Number Of Perfect Matchings Research Articles

Boost clustering with Gaussian Boson Sampling: a quantum-classical hybrid approach

Gaussian Boson Sampling (GBS) is a recently developed paradigm of quantum computing consisting of sending a Gaussian state through a linear interferometer and then counting the number of photons in each output mode. When the system encodes a symmetric matrix, GBS can be viewed as a tool to sample subgraphs: the most sampled are those with a large number of perfect matchings, and thus are the densest ones. This property has been the foundation of the novel clustering approach we propose in this work, called GBS-based clustering, which relies solely on GBS, without the need of classical algorithms. The GBS-based clustering has been tested on several datasets and benchmarked with two well-known classical clustering algorithms. Results obtained by using a GBS simulator show that on average, our approach outperforms the two classical algorithms in two out of the three chosen metrics, proposing itself as a viable quantum-classical hybrid clustering option.

Further results on enumeration of perfect matchings of Cartesian product graphs

Abstract Counting perfect matchings is an interesting and challenging combinatorial task. It has important applications in statistical physics and chemistry. As the general problem is #P-complete, it is usually tackled by randomized heuristics and approximation schemes. Let G G and H H be two graphs. Denote by G × H G\times H the Cartesian product of graphs G G and H H . The number of perfect matchings of some types of Cartesian product G × H G\times H is investigated by the Pfaffian method, where G ≅ P 2 G\cong {P}_{2} , P 3 {P}_{3} , P 4 {P}_{4} or C 4 {C}_{4} , and H ≅ T H\cong T or H H is a non-bipartite graph with a unique cycle. In this article, we construct a Pfaffian orientation of the graph G × P 2 G\times {P}_{2} , where G G is a non-bipartite graph with k k odd cycles. We obtain the counting formula on the number of perfect matchings of G × P 2 G\times {P}_{2} .

Killing a Vortex

Killing a Vortex

Counting deranged matchings

Let pm(G) denote the number of perfect matchings of a graph G, and let Kr×2n/r denote the complete r-partite graph where each part has size 2n/r. Johnson, Kayll, and Palmer conjectured that for any perfect matching M of Kr×2n/r, we have for 2n divisible by rpm(Kr×2n/r−M)pm(Kr×2n/r)∼e−r/(2r−2).This conjecture can be viewed as a common generalization of counting the number of derangements on n letters, and of counting the number of deranged matchings of K2n. We prove this conjecture. In fact, we prove the stronger result that if R is a uniformly random perfect matching of Kr×2n/r, then the number of edges that R has in common with M converges to a Poisson distribution with parameter r2r−2.

Anti-Ramsey number for perfect matchings in 3-regular bipartite graphs

The anti-Ramsey number AR(G,H) is the maximum number of colors in an edge-coloring of a graph in the family G without any rainbow copy of H. The anti-Ramsey number for matchings has been studied extensively in several graph families, while the problem for perfect matchings was only solved in complete graphs. The anti-Ramsey number of matchings in the family of regular bipartite graphs of order large enough was determined by Li and Xu. Jin improved their bounds on the order of 3-regular bipartite graphs. In this paper, we consider the problem for perfect matchings in 3-regular bipartite graphs. Let G be the family of 3-regular bipartite graphs with m vertices in each partite set. First, we characterize the structure of extremal graphs for the Turán number of perfect matchings in graphs with maximum degree three. Using this characterization, we show that AR(G,mK2)=3m−3. Moreover, we show that AR(F,mK2)=3m−5, where F is the subfamily of 3-edge-connected graphs in G. In order to prove this result, we characterize the structure of bipartite graphs, which contains no perfect matchings, with maximum degree three and size Turán number minus one.

