering, we determine the maximal and second maximal cyclo-polyphenacenes as well as the minimal, the second minimal, and up to the seventh minimal cyclo-polyphenacenes. * Corresponding address. 293 0259-9791/05/0800-0293/0
In this paper, we first recall some known architectures of polyhedral links . Motivated by these ... more In this paper, we first recall some known architectures of polyhedral links . Motivated by these architectures we introduce the notions of polyhedral links based on edge covering, vertex covering, and mixed edge and vertex covering, which include all polyhedral links in as special cases. The analysis of chirality of polyhedral links is very important in stereochemistry and the Jones polynomial is powerful in differentiating the chirality . Then we give a detailed account of a result on the computation of the Jones polynomial of polyhedral links based on edge covering developed by the present authors in and, at the same time, by using this method we obtain some new computational results on polyhedral links of rational type and uniform polyhedral links with small edge covering units. These new computational results are helpful to judge the chirality of of polyhedral links based on edge covering. Finally, we give some remarks and pose some problems for further study. We introduce the notions of polyhedral links based on edge covering, vertex covering, and mixed edge and vertex covering. These new architectures include polyhedral links in [10-16] as special cases. We hope that these new architectures will become the potential synthetical objects of chemists and biologists. Then we review a general method on the computation of the Jones polynomial of polyhedral links based on the edge covering developed by the present authors in , at the same time, by using this method to obtain some new computational results on polyhedral links of rational type and uniform polyhedral links with small edge covering units. These new computational results are helpful to judge the chirality of polyhedral links based on edge covering. Finally, we give some remarks and pose some open problems for further study.
Denote by T n the set of polyomino chains with n squares. For any T n ∈ T n , let m k (T n ) and ... more Denote by T n the set of polyomino chains with n squares. For any T n ∈ T n , let m k (T n ) and i k (T n ) be the number of k-matchings and k-independent sets of T n , respectively. In this paper, we show that for any polyomino chain T n ∈ T n and any k 0, m k (L n ) m k (T n ) m k (Z n ) and i k (L n ) i k (T n ) i k (Z n ), with the left equalities holding for all k only if T n = L n , and the right equalities holding for all k only if T n = Z n , where L n and Z n are the linear chain and the zig-zag chain, respectively.
As the extension of the previous work by Ciucu and the present authors (J. Combin. Theory Ser. A ... more As the extension of the previous work by Ciucu and the present authors (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.
A "perfect matching" of a graph G with n vertices is a set of ⌊ n 2 ⌋ independent edges of G. In ... more A "perfect matching" of a graph G with n vertices is a set of ⌊ n 2 ⌋ independent edges of G. In this paper we succeeded in determining the trees whose complements have the extremal number of "perfect matchings" for two different group of trees, and some further problems are posed.
Suppose that G is a simple graph. We prove that if G contains small number of cycles of even leng... more Suppose that G is a simple graph. We prove that if G contains small number of cycles of even length then the matching polynomial of G can be expressed in terms of the characteristic polynomials of the skew adjacency matrix A(G e ) of an arbitrary orientation G e of G and the minors of A(G e ). In addition to a formula previously discovered by Godsil and Gutman, we obtain a different formula for the matching polynomial of a general graph.
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