Extremal Polyomino Chains on k-matchings and k-independent Sets

Discrete Applied Mathematics, 1984

An explicit recurrence is derived for the matching polynomial of the pentagonal chain. From this, explicit formulae are obtained for the number of defect-d matchings in the pentagonal chain, for various values of d.

Journal of Algebraic Combinatorics, 2006

We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky. Fomin and Zelivinsky were able to prove Laurentness and conjectured that the coefficients were 1. Our proof establishes a bijection between the terms of the Laurent polynomial and the perfect matchings of certain graphs, generalizing the theory of Aztec Diamonds. In particular, this shows that the coefficients of this polynomial, and polynomials obtained by specializing its variables, are positive, a conjecture of Fomin and Zelevinsky.

The matching polynomial (also called reference and acyclic polynomial) was discovered in chemistry, physics and mathematics at least six times. We demonstrate that the matching polynomial of a bipartite graph coincides with the rook polynomial of a certain board. The basic notions of rook theory 17 are described. It is also shown that the matching polynomial cannot always discriminate between planar isospectral molecules.

2007

In this paper we review the asymptotic matching conjectures for r- regular bipartite graphs, and their connections in estimating the monomer- dimer entropies in d-dimensional integer lattice and Bethe lattices. We prove new rigorous upper and lower bounds for the monomer-dimer entropies, which support these conjectures. We describe a general construction of infinite families of r-regular tori graphs and give

Advances in Mathematics, 2006

In 1979 Kazhdan and Lusztig defined, for every Coxeter group W, a family of polynomials, indexed by pairs of elements of W, which have become known as the Kazhdan–Lusztig polynomials of W, and which have proven to be of importance in several areas of mathematics. In this paper, we show that the combinatorial concept of a special matching plays a

European Journal of Combinatorics, 2007

We consider the class of L-convex polyominoes, i.e. those polyominoes in which any two cells can be connected with an "L" shaped path in one of its four cyclic orientations. The paper proves bijectively that the number f n of L-convex polyominoes with perimeter 2(n + 2) satisfies the linear recurrence relation f n+2 = 4 f n+1 − 2 f n , by first establishing a recurrence of the same form for the cardinality of the "2-compositions" of a natural number n, a simple generalization of the ordinary compositions of n. Then, such 2-compositions are studied and bijectively related to certain words of a regular language over four letters which is in turn bijectively related to L-convex polyominoes. In the last section we give a solution to the open problem of determining the generating function of the area of L-convex polyominoes.

2005

In the first part of this paper it is shown how to use ear decomposition techniques in proving existence and establishing lower bounds to the number of perfect matchings in lattice animals. A correspondence is then established between perfect matchings in certain classes of benzenoid graphs and paths in the rectangular lattice that satisfy certain diagonal constraints. This correspondence is used to give explicit formulas for the number of perfect matchings in hexagonal benzenoid graphs and to derive some identities involving Fibonacci numbers and binomial coefficients. Some of the results about benzenoid graphs are also translated into the context of polyominoes.

Linear Algebra and its Applications, 2003

The energy of a graph G is defined as . . . , p) are the eigenvalues of the adjacency matrix of G. We show that among all polygonal chains with polygons of 4n − 2 vertices (n 2), the linear polygonal chain has minimal energy.

Journal of Physics A: Mathematical and General, 2001

We study a proper subset of polyominoes, called polygonal polyominoes, which are defined to be self-avoiding polygons containing any number of holes, each of which is a self-avoiding polygon. The staircase polygon subset, with staircase holes, is also discussed. The internal holes have no common vertices with each other, nor any common vertices with the surrounding polygon. There are no 'holes-within-holes'. We use the finite-lattice method to count the number of polygonal polyominoes on the square lattice. Series have been derived for both the perimeter and area generating functions. It is known that while the critical point is unchanged by a finite number of holes, when the number of holes is unrestricted the critical point changes. The area generating function coefficients grow exponentially, with a growth constant greater than that for polygons with a finite number of holes, but less than that of polyominoes. We provide an estimate for this growth constant and prove that it is strictly less than that for polyominoes. Also, we prove that, enumerating by perimeter, the generating function of polygonal polyominoes has zero radius of convergence and furthermore we calculate the dominant asymptotics of its coefficients using rigorous bounds.

Molecules, 2021

Multiple zigzag chains Zm,n of length n and width m constitute an important class of regular graphene flakes of rectangular shape. The physical and chemical properties of these basic pericondensed benzenoids can be related to their various topological invariants, conveniently encoded as the coefficients of a combinatorial polynomial, usually referred to as the ZZ polynomial of multiple zigzag chains Zm,n. The current study reports a novel method for determination of these ZZ polynomials based on a hypothesized extension to John–Sachs theorem, used previously to enumerate Kekulé structures of various benzenoid hydrocarbons. We show that the ZZ polynomial of the Zm,n multiple zigzag chain can be conveniently expressed as a determinant of a Toeplitz (or almost Toeplitz) matrix of size m2×m2 consisting of simple hypergeometric polynomials. The presented analysis can be extended to generalized multiple zigzag chains Zkm,n, i.e., derivatives of Zm,n with a single attached polyacene chain ...

Cornell University - arXiv, 2021

We study various versions of the univariate and multivariate matching and rook polynomials. We show that there is most general multivariate matching polynomial, which is, up the some simple substitutions and multiplication with a prefactor, the original multivariate matching polynomial introduced by C. Heilmann and E. Lieb. We follow here a line of investigation which was very successfully pursued over the years by, among others, W. Tutte, B. Bollobas and O. Riordan, and A. Sokal in studying the chromatic and the Tutte polynomial.

European Journal of Combinatorics, 2007

We consider the class of L-convex polyominoes, i.e. those polyominoes in which any two cells can be connected with an "L" shaped path in one of its four cyclic orientations. The paper proves bijectively that the number f n of L-convex polyominoes with perimeter 2(n + 2) satisfies the linear recurrence relation f n+2 = 4 f n+1 − 2 f n , by first establishing a recurrence of the same form for the cardinality of the "2-compositions" of a natural number n, a simple generalization of the ordinary compositions of n. Then, such 2-compositions are studied and bijectively related to certain words of a regular language over four letters which is in turn bijectively related to L-convex polyominoes. In the last section we give a solution to the open problem of determining the generating function of the area of L-convex polyominoes.

Match Communications in Mathematical and in Computer Chemistry, 2015

The matching energy of a graph is defined as the sum of the absolute values of the zeros of its matching polynomial. Let G n,m be the set of connected graphs of order n and with m edges. In this note we determined the extremal graphs from G n,m with n ≤ m ≤ 2n − 4 minimizing the matching energy. Also we determined the minimal matching energy of graphs from G n,m where m = n−1+t and 1 ≤ t ≤ β −1 and with a given matching number β. Moreover, the above extremal graphs have been completely characterized.

Journal of Combinatorial Theory, Series A, 2012

Plane polyominoes are edge-connected sets of cells on the orthogonal lattice Z 2 , considered identical if their cell sets are equal up to an integral translation. We introduce a novel injection from the set of polyominoes with n cells to the set of permutations of [n], and classify the families of convex polyominoes and tree-like convex polyominoes as classes of permutations that avoid some sets of forbidden patterns. By analyzing the structure of the respective permutations of the family of tree-like convex polyominoes, we are able to find the generating function of the sequence that enumerates this family, conclude that this sequence satisfies the linear recurrence a n = 6a n−1 −14a n−2 +16a n−3 −9a n−4 +2a n−5 , and compute the closed-form formula a n = 2 n+2 − (n 3 − n 2 + 10n + 4)/2.