Enumeration of perfect matchings of the middle graph of a graph G with △(G)≤4

Let G be a connected graph with n vertices and m edges, and L(G) and M(G) be the line graph and middle graph of G, respectively. Ciucu et al. (2012) [2] and Dong et al. (2013) [3] proved independently that if m is even and the maximum degree Δ(G) of G satisfies Δ(G)≤3, then L(G) has 2m−n+1 perfect matchings. Recently, Li et al. (2023) [6] showed that if G is a connected cubic graph and m is even, then the number of perfect matchings of M(G) equals 2n2+13n4. In this paper, we extend the results above and show that if G is a connected graph with △(G)≤4 and m+n is even, then the number of perfect matchings of M(G) equals 2m−n+13m−n2.

On the number of perfect matchings in random polygonal chains

Abstract Let G G be a graph. A perfect matching of G G is a regular spanning subgraph of degree one. Enumeration of perfect matchings of a (molecule) graph is interest in chemistry, physics, and mathematics. But the enumeration problem of perfect matchings for general graphs (even in bipartite graphs) is non-deterministic polynomial (NP)-hard. Xiao et al. [C. Xiao, H. Chen, L. Liu, Perfect matchings in random pentagonal chains, J. Math. Chem. 55 (2017), 1878–1886] have studied the problem of perfect matchings for random odd-polygonal chain (i.e., with odd polygons). In this article, we further present simple counting formulae for the expected value of the number of perfect matchings in random even-polygonal chains (i.e., with even polygons). Based on these formulae, we obtain the average values of the number for perfect matchings with respect to the set of all even-polygonal chains with n n polygons.

The Structural Properties of (2, 6)-Fullerenes

A (2,6)-fullerene F is a 2-connected cubic planar graph whose faces are only 2-length and 6-length. Furthermore, it consists of exactly three 2-length faces by Euler’s formula. The (2,6)-fullerene comes from Došlić’s (k,6)-fullerene, a 2-connected 3-regular plane graph with only k-length faces and hexagons. Došlić showed that the (k,6)-fullerenes only exist for k=2, 3, 4, or 5, and some of the structural properties of (k,6)-fullerene for k=3, 4, or 5 were studied. The structural properties, such as connectivity, extendability, resonance, and anti−Kekulé number, are very useful for studying the number of perfect matchings in a graph, and thus for the study of the stability of the molecular graphs. In this paper, we study the properties of (2,6)-fullerene. We discover that the edge-connectivity of (2,6)-fullerenes is 2. Every (2,6)-fullerene is 1-extendable, but not 2-extendable (F is called n-extendable (|V(F)|≥2n+2) if any matching of n edges is contained in a perfect matching of F). F is said to be k-resonant (k≥1) if the deleting of any i (0≤i≤k) disjoint even faces of F results in a graph with at least one perfect matching. We have that every (2,6)-fullerene is 1-resonant. An edge set, S, of F is called an anti−Kekulé set if F−S is connected and has no perfect matchings, where F−S denotes the subgraph obtained by deleting all edges in S from F. The anti−Kekulé number of F, denoted by ak(F), is the cardinality of a smallest anti−Kekulé set of F. We have that every (2,6)-fullerene F with |V(F)|>6 has anti−Kekulé number 4. Further we mainly prove that there exists a (2,6)-fullerene F having fF hexagonal faces, where fF is related to the two parameters n and m.

Excluding a planar matching minor in bipartite graphs

The notion of matching minors is a specialisation of minors fit for the study of graphs with perfect matchings. Matching minors have been used to give a structural description of bipartite graphs on which the number of perfect matchings can be computed efficiently, based on a result of Little, by McCuaig et al. in 1999.In this paper we generalise basic ideas from the graph minor series by Robertson and Seymour to the setting of bipartite graphs with perfect matchings. We introduce a version of Erdős-Pósa property for matching minors and find a direct link between this property and planarity. From this, it follows that a class of bipartite graphs with perfect matchings has bounded perfect matching width if and only if it excludes a planar matching minor. We also present algorithms for bipartite graphs of bounded perfect matching width for a matching version of the disjoint paths problem, matching minor containment, and for counting the number of perfect matchings. From our structural results, we obtain that recognising whether a bipartite graph G contains a fixed planar graph H as a matching minor, and that counting the number of perfect matchings of a bipartite graph that excludes a fixed planar graph as a matching minor are both polynomial time solvable.

