Pfaffian Orientations Research Articles

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} .

A note on graphs with purely imaginary per-spectrum

In 1983, Borowiecki and Jóźwiak posed the problem “Characterize those graphs which have purely imaginary per-spectrum.” This problem is still open. The most general result, although a partial solution, was given in 2004 by Yan and Zhang, who show that if G is a bipartite graph containing no subgraph which is an even subdivision of K2,3, then it has purely imaginary per-spectrum. Zhang and Li in 2012 proved that such graphs are planar and admit a Pfaffian orientation. In this article, we describe how to construct graphs with purely imaginary per-spectrum having a subgraph which is an even subdivision of K2,3 (planar and nonplanar) using coalescence of rooted graphs.

The Pfaffian property of Cayley graphs on dihedral groups

Which graphs admit a Pfaffian orientation? This is a significant and interesting problem in matching theory because the number of perfect matchings of a graph can be computed in polynomial time if it admits a Pfaffian orientation. In this paper, we completely characterize the Pfaffian property of Cayley graphs on dihedral groups.

Perfect matchings on a type of lattices with toroidal boundary

Enumeration of perfect matchings on graphs has a longstanding interest in combinatorial mathematics. In this paper, we obtain some explicit expressions of the number of perfect matchings for a type of Archimedean lattices with toroidal boundary by applying Tesler's crossing orientations to obtain some Pfaffian orientations and enumerating their Pfaffians.

Topologically robust zero-sum games and Pfaffian orientation: How network topology determines the long-time dynamics of the antisymmetric Lotka-Volterra equation

To explore how the topology of interaction networks determines the robustness of dynamical systems, we study the antisymmetric Lotka-Volterra equation (ALVE). The ALVE is the replicator equation of zero-sum games in evolutionary game theory, in which the strengths of pairwise interactions between strategies are defined by an antisymmetric matrix such that typically some strategies go extinct over time. Here we show that there also exist topologically robust zero-sum games, such as the rock-paper-scissors game, for which all strategies coexist for all choices of interaction strengths. We refer to such zero-sum games as coexistence networks and construct coexistence networks with an arbitrary number of strategies. By mapping the long-time dynamics of the ALVE to the algebra of antisymmetric matrices, we identify simple graph-theoretical rules by which coexistence networks are constructed. Examples are triangulations of cycles characterized by the golden ratio $\varphi = 1.6180...$, cycles with complete subnetworks, and non-Hamiltonian networks. In graph-theoretical terms, we extend the concept of a Pfaffian orientation from even-sized to odd-sized networks. Our results show that the topology of interaction networks alone can determine the long-time behavior of nonlinear dynamical systems, and may help to identify robust network motifs arising, for example, in ecology.

An $$O(|E(G)|^2)$$ O ( | E ( G ) | 2 ) algorithm for recognizing Pfaffian graphs of a type of bipartite graphs

A graph $$G=(V,E)$$ with even number vertices is called Pfaffian if it has a Pfaffian orientation, namely it admits an orientation such that the number of edges of any M-alternating cycle which have the same direction as the traversal direction is odd for some perfect matching M of the graph G. In this paper, we obtain a necessary and sufficient condition of Pfaffian graphs in a type of bipartite graphs. Then, we design an $$O(|E(G)|^2)$$ algorithm for recognizing Pfaffian graphs in this class and constructs a Pfaffian orientation if the graph is Pfaffian. The results improve and generalize some known results.

A Sufficient Condition for Pfaffian Graphs on the Torus

The significance of Pfaffian orientations stems from the fact that if a graph has such an orientation, then the number of perfect matchings (as known as the dimer problem) can be computed in polynomial time. A graph is called Pfaffian if it has a Pfaffian orientation. The Pfaffian property of a graph is not well-characterized. We do not even know whether the property of an undirected graph that it has a Pfaffian orientation is in NP. In this paper, we consider Pfaffian graphs on torus. More specific, we prove that for a non-bipartite graph embedding on the torus, if the boundaries of all faces are even, then it is Pfaffian.

Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

We study the ideals of the partition, Brauer, and Jones monoid, establishing various combinatorial results on generating sets and idempotent generating sets via an analysis of their Graham–Houghton graphs. We show that each proper ideal of the partition monoid Pn is an idempotent generated semigroup, and obtain a formula for the minimal number of elements (and the minimal number of idempotent elements) needed to generate these semigroups. In particular, we show that these two numbers, which are called the rank and idempotent rank (respectively) of the semigroup, are equal to each other, and we characterize the generating sets of this minimal cardinality. We also characterize and enumerate the minimal idempotent generating sets for the largest proper ideal of Pn, which coincides with the singular part of Pn. Analogous results are proved for the ideals of the Brauer and Jones monoids; in each case, the rank and idempotent rank turn out to be equal, and all the minimal generating sets are described. We also show how the rank and idempotent rank results obtained, when applied to the corresponding twisted semigroup algebras (the partition, Brauer, and Temperley–Lieb algebras), allow one to recover formulae for the dimensions of their cell modules (viewed as cellular algebras) which, in the semisimple case, are formulae for the dimensions of the irreducible representations of the algebras. As well as being of algebraic interest, our results relate to several well-studied topics in graph theory including the problem of counting perfect matchings (which relates to the problem of computing permanents of {0,1}-matrices and the theory of Pfaffian orientations), and the problem of finding factorizations of Johnson graphs. Our results also bring together several well-known number sequences such as Stirling, Bell, Catalan and Fibonacci numbers.

Counting the Number of Perfect Matchings in K 5-Free Graphs

Counting the number of perfect matchings in graphs is a computationally hard problem. However, in the case of planar graphs, and even for K3,3-free graphs, the number of perfect matchings can be computed efficiently. The technique to achieve this is to compute a Pfaffian orientation of a graph. In the case of K5-free graphs, this technique will not work because some K5-free graphs do not have a Pfaffian orientation. We circumvent this problem and show that the number of perfect matchings in K5-free graphs can be computed in polynomial time. We also parallelize the sequential algorithm and show that the problem is in TC2. We remark that our results generalize to graphs without singly-crossing minor.

Pfaffian orientations and perfect matchings of scale-free networks

Counting perfect matchings is of great interest in theoretical computer science, mathematics, among other disciplines. However, explicitly determining the number of perfect matchings of general graphs is a challenge. In this paper, we present a first analytical study of the enumeration problem of perfect matchings on scale-free networks, which are ubiquitous in real-life systems. We focus on two iteratively growing scale-free networks: one is fractal, while the other is non-fractal. For each network, we construct an orientation and prove that it is Pfaffian. Then, based on the connection between the number of perfect matchings and determinant of the skew adjacency matrix corresponding to a Pfaffian orientation, we derive the recursive expressions for the number of perfect matchings of the networks at two successive iterations. Furthermore, using the recursive relations, we show that for both scale-free networks, their entropy is zero, which is in sharp contrast to the previously obtained results for other networks without scale-free behavior, but having the same average degree as the studied networks, such as square lattice and honeycomb lattice, whose entropies are larger than zero. Our results indicate from another angle that scale-free networks have a distinct architecture in comparison with those lattices lacking the power-law property.

On the Permanental Polynomials of Matrices

An \(m\times n\,\{0,1\}\)-matrix \(A\) is said to be totally convertible if there exists a matrix \(B\) obtained from \(A\) by changing some 1’s in \(A\) to \(-1\)’s such that for any submatrix \(A^{\prime }\) of \(A\) of order \(m\), the corresponding submatrix \(B^{\prime }\) of \(B\) satisfies \(\mathrm{per}(xI-A^{\prime })=\det (xI-B^{\prime })\). In this paper, motivated by the well-known Polya’s problem, our object is to characterize those totally convertible matrices. Associate a matrix \(A\) with a bipartite graph \(G^{*}_A\). We first prove that a square matrix \(A\) is totally convertible if and only if \(G^{*}_A\) is Pfaffian, and then we generalize this result to an \(m\times n\)\(\{0,1\}\)-matrix. Moreover, the characterization of a totally convertible matrix provides an equivalent condition to compute the permanental polynomial of a bipartite graph by the characteristic polynomial of the skew adjacency matrix of its orientation graph. As applications, we give some explicit expressions of the permanental polynomials of two totally convertible matrices by the technique of Pfaffian orientation.

