On the non-ergodicity of the Swendsen-Wang-Kotecky algorithm on the kagome lattice

We study the properties of the Wang-Swendsen-Kotecky cluster Monte Carlo algorithm for simulating the 3-state kagome-lattice Potts antiferromagnet at zero temperature. We prove that this algorithm is not ergodic for symmetric subsets of the kagome lattice with fully periodic boundary conditions: given an initial configuration, not all configurations are accessible via Monte Carlo steps. The same conclusion holds for single-site dynamics.Comment: Latex2e. 22 pages. Contains 11 figures using pstricks package. Uses iopart.sty. Final version accepted in journa

Arrow ribbon graphs

We introduce an additional structure on ribbon graphs, arrow structure. We extend the Bollob\'as-Riordan polynomial to ribbon graph with this structure. The extended polynomial satisfies the contraction-deletion relations and naturally behaves with respect to the partial duality of ribbon graphs. We construct an arrow ribbon graph from a virtual link whose extended Bollob\'as-Riordan polynomial specializes to the arrow polynomial of the virtual link recently introduced by H.Dye and L.Kauffman. This result generalizes the classical Thistlethwaite theorem to the arrow polynomial of virtual links.Comment: to appear in Journal of Knot Theory and Its Ramification

Spectral partitions on infinite graphs

Statistical models on infinite graphs may exhibit inhomogeneous thermodynamic behaviour at macroscopic scales. This phenomenon is of geometrical origin and may be properly described in terms of spectral partitions into subgraphs with well defined spectral dimensions and spectral weights. These subgraphs are shown to be thermodynamically homogeneous and effectively decoupled.Comment: 8 pages, to appear on Journal of Physics

The Complexity of Separating Points in the Plane

We study the following separation problem: given n connected curves and two points s and t in the plane, compute the minimum number of curves one needs to retain so that any path connecting s to t intersects some of the retained curves. We give the first polynomial (O(n3)) time algorithm for the problem, assuming that the curves have reasonable computational properties. The algorithm is based on considering the intersection graph of the curves, defining an appropriate family of closed walks in the intersection graph that satisfies the 3-path-condition, and arguing that a shortest cycle in the family gives an optimal solution. The 3-path-condition has been used mainly in topological graph theory, and thus its use here makes the connection to topology clear. We also show that the generalized version, where several input points are to be separated, is NP-hard for natural families of curves, like segments in two directions or unit circles

On the degree conjecture for separability of multipartite quantum states

We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein {\it et al.} [Phys. Rev. A \textbf{73}, 012320 (2006)]. The conjecture states that a multipartite quantum state is separable if and only if the degree matrix of the graph associated with the state is equal to the degree matrix of the partial transpose of this graph. We call this statement to be the strong form of the conjecture. In its weak version, the conjecture requires only the necessity, that is, if the state is separable, the corresponding degree matrices match. We prove the strong form of the conjecture for {\it pure} multipartite quantum states, using the modified tensor product of graphs defined in [J. Phys. A: Math. Theor. \textbf{40}, 10251 (2007)], as both necessary and sufficient condition for separability. Based on this proof, we give a polynomial-time algorithm for completely factorizing any pure multipartite quantum state. By polynomial-time algorithm we mean that the execution time of this algorithm increases as a polynomial in $m,$ where $m$ is the number of parts of the quantum system. We give a counter-example to show that the conjecture fails, in general, even in its weak form, for multipartite mixed states. Finally, we prove this conjecture, in its weak form, for a class of multipartite mixed states, giving only a necessary condition for separability.Comment: 17 pages, 3 figures. Comments are welcom

Analysis of Nonlinear Synchronization Dynamics of Oscillator Networks by Laplacian Spectral Methods

We analyze the synchronization dynamics of phase oscillators far from the synchronization manifold, including the onset of synchronization on scale-free networks with low and high clustering coefficients. We use normal coordinates and corresponding time-averaged velocities derived from the Laplacian matrix, which reflects the network's topology. In terms of these coordinates, synchronization manifests itself as a contraction of the dynamics onto progressively lower-dimensional submanifolds of phase space spanned by Laplacian eigenvectors with lower eigenvalues. Differences between high and low clustering networks can be correlated with features of the Laplacian spectrum. For example, the inhibition of full synchoronization at high clustering is associated with a group of low-lying modes that fail to lock even at strong coupling, while the advanced partial synchronizationat low coupling noted elsewhere is associated with high-eigenvalue modes.Comment: Revised version: References added, introduction rewritten, additional minor changes for clarit

Local chromatic number of quadrangulations of surfaces

The local chromatic number of a graph G, as introduced in [4], is the minimum integer k such that G admits a proper coloring (with an arbitrary number of colors) in which the neighborhood of each vertex uses less than k colors. In [17] a connection of the local chromatic number to topological properties of (a box complex of) the graph was established and in [18] it was shown that a topological condition implying the usual chromatic number being at least four has the stronger consequence that the local chromatic number is also at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs’ result to arbitrary non-orientable surfaces replacing the condition of the graph being not bipartite by a more technical condition of an odd quadrangulation. This paper investigates when these general results are true for the local chromatic number instead of the chromatic number. Surprisingly, we find out that (unlike in the case of the chromatic number) this depends on the genus of the surface. For the non-orientable surfaces of genus at most four, the local chromatic number of any odd quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5 or higher. We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for the usual chromatic number

Drawing bobbin lace graphs, or, Fundamental cycles for a subclass of periodic graphs

In this paper, we study a class of graph drawings that arise from bobbin lace patterns. The drawings are periodic and require a combinatorial embedding with specific properties which we outline and demonstrate can be verified in linear time. In addition, a lace graph drawing has a topological requirement: it contains a set of non-contractible directed cycles which must be homotopic to $(1,0)$, that is, when drawn on a torus, each cycle wraps once around the minor meridian axis and zero times around the major longitude axis. We provide an algorithm for finding the two fundamental cycles of a canonical rectangular schema in a supergraph that enforces this topological constraint. The polygonal schema is then used to produce a straight-line drawing of the lace graph inside a rectangular frame. We argue that such a polygonal schema always exists for combinatorial embeddings satisfying the conditions of bobbin lace patterns, and that we can therefore create a pattern, given a graph with a fixed combinatorial embedding of genus one.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Diffusion dynamics on multiplex networks

We study the time scales associated to diffusion processes that take place on multiplex networks, i.e. on a set of networks linked through interconnected layers. To this end, we propose the construction of a supra-Laplacian matrix, which consists of a dimensional lifting of the Laplacian matrix of each layer of the multiplex network. We use perturbative analysis to reveal analytically the structure of eigenvectors and eigenvalues of the complete network in terms of the spectral properties of the individual layers. The spectrum of the supra-Laplacian allows us to understand the physics of diffusion-like processes on top of multiplex networks.Comment: 6 Pages including supplemental material. To appear in Physical Review Letter