Di-Forcing Polynomials for Cyclic Ladder Graphs CLn

The cyclic ladder graph CLn is the Cartesian product of cycles Cn and paths P2, that is CLn=Cn×P2, (n≥3). The di-forcing polynomial of CLn is a binary enumerative polynomial of all perfect matching forcing and anti-forcing numbers. In this paper, we derive recursive formulas for the di-forcing polynomial of cyclic ladder graph CLn by classifying and counting the matching cases of the associated edges of a given vertex, from which we obtain the number of perfect matching, the forcing and anti-forcing polynomials, and the generating function and by computing some di-forcing polynomials of the lower order CLn.

On the number of Perfect Matchings of Tubular Fullerene Graphs

On the number of Perfect Matchings of Tubular Fullerene Graphs

The Number of Perfect Matchings in (3,6)-Fullerene

A [see formula in PDF]-fullerene is a connected cubic plane graph whose faces are only triangles and hexagons, and has the connectivity [see formula in PDF] or [see formula in PDF]. The [see formula in PDF]-fullerenes with connectivity [see formula in PDF] are the tubes consisting of [see formula in PDF] concentric hexagonal layers such that each layer consists of two hexagons, capped on each end by two adjacent triangles, denoted by [see formula in PDF]. A [see formula in PDF]-fullerene [see formula in PDF] with [see formula in PDF] vertices has exactly [see formula in PDF] perfect matchings. The structure of a [see formula in PDF]-fullerene [see formula in PDF] with connectivity [see formula in PDF] can be determined by only three parameters [see formula in PDF], [see formula in PDF] and[see formula in PDF], thus we denote it by [see formula in PDF], where [see formula in PDF] is the radius (number of rings), [see formula in PDF] is the size (number of spokes in each layer, [see formula in PDF], [see formula in PDF] is even), and [see formula in PDF] is the torsion ([see formula in PDF]). In this paper, the counting formula of the perfect matchings in [see formula in PDF]is given, and the number of perfect matchings is obtained. Therefore, the correctness of the conclusion that every bridgeless cubic graph with [see formula in PDF] vertices has at least [see formula in PDF] perfect matchings proposed by Esperet et al is verified for [see formula in PDF]-fullerene [see formula in PDF].

Enumeration of Perfect Matchings of the Cartesian Products of Graphs

A subgraph $ H $ of a graph $G$ is nice if $ G-V(H) $ has a perfect matching. An even cycle $ C $ in an oriented graph is oddly oriented if for either choice of direction of traversal around $ C $, the number of edges of $C$ directed along the traversal is odd. An orientation $ D $ of a graph $ G $ with an even number of vertices is Pfaffian if every nice cycle of $ G $ is oddly oriented in $ D $. Let $ P_{n} $ denote a path on $ n $ vertices. The Pfaffian graph $G \times P_{2n} $ was determined by Lu and Zhang [The Pfaffian property of Cartesian products of graphs, J. Comb. Optim. 27 (2014) 530--540]. In this paper, we characterize the Pfaffian graph $ G \times P_{2n+1} $ with respect to the forbidden subgraphs of $G$. We first give sufficient and necessary conditions under which $G\times P_{2n+1}$ ($n\geqslant 2$) is Pfaffian. Then we characterize the Pfaffian graph $ G \times P_{3} $ when $G$ is a bipartite graph, and we generalize this result to the the case $G$ contains exactly one odd cycle. Following these results, we enumerate the number of perfect matchings of the Pfaffian graph $G \times P_{n}$ in terms of the eigenvalues of the orientation graph of $G$, and we also count perfect matchings of some Pfaffian graph $G \times P_{n}$ by the eigenvalues of $G$.

