Genus Ranges of Chord Diagrams

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary circle at the ends of each chord. Variations of this construction are considered here, where bands