Number Of Perfect Matchings Research Articles

Counting Perfect Matchings in Hexagonal Systems Associated with Benzenoids

Many hydrocarbons involve hexagonal rings; benzene, consisting of a single hexagon of carbons with hydrogens attached, is the original example. It turns out that the number of perfect matchings in certain associated graphs is relevant to the chemistry of these hydrocarbons. Furthermore, standard methods of undergraduate linear algebra and discrete mathematics can be used to count the matchings, and familiar counting numbers show up. In this article we present this easily accessible but seemingly little known connection between chemistry and mathematics. According to the chemistry folklore, the German chemist August Kekule (18291896) discovered the molecular structure of benzene after he dreamed of a snake swallowing its own tail. Apparently, the dream led to his conjecture that benzene consists of 6 carbon atoms, each linked to 1 hydrogen atom via a carbon-hydrogen bond, and that the carbon atoms are linked to each other via a cycle of length 6 consisting of alternating single and double carbon-carbon bonds. FIGURE 1 illustrates a molecular model of benzene. Kekule's discovery initiated the study of special types of graphs used to model benzene-like molecules called benzenoids.

New lower bound on the number of perfect matchings in fullerene graphs

A fullerene graph is a cubic and 3-connected plane graph (or spherical map) that has exactly 12 faces of size 5 and other faces of size 6, which can be regarded as the molecular graph of a fullerene. T. Doslic [3] obtained that a fullerene graph with p vertices has at least (p+2)/2 perfect matchings by applying the recently developed decomposition techniques in matching theory of graphs. This note gets a better lower bound ⌈3(p+2)/4⌉ of the number of perfect matchings of a fullerene graph by finding its 2-extendability. This property further implies a chemical consequence that every derivative of a fullerene by substituting any two pairs of adjacent carbon atoms permits a Kekule structure.

Random Matchings Which Induce Hamilton Cycles and Hamiltonian Decompositions of Random Regular Graphs

Select four perfect matchings of 2n vertices, independently at random. We find the asymptotic probability that each of the first and second matchings forms a Hamilton cycle with each of the third and fourth. This is generalised to embrace any fixed number of perfect matchings, where a prescribed set of pairs of matchings must each produce Hamilton cycles (with suitable restrictions on the prescribed set of pairs). We also show how the result with four matchings implies that a random d-regular graph for fixed even d⩾4 asymptotically almost surely decomposes into d/2 Hamilton cycles. This completes a general result on the edge-decomposition of a random regular graph into regular spanning subgraphs of given degrees together with Hamilton cycles and verifies conjectures of Janson and of Robinson and Wormald.

On the Number of Perfect Matchings and Hamilton Cycles in $\epsilon$-Regular Non-bipartite Graphs

A graph $G=(V,E)$ on $n$ vertices is super $\epsilon$-regular if (i) all vertices have degree in the range $[(d-\epsilon)n,(d+\epsilon)n]$, $dn$ being the average degree, and (ii) for every pair of disjoint sets $S,T\subseteq V,\,|S|,|T|\geq \epsilon n$, $e(S,T)$ is in the range $[(d-\epsilon)|S||T|,(d+\epsilon)|S||T|]$. We show that the number of perfect matchings lies in the range $$[((d-2\epsilon)^\nu {{n!}\over {\nu !2^\nu}},(d+2\epsilon)^\nu {{n!}\over {\nu !2^\nu}}],$$ where $\nu ={{n}\over {2}}$, and the number of Hamilton cycles lies in the range $[(d-2\epsilon)^nn!,(d+2\epsilon)^nn!]$.

Matchings in Graphs on Non-orientable Surfaces

We generalize Kasteleyn's method of enumerating the perfect matchings in a planar graph to graphs embedding on an arbitrary compact boundaryless 2-manifold S. Kasteleyn stated that perfect matchings in a graph embedding on a surface of genus g could be enumerated as a linear combination of 4g Pfaffians of modified adjacency matrices of the graph. We give an explicit construction that not only does this, but also does it for graphs embedding on non-orientable surfaces. If a graph embeds on the connected sum of a genus g surface with a projective plane (respectively, Klein bottle), the number of perfect matchings can be computed as a linear combination of 22g+1 (respectively, 22g+2) Pfaffians. Thus for any S, this is 22−χ(S) Pfaffians. We also introduce “crossing orientations,” the analogue of Kasteleyn's “admissible orientations” in our context, describing how the Pfaffian of a signed adjacency matrix of a graph gives the sign of each perfect matching according to the number of edge-crossings in the matching. Finally, we count the perfect matchings of an m×n grid on a Möbius strip.