Oriented Euler complexes and signed perfect matchings

This paper presents "oriented pivoting systems" as an abstract framework for complementary pivoting. It gives a unified simple proof that the endpoints of complementary pivoting paths have opposite sign. A special case are the Nash equilibria of a bimatrix game at the ends of Lemke-Howson paths, which have opposite index. For Euler complexes or "oiks", an orientation is defined which extends the known concept of oriented abstract simplicial manifolds. Ordered "room partitions" for a family of oriented oiks come in pairs of opposite sign. For an oriented oik of even dimension, this sign property holds also for unordered room partitions. In the case of a two-dimensional oik, these are perfect matchings of an Euler graph, with the sign as defined for Pfaffian orientations of graphs. A near-linear time algorithm is given for the following problem: given a graph with an Eulerian orientation with a perfect matching, find another perfect matching of opposite sign. In contrast, the complementary pivoting algorithm for this problem may be exponential.

Pfaffian orientations for a type of bipartite graph

An even cycle C of a graph G is nice if the graph G−V(C) has a perfect matching. An orientation of G is a Pfaffian orientation if every nice cycle of G has an odd number of edges directed in either direction of the cycle. A graph is Pfaffian if it has a Pfaffian orientation. If a graph is Pfaffian, then the number of perfect matchings of it can be computed in polynomial time. In this paper, we focus on a special type of 1-extendable bipartite graph with maximum degree Δ(G)=|V(G)|/2. We characterize some properties of Pfaffian graphs in this type. According to the properties, we find an algorithm in time O(|E(G)|2) to determine whether a graph G in this type is Pfaffian or not. Furthermore, if G is Pfaffian, this algorithm also constructs a Pfaffian orientation of it.

Pfaffian graphs embedding on the torus

An orientation of a graph G with even number of vertices is Pfaffian if every even cycle C such that G − V (C) has a perfect matching has an odd number of edges directed in either direction of the cycle. The significance of Pfaffian orientations stems from the fact that if a graph G has one, then the number of perfect matchings of G can be computed in polynomial time. There is a classical result of Kasteleyn that every planar graph has a Pfaffian orientation. Little proved an elegant characterization of bipartite graphs that admit a Pfaffian orientation. Robertson, Seymour and Thomas (1999) gave a polynomial-time recognition algorithm to test whether a bipartite graph is Pfaffian by a structural description of bipartite graphs. In this paper, we consider the Pfaffian property of graphs embedding on the orientable surface with genus one (i.e., the torus). Some sufficient conditions for Pfaffian graphs on the torus are obtained. Furthermore, we show that all quadrilateral tilings on the torus are Pfaffian if and only if they are not bipartite graphs.

A generalization of Littleʼs Theorem on Pfaffian orientations

Little (1975) [12] showed that, in a certain sense, the only minimal non-Pfaffian bipartite matching covered graph is the brace K3,3. Using a stronger notion of minimality than the one used by Little, we show that every minimal non-Pfaffian brick G contains two disjoint odd cycles C1 and C2 such that the subgraph G−V(C1∪C2) has a perfect matching. This implies that the only minimal non-Pfaffian solid matching covered graph is the brace K3,3. (A matching covered graph G is solid if, for any two disjoint odd cycles C1 and C2 of G, the subgraph G−V(C1∪C2) has no perfect matching. Solid matching covered graphs constitute a natural generalization of the class of bipartite graphs, see Carvalho et al., 2004 [5].)

