Locally finite graphs with ends: a topological approach

This paper is intended as an introductory survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. This approach has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. The shift of paradigm it proposes is thus as much an answer to old questions as a source of new ones; many concrete problems of both types are suggested in the paper. This paper attempts to provide an entry point to this field for readers that have not followed the literature that has emerged in the last 10 years or so. It takes them on a quick route through what appear to be the most important lasting results, introduces them to key proof techniques, identifies the most promising open problems, and offers pointers to the literature for more detail.Comment: Introductory survey. This post-publication update is the result of a thorough revision undertaken when I lectured on this material in 2012. The emphasis was on correcting errors and, occasionally, improving the presentation. I did not attempt to bring the material as such up to the current level of knowledg

Toroidal Dimer Model and Temperley's Bijection

Temperley's bijection relates the toroidal dimer model to cycle rooted spanning forests ($CRSF$) on the torus. The height function of the dimer model and the homology class of $CRSF$ are naturally related. When the size of the torus tends to infinity, we show that the measure on $CRSF$ arising from the dimer model converges to a measure on (disconnected) spanning forests or spanning trees. There is a phase transition, which is determined by the average height change

Algorithmic releases on spanning trees of Jahangir graphs

In this paper algebraic and combinatorial properties and a computation of the number of the spanning trees are developed for a Jahangir graph.Comment: 17 pages, 4 figure

Learning Chordal Markov Networks by Constraint Satisfaction

We investigate the problem of learning the structure of a Markov network from data. It is shown that the structure of such networks can be described in terms of constraints which enables the use of existing solver technology with optimization capabilities to compute optimal networks starting from initial scores computed from the data. To achieve efficient encodings, we develop a novel characterization of Markov network structure using a balancing condition on the separators between cliques forming the network. The resulting translations into propositional satisfiability and its extensions such as maximum satisfiability, satisfiability modulo theories, and answer set programming, enable us to prove optimal certain network structures which have been previously found by stochastic search

Spanning Trees and Mahler Measure

The complexity of a finite connected graph is its number of spanning trees; for a non-connected graph it is the product of complexities of its connected components. If $G$ is an infinite graph with cofinite free ${\mathbb Z}^d$-symmetry, then the logarithmic Mahler measure $m(\Delta)$ of its Laplacian polynomial $\Delta$ is the exponential growth rate of the complexity of finite quotients of $G$. It is bounded below by $m(\Delta({\mathbb G}_d))$, where ${\mathbb G}_d$ is the grid graph of dimension $d$. The growth rates $m(\Delta({\mathbb G}_d))$ are asymptotic to $\log 2d$ as $d$ tends to infinity. If $m(\Delta(G))\ne 0$, then $m(\Delta(G)) \ge \log 2$. An application to determinant growth rates of families of alternating links arising from planar graphs is given.Comment: 12 pages, 1 figur

Non-amenable products are not treeable

Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism group of Y has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct product X times Y. This implies that the minimal spanning forest corresponding to i.i.d. edge-weights in such a product, has infinitely many connected components almost surely.Comment: 8 page

Some physical and chemical indices of clique-inserted-lattices

The operation of replacing every vertex of an $r$-regular lattice $H$ by a complete graph of order $r$ is called clique-inserting, and the resulting lattice is called the clique-inserted-lattice of $H$. For any given $r$-regular lattice, applying this operation iteratively, an infinite family of $r$-regular lattices is generated. Some interesting lattices including the 3-12-12 lattice can be constructed this way. In this paper, we reveal the relationship between the energy and resistance distance of an $r$-regular lattice and that of its clique-inserted-lattice. As an application, the asymptotic energy per vertex and average resistance distance of the 3-12-12 and 3-6-24 lattices are computed. We also give formulae expressing the numbers of spanning trees and dimers of the $k$-th iterated clique-inserted lattices in terms of that of the original lattice. Moreover, we show that new families of expander graphs can be constructed from the known ones by clique-inserting

Combinatorial Aspects of Elliptic Curves II: Relationship between Elliptic Curves and Chip-Firing Games on Graphs

Let q be a power of a prime and E be an elliptic curve defined over F_q. In "Combinatorial aspects of elliptic curves" [17], the present author examined a sequence of polynomials which express the N_k's, the number of points on E over the field extensions F_{q^k}, in terms of the parameters q and N_1 = #E(F_q). These polynomials have integral coefficients which alternate in sign, and a combinatorial interpretation in terms of spanning trees of wheel graphs. In this sequel, we explore further ramifications of this connection. In particular, we highlight a relationship between elliptic curves and chip-firing games on graphs by comparing the groups structures of both. As a coda, we construct a cyclic rational language whose zeta function is dual to that of an elliptic curve.Comment: 24 pages, 2 figures, part of author's Ph.D. Thesis, presented at FPSAC 200

Limiting entropy of determinantal processes

We extend Lyons's tree entropy theorem to general determinantal measures. As a byproduct we show that the sofic entropy of an invariant determinantal measure does not depend on the chosen sofic approximation

Laplacian Simplices II: A Coding Theoretic Approach

This paper further investigates \emph{Laplacian simplices}. A construction by Braun and the first author associates to a simple connected graph $G$ a simplex \cP_G whose vertices are the rows of the Laplacian matrix of $G$. In this paper we associate to a reflexive \cP_G a duality-preserving linear code \cC(\cP_G). This new perspective allows us to build upon previous results relating graphical properties of $G$ to properties of the polytope \cP_G. In particular, we make progress towards a graphical characterization of reflexive \cP_G using techniques from Ehrhart theory. We provide a systematic investigation of \cC(\cP_G) for cycles, complete graphs, and graphs with a prime number of vertices. We construct an asymptotically good family of MDS codes. In addition, we show that any rational rate is achievable by such construction