The Clar covering polynomial of hexagonal systems III

The Clar covering polynomial is a recently proposed concept of hexagonal systems by which some important topological indices such as perfect matching count, Clar number, first Herndon number, etc., can be easily obtained. In this paper we establish a relationship between the Clar covering polynomial and sextet polynomial. A lower bound of the Clar number and some properties of coefficients of the Clar covering polynomial are thus deduced. It is mentioned that the summation of coefficients of Clar covering polynomial can be used to calculate the number of perfect matchings of a certain kind of polyominoes relating to crystal physics.

Higher Dimensional Aztec Diamonds and a (2d + 2)-Vertex Model

Motivated by the close relationship between the number of perfect matchings of the Aztec diamond graph introduced in [5] and the free energy of the square-ice model, we consider a higher dimensional analog of this phenomenon. For d ≥ 1, we construct d-uniform hypergraphs which generalize the Aztec diamonds and we consider a companion d-dimensional statistical model (called the 2d + 2-vertex model) whose free energy is given by the logarithm of the number of perfect matchings of our hypergraphs. We prove that the limit defining the free energy per site of the 2d + 2-vertex model exists and we obtain bounds for it. As a consequence, we obtain an especially good asymptotical approximation for the number of matchings of our hypergraphs.

A Parallel Algorithm for Finding a Perfect Matching in a Planar Graph

We present au intuitive and simple O( log 2 n log Φ) time, O(n2 M(n)) processor parallel algorithm for finding a perfect matching in a planar graph, where n is the order, Φ is the number of the distinct perfect matchings of the graph and M(n) is the best sequential complexity of matrix multiplication of size n. The algorithm runs on a CRCW PRAM. Clearly, our algorithm belongs to the class NC if the number of perfect matchings is polynomially bounded in the size of the input graph.

A Chernoff Bound for Random Walks on Expander Graphs

We consider a finite random walk on a weighted graph G; we show that the fraction of time spent in a set of vertices A converges to the stationary probability $\pi (A)$ with error probability exponentially small in the length of the random walk and the square of the size of the deviation from $\pi (A)$. The exponential bound is in terms of the expansion of G and improves previous results of [D. Aldous, Probab. Engrg. Inform. Sci., 1 (1987), pp. 33--46], [L. Lovász and M. Simonovits, {\it Random Structures Algorithms}, 4 (1993), pp. 359--412], [M. Ajtai, J. Komlós, and E. Szemerédi, Deterministic simulation of logspace, in Proc. 19th ACM Symp. on Theory of Computing, 1987]. We show that taking the sample average from one trajectory gives a more efficient estimate of $\pi (A)$ than the standard method of generating independent sample points from several trajectories. Using this more efficient sampling method, we improve the algorithms of Jerrum and Sinclair for approximating the number of perfect matchings in a dense graph and for approximating the partition function of a ferromagnetic Ising system, and we give an efficient algorithm to estimate the entropy of a random walk on an unweighted graph.

On the random generation and counting of matchings in dense graphs

In this work we present a fully randomized approximation scheme for counting the number of perfect matchings in a dense bipartite graphs, that is equivalent to get a fully randomized approximation scheme to the permanent of a dense boolean matrix. We achieve this known solution, through novel extensions in the theory of suitable non-reversible, Markov chains which mix rapidly and have a near-uniform distribution.

Perfect Matchings in $\epsilon$-regular Graphs

A super $(d,\epsilon)$-regular graph on $2n$ vertices is a bipartite graph on the classes of vertices $V_1$ and $V_2$, where $|V_1|=|V_2|=n$, in which the minimum degree and the maximum degree are between $ (d-\epsilon)n$ and $ (d+\epsilon) n$, and for every $U \subset V_1, W \subset V_2$ with $|U| \geq \epsilon n$, $|W| \geq \epsilon n$, $|{{e(U,W) }\over{|U||W|}}-{{e(V_1,V_2)}\over{|V_1||V_2|}}| < \epsilon.$ We prove that for every $1>d >2 \epsilon >0$ and $n>n_0(\epsilon)$, the number of perfect matchings in any such graph is at least $(d-2\epsilon)^n n!$ and at most $(d+2 \epsilon)^n n!$. The proof relies on the validity of two well known conjectures for permanents; the Minc conjecture, proved by Brégman, and the van der Waerden conjecture, proved by Falikman and Egorichev.