The Pfaffian property of Cartesian products of graphs

Suppose that G=(V,E) is a graph with even vertices. An even cycle C is a nice cycle of G if G?V(C) has a perfect matching. An orientation of G is a Pfaffian orientation if each nice cycle C has an odd number of edges directed in either direction of the cycle. Let P n and C n denote the path and the cycle on n vertices, respectively. In this paper, we characterize the Pfaffian property of Cartesian products G×P 2n and G×C 2n for any graph G in terms of forbidden subgraphs of G. This extends the results in (Yan and Zhang in Discrete Appl Math 154:145---157, 2006).

Computing the permanental polynomials of bipartite graphs by Pfaffian orientation

The permanental polynomial of a graph G is π(G,x)≜per(xI−A(G)). From the result that a bipartite graph G admits an orientation Ge such that every cycle is oddly oriented if and only if it contains no even subdivision of K2,3, Yan and Zhang showed that the permanental polynomial of such a bipartite graph G can be expressed as the characteristic polynomial of the skew adjacency matrix A(Ge). In this note we first prove that this equality holds only if the bipartite graph G contains no even subdivision of K2,3. Then we prove that such bipartite graphs are planar. Unexpectedly, we mainly show that a 2-connected bipartite graph contains no even subdivision of K2,3 if and only if it is planar 1-cycle resonant. This implies that each cycle is oddly oriented in any Pfaffian orientation of a 2-connected bipartite graph containing no even subdivision of K2,3. Accordingly, we give a way to compute the permanental polynomials of such graphs by Pfaffian orientation.

Dimer statistics of honeycomb lattices on Klein bottle, Möbius strip and cylinder

Dimer statistics is a central problem in statistical physics. In this paper the enumerations of close-packed dimers of honeycomb lattices on Klein bottle, Möbius strip and cylinder are considered. By establishing a Pfaffian orientation or a crossing orientation, and then computing the determinants of the skew-symmetric matrices of the resulting orientation graphs, we obtain explicit expressions of the number of close-packed dimers of the Klein-bottle polyhex, the Möbius polyhex and the cylindrical polyhex.

Geometric representations of binary codes and computation of weight enumerators

For every linear binary code C , we construct a geometric triangular configuration Δ so that the weight enumerator of C is obtained by a simple formula from the weight enumerator of the cycle space of Δ. The triangular configuration Δ thus provides a geometric representation of C which carries its weight enumerator. This is the first step in the suggestion by M. Loebl, to extend the theory of Pfaffian orientations from graphs to general linear binary codes. Then we carry out also the second step by constructing, for every triangular configuration Δ, a triangular configuration Δ ′ and a bijection between the cycle space of Δ and the set of the perfect matchings of Δ ′ .

Pfaffian Orientation and Enumeration of Perfect Matchings for some Cartesian Products of Graphs

The importance of Pfaffian orientations stems from the fact that if a graph $G$ is Pfaffian, then the number of perfect matchings of $G$ (as well as other related problems) can be computed in polynomial time. Although there are many equivalent conditions for the existence of a Pfaffian orientation of a graph, this property is not well-characterized. The problem is that no polynomial algorithm is known for checking whether or not a given orientation of a graph is Pfaffian. Similarly, we do not know whether this property of an undirected graph that it has a Pfaffian orientation is in NP. It is well known that the enumeration problem of perfect matchings for general graphs is NP-hard. L. Lovász pointed out that it makes sense not only to seek good upper and lower bounds of the number of perfect matchings for general graphs, but also to seek special classes for which the problem can be solved exactly. For a simple graph $G$ and a cycle $C_n$ with $n$ vertices (or a path $P_n$ with $n$ vertices), we define $C_n$ (or $P_n)\times G$ as the Cartesian product of graphs $C_n$ (or $P_n$) and $G$. In the present paper, we construct Pfaffian orientations of graphs $C_4\times G$, $P_4\times G$ and $P_3\times G$, where $G$ is a non bipartite graph with a unique cycle, and obtain the explicit formulas in terms of eigenvalues of the skew adjacency matrix of $\overrightarrow{G}$ to enumerate their perfect matchings by Pfaffian approach, where $\overrightarrow{G}$ is an arbitrary orientation of $G$.