Sequential importance sampling for estimating expectations over the space of perfect matchings

This paper makes three contributions to estimating the number of perfect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a (1±ϵ)-approximation for the number of perfect matchings of a λ-dense bipartite graph, using O(n1−2λλϵ−2) samples. With size n on each side and for 1 2>λ>0, a λ-dense bipartite graph has all degrees greater than (λ+1 2)n. Second, practical applications of the algorithm requires many calls to matching algorithms. A novel preprocessing step is provided which makes significant improvements. Third, three applications are provided. The first is for counting Latin squares, the second is a practical way of computing the greedy algorithm for a card guessing game with feedback and the third is for stochastic block models. In all three examples, sequential importance sampling allows treating practical problems of reasonably large sizes.

On the number of perfect matchings in the line graph of a traceable graph

On the number of perfect matchings in the line graph of a traceable graph

Counting the maximal and perfect matchings in benzenoid chains

Došlić et al. obtained general generating functions for the numbers of maximal matchings in three classes of benzenoid chains. By using the Hosoya vector and k-matching vector, Cruz et al. and Oz et al. researched the Hosoya index and the k-matching number of benzenoids, respectively. Inspired by these results, by using the maximal matching vector, we show that the number of maximal matchings of a benzenoid chain with n hexagons equals to the product of n certain matrices. In addition, we obtain the numbers of perfect matchings of all benzenoid chains.

The number of perfect matchings, and the nesting properties, of random regular graphs

AbstractWe prove that the number of perfect matchings in is asymptotically normal when is even, as , and . This is the first distributional result of spanning subgraphs of when . Moreover, we prove that and can be coupled so that is a subgraph of with high probability when and . Furthermore, if , , and then and can be coupled so that asymptotically almost surely (a.a.s.) is a subgraph of .

Two counterparts of the TFK formula for cylinder graphs

The classical 1961 solution to the problem of determining the number of perfect matchings (or dimer coverings) of a rectangular grid graph — due independently to Temperley and Fisher, and Kasteleyn, who found the TFK formula — consists of changing the sign of some of the entries in the adjacency matrix so that the Pfaffian of the new matrix gives the number of perfect matchings, and then evaluating this Pfaffian. Another classical method is to use the Lindström-Gessel-Viennot theorem on non-intersecting lattice paths to express the number of perfect matchings as a determinant, and then evaluate this determinant. In this paper we present a new method, which relies on the Cauchy-Binet theorem, and use it to solve the two dimensional dimer problem for cylinder graphs on the square and on the hexagonal lattice.We provide explicit product formulas for these two kinds of cylinder graphs. One advantage of our formula for the square lattice compared to the TFK formula is that ours has a linear number of factors, while the number of factors in the former is quadratic. Our result for the hexagonal lattice yields a formula for the number of periodic stepped surfaces that fit in an infinite tube of given cross-section. This can be regarded as a counterpart of MacMahon's boxed plane partition theorem.

Complete forcing numbers of catacondensed phenylene systems

Combining ?forcing? and ?global? idea, Xu et al. proposed the concepts: complete forcing set and complete forcing number of perfect matchings of graph. In this paper, we give explicit formulae for the complete forcing numbers of phenylene chains and catacondensed phenylene systems, respectively. Moreover, we present an algorithm to find the minimum complete forcing sets of these graphs.

COMBINATORIAL PROPERTIES FOR A CLASS OF SIMPLICIAL COMPLEXES EXTENDED FROM PSEUDO-FRACTAL SCALE-FREE WEB

Simplicial complexes are a popular tool used to model higher-order interactions between elements of complex social and biological systems. In this paper, we study some combinatorial aspects of a class of simplicial complexes created by a graph product, which is an extension of the pseudo-fractal scale-free web. We determine explicitly the independence number, the domination number, and the chromatic number. Moreover, we derive closed-form expressions for the number of acyclic orientations, the number of root-connected acyclic orientations, the number of spanning trees, as well as the number of perfect matchings for some particular cases.