Maximising the Permanent of (0,1)-Matrices and the Number of Extensions of Latin Rectangles

Let $k\geq2$, $m\geq5$ and $n=mk$ be integers. By finding bounds for certain rook polynomials, we identify the $k\times n$ Latin rectangles with the most extensions to $(k+1)\times n$ Latin rectangles. Equivalently, we find the $(n-k)$-regular subgraphs of $K_{n,n}$ which have the greatest number of perfect matchings, and the $(0,1)$-matrices with exactly $k$ zeroes in every row and column which maximise the permanent. Without the restriction on $n$ being a multiple of $k$ we solve the above problem (and the corresponding minimisation problem) for $k=2$. We also provide some computational results for small values of $n$ and $k$. Our results partially settle two open problems of Minc and conjectures by Merriell, and Godsil and McKay.

On lower bounds of number of perfect matchings in fullerene graphs

Couting perfect matchings in graphs is a very difficult problem. Some recently developed decomposition techniques allowed us to estimate the lower bound of the number of perfect matchings in certain classes of graphs. By applying these techniques, it will be shown that every fullerene graph with p vertices contains at least p/2+1 perfect matchings. It is a significant improvement over a previously published estimate, which claimed at least three perfect matchings in every fullerene graph. As an interesting chemical consequence, it is noted that every bisubstituted derivative of a fullerene still permits a Kekule structure.

On the Number of Group-Weighted Matchings

Let G be a bipartite graph with a bicoloration {A,B}, |A|=|B|. Let E(G) → A x B denote the edge set of G, and let m(G) denote the number of perfect matchings of G. Let K be a (multiplicative) finite abelsian group |K| = k, and let w:E(G) → K be a weight assignment on the edges of G. FOr S → E(G) let w(S) = ∏e∈Sw(e). A perfect matching M of G is a w-matching if w(M)=1. We shall be interested in m(G,w), the number of w-matchings of G.It is shown that if deg(a) ≥ d for all a ∈ A, then either G has no w-matchings, or G has at least (d - k + 1)! w-matchings.

Perfect matchings and ears in elementary bipartite graphs

We give lower and upper bounds for the number of reducible ears as well as upper bounds for the number of perfect matchings in an elementary bipartite graph. An application to chemical graphs is also discussed. In addition, a method to construct all minimal elementary bipartite graphs is described.

Perfect Matchings of Polyomino Graphs

This paper gives necessary and sufficient conditions for a polyomino graph to have a perfect matching and to be elementary, respectively. As an application, we can decompose a non-elementary polyomino with perfect matchings into a number of elementary subpolyominoes so that the number of perfect matchings of the original non-elementary polyomino is equal to the product of those of the elementary subpolyominoes.

A note on the number of perfect matchings of bipartite graphs

Let G be a bipartite graph with 2 n vertices, A its adjacency matrix and K the number of perfect matchings. For plane bipartite graphs each interior face of which is surrounded by a circuit of length 4 s + 2, s ϵ {1, 2, …}, an elegant formula, i.e. det A = (−1) n K 2, had been rigorously proved by Cvetković et al. (1982). For general bipartite graphs, this note contains a necessary and sufficient condition for the above relation to hold. A fast algorithm to check if a plane bipartite graph has such a relation is given.

Enumeration of Perfect Matchings in Graphs with Reflective Symmetry

A plane graph is called symmetric if it is invariant under the reflection across some straight line. We prove a result that expresses the number of perfect matchings of a large class of symmetric graphs in terms of the product of the number of matchings of two subgraphs. When the graph is also centrally symmetric, the two subgraphs are isomorphic and we obtain a counterpart of Jockusch's squarishness theorem. As applications of our result, we enumerate the perfect matchings of several families of graphs and we obtain new solutions for the enumeration of two of the ten symmetry classes of plane partitions (namely, transposed complementary and cyclically symmetric, transposed complementary) contained in a given box. Finally, we consider symmetry classes of perfect matchings of the Aztec diamond graph and we solve the previously open problem of enumerating the matchings that are invariant under a rotation by 90 degrees.

Perfect Matchings in Random r-regular, s-uniform Hypergraphs

We show that r-regular, s-uniform hypergraphs contain a perfect matching with high probability (whp), provided The Proof is based on the application of a technique of Robinson and Wormald [7, 8]. The space of hypergraphs is partitioned into subsets according to the number of small cycles in the hypergraph. The difference in the expected number of perfect matchings between these subsets explains most of the variance of the number of perfect matchings in the space of hypergraphs, and is sufficient to prove existence (whp), using the Chebychev Inequality.

Expressions for the perfect matching numbers of cubicl x m x n lattices and their asymptotic values

A general expression of the perfect matching number for thel x m x n cubic lattice was conjectured and examined for infinitely large systems. The asymptotic value of the square of the perfect matching number was calculated by numerical integration. The present treatment will give a key to obtain the true analytic solution of the perfect matching numbers for the 3-dimensional lattices.