Results and open problems in matchings in regular graphs

This survey paper deals with upper and lower bounds on the number of $k$-matchings in regular graphs on $N$ vertices. For the upper bounds we recall the upper matching conjecture which is known to hold for perfect matchings. For the lower bounds we first survey the known results for bipartite graphs, and their continuous versions as the van der Waerden and Tverberg permanent conjectures and its variants. We then discuss non-bipartite graphs. Little is known beyond the recent proof of the Lov\'asz-Plummer conjecture on the exponential growth of perfect matchings in cubic bridgeless graphs. We discuss the problem of the minimum of haffnians on the convex set of matrices, whose extreme points are the adjacency matrices of subgraphs of the complete graph corresponding to perfect matchings. We also consider infinite regular graphs. The analog of $k$-matching is the $p$-monomer entropy, where $p\in [0,1]$ is the density of the number of matchings.Comment: 15 pages, 3 figure

Entropy of Symmetric Graphs

A graph $G$ is called \emph{symmetric with respect to a functional $F_G(P)$} defined on the set of all the probability distributions on its vertex set if the distribution $P^*$ maximizing $F_G(P)$ is uniform on $V(G)$. Using the combinatorial definition of the entropy of a graph in terms of its vertex packing polytope and the relationship between the graph entropy and fractional chromatic number, we prove that vertex transitive graphs are symmetric with respect to graph entropy. As the main result of this paper, we prove that a perfect graph is symmetric with respect to graph entropy if and only if its vertices can be covered by disjoint copies of its maximum-size clique. Particularly, this means that a bipartite graph is symmetric with respect to graph entropy if and only if it has a perfect matching.Comment: 20 pages, 3 figure

LDPC codes constructed from cubic symmetric graphs

Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction of LDPC codes from cubic symmetric graphs. The constructed codes are $(3,3)$-regular and the vast majority of the corresponding Tanner graphs have girth greater than four. We analyse properties of the obtained codes and present bounds for the code parameters, the dimension and the minimum distance. Furthermore, we give an expression for the variance of the syndrome weight of the constructed codes. Information on the LDPC codes constructed from bipartite cubic symmetric graphs with less than 200 vertices is presented as well. Some of the constructed codes are optimal, and some have an additional property of being self-orthogonal or linear codes with complementary dual (LCD codes).Comment: 17 page

A positivity property of the dimer entropy of graphs

The entropy of a monomer-dimer system on an infinite bipartite lattice can be written as a mean-field part plus a series expansion in the dimer density. In a previous paper it has been conjectured that all coefficients of this series are positive. Analogously on a connected regular graph with $v$ vertices, the "entropy" of the graph ${\rm ln} N(i)/v$, where $N(i)$ is the number of ways of setting down $i$ dimers on the graph, can be written as a part depending only on the number of the dimer configurations over the completed graph plus a Newton series in the dimer density on the graph. In this paper, we investigate for which connected regular graphs all the coefficients of the Newton series are positive (for short, these graphs will be called positive). In the class of connected regular bipartite graphs, up to $v=20$, the only non positive graphs have vertices of degree $3$. From $v=14$ to $v=30$, the frequency of the positivity violations in the $3$-regular graphs decreases with increasing $v$. In the case of connected $4$-regular bipartite graphs, the first violations occur in two out of the $2806490$ graphs with $v=22$. We conjecture that for each degree $r$ the frequency of the violations, in the class of the $r-$regular bipartite graphs, goes to zero as $v$ tends to infinity. This graph-positivity property can be extended to non-regular or non-bipartite graphs. We have examined a large number of rectangular grids of size $N_x \times N_y$ both with open and periodic boundary conditions. We have observed positivity violations only for $min(N_x, N_y) = 3$ or $4$.Comment: 13 pages, 1 figur

Agglomerative Percolation on Bipartite Networks: A Novel Type of Spontaneous Symmetry Breaking

Ordinary bond percolation (OP) can be viewed as a process where clusters grow by joining them pairwise, by adding links chosen randomly one by one from a set of predefined `virtual' links. In contrast, in agglomerative percolation (AP) clusters grow by choosing randomly a `target cluster' and joining it with all its neighbors, as defined by the same set of virtual links. Previous studies showed that AP is in different universality classes from OP for several types of (virtual) networks (linear chains, trees, Erdos-Renyi networks), but most surprising were the results for 2-d lattices: While AP on the triangular lattice was found to be in the OP universality class, it behaved completely differently on the square lattice. In the present paper we explain this striking violation of universality by invoking bipartivity. While the square lattice is a bipartite graph, the triangular lattice is not. In conformity with this we show that AP on the honeycomb and simple cubic (3-d) lattices -- both of which are bipartite -- are also not in the OP universality classes. More precisely, we claim that this violation of universality is basically due to a Z_2 symmetry that is spontaneously broken at the percolation threshold. We also discuss AP on bipartite random networks and suitable generalizations of AP on k-partite graphs.Comment: 11 pages, including 18 figure

Higher order expansions for the entropy of a dimer or a monomer-dimer system on d-dimensional lattices

Recently an expansion as a power series in 1/d has been presented for the specific entropy of a complete dimer covering of a d-dimensional hypercubic lattice. This paper extends from 3 to 10 the number of terms known in the series. Likewise an expansion for the entropy, dependent on the dimer-density p, of a monomer-dimer system, involving a sum sum_k a_k(d) p^k, has been recently offered. We herein extend the number of the known expansion coefficients from 6 to 20 for the hyper-cubic lattices of general dimension d and from 6 to 24 for the hyper-cubic lattices of dimensions d < 5 . We show that this extension can lead to accurate numerical estimates of the p-dependent entropy for lattices with dimension d > 2. The computations of this paper have led us to make the following marvelous conjecture: "In the case of the hyper-cubic lattices, all the expansion coefficients, a_k(d), are positive"! This paper results from a simple melding of two disparate research programs: one computing to high orders the Mayer series coefficients of a dimer gas, the other studying the development of entropy from these coefficients. An effort is made to make this paper self-contained by including a review of the earlier works.Comment: 17 pages, no figure

Positivity of the virial coefficients in lattice dimer models and upper bounds on the number of matchings on graphs

Using a relation between the virial expansion coefficients of the pressure and the entropy expansion coefficients in the case of the monomer-dimer model on infinite regular lattices, we have shown that, on hypercubic lattices of any dimension, the virial coefficients are positive through the 20th order. We have observed that all virial coefficients so far known for this system are positive also on infinite regular lattices with different structure. We are thus led to conjecture that the virial expansion coefficients $m_k$ are always positive. These considerations can be extended to the study of related bounds on finite graphs generalizing the infinite regular lattices, namely the finite grids and the regular biconnected graphs. The validity of the bounds $\Delta^k {\rm ln}(i! N(i)) \le 0$ for $k \ge 2$, where $N(i)$ is the number of configurations of $i$ dimers on the graph and $\Delta$ is the forward difference operator, is shown to correspond to the positivity of the virial coefficients. Our tests on many finite lattice graphs indicate that on large lattices these bounds are satisfied, giving support to the conjecture on the positivity of the virial coefficients. An exhaustive survey of some classes of regular biconnected graphs with a not too large number $v$ of vertices shows only few violations of these bounds. We conjecture that the frequency of the violations vanishes as $v \to \infty$. We find rigorous upper bounds on $N(i)$ valid for arbitrary graphs and for regular graphs. The similarity between the Heilman-Lieb inequality and the one conjectured above suggests that one study the stricter inequality $m_k \ge \frac{1}{2k}$ for the virial coefficients, which is valid for all the known coefficients of the infinite regular lattice models.Comment: 22 pages, 1 figur

Iterated altans and their properties

Recently a class of molecular graphs, called altans, became a focus of attention of several theoretical chemists and mathematicians. In this paper we study primary iterated altans and show, among other things, their connections with nanotubes and nanocaps. The question of classification of bipartite altans is also addressed. Using the results of Gutman we are able to enumerate Kekul\'e structures of several nanocaps of arbitrary length.Comment: 13 pages, 12 figure

MAP Estimation, Message Passing, and Perfect Graphs

Efficiently finding the maximum a posteriori (MAP) configuration of a graphical model is an important problem which is often implemented using message passing algorithms. The optimality of such algorithms is only well established for singly-connected graphs and other limited settings. This article extends the set of graphs where MAP estimation is in P and where message passing recovers the exact solution to so-called perfect graphs. This result leverages recent progress in defining perfect graphs (the strong perfect graph theorem), linear programming relaxations of MAP estimation and recent convergent message passing schemes. The article converts graphical models into nand Markov random fields which are straightforward to relax into linear programs. Therein, integrality can be established in general by testing for graph perfection. This perfection test is performed efficiently using a polynomial time algorithm. Alternatively, known decomposition tools from perfect graph theory may be used to prove perfection for certain families of graphs. Thus, a general graph framework is provided for determining when MAP estimation in any graphical model is in P, has an integral linear programming relaxation and is exactly recoverable by message passing.Comment: Appears in Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence (UAI2009

Cubical coloring -- fractional covering by cuts and semidefinite programming

We introduce a new graph invariant that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that circular cliques play for the circular chromatic number. The fact that the defined parameter attains on these graphs the `correct' value suggests that the definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engstr\"om, F\"arnqvist, Jonsson, and Thapper. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246--265]).Comment: 17 page