Related papers: Dimer-monomer model on the generalized Tower of Ha…
The number of dimer-monomers (matchings) of a graph $G$ is an important graph parameter in statistical physics. Following recent research, we study the asymptotic behavior of the number of dimer-monomers $m(G)$ on the Towers of Hanoi graphs…
The mapping spaces of the $r$-truncated versions of the $E_n$ operads appear as the $r$-th stage of the Taylor tower for long embedding spaces. It has been shown that their rational homotopy groups can be expressed through graph homology in…
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…
We study the richness of the ensemble of graphical structures (i.e., unlabeled graphs) of the one-dimensional random geometric graph model defined by $n$ nodes randomly scattered in $[0,1]$ that connect if they are within the connection…
The number of spanning trees of a graph is an important invariant related to topological and dynamic properties of the graph, such as its reliability, communication aspects, synchronization, and so on. However, the practical enumeration of…
Let $\mathcal{H}$ be a hypergraph with $n$ vertices. Suppose that $d_1,d_2,\ldots,d_n$ are degrees of the vertices of $\mathcal{H}$. The $t$-th graph entropy based on degrees of $\mathcal{H}$ is defined as $$ I_d^t(\mathcal{H})…
A numerical parameter, referred to as a topological index, is used to represent the molecular structure of a compound by analyzing its graph-theoretical characteristics. Topological indices are predictive methods for the physicochemical…
We consider the monomer-dimer model on sequences of random graphs locally convergent to trees. We prove that the monomer density converges almost surely, in the thermodynamic limit, to an analytic function of the monomer activity. We…
By using the transfer matrix approach, we investigate the asymptotic behavior of the entropy of flexible chains with $M$ monomers each placed on stripes. In the limit of high density of monomers, we study the behavior of the entropy as a…
In this paper, we study the entropy of a hard random geometric graph (RGG), a commonly used model for spatial networks, where the connectivity is governed by the distances between the nodes. Formally, given a connection range $r$, a hard…
The famous Tower of Hanoi puzzle involves moving $n$ discs of distinct sizes from one of $p\geq 3$ pegs (traditionally $p=3$) to another of the pegs, subject to the constraints that only one disc may be moved at a time, and no disc can ever…
Considering the symmetries and self similarity properties of the corresponding labeled graphs, it is shown that the minimal number of moves in the Tower of Hanoi game with $p =4$ pegs and $n \geq p$ disks satisfies the recursive formula $…
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…
In this paper we review the asymptotic matching conjectures for $r$-regular bipartite graphs, and their connections in estimating the monomer-dimer entropies in $d$-dimensional integer lattice and Bethe lattices. We prove new rigorous upper…
We show how the theory of the critical behaviour of $d$-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain…
A characterization of topological order in terms of bi-partite entanglement was proposed recently [A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006); M. Levin and X.-G. Wen, ibid, 110405]. It was argued that in a topological…
We outline the most recent theory for the computation of the exponential growth rate of the number of configurations on a multi-dimensional grid. As an application we compute the monomer-dimer constant for the 2-dimensional grid to 8…
We normalize the combinatorial Laplacian of a graph by the degree sum, look at its eigenvalues as a probability distribution and then study its Shannon entropy. Equivalently, we represent a graph with a quantum mechanical state and study…
The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition $h_*$ for the topological entropy of $T$. We prove that $h_*$ is not smaller than the…
Finite dimensional representations of the Hanoi Towers group are used to calculate the spectra of the finite graphs associated to the Hanoi Towers Game on three pegs (the group serves as a renorm group for the game). These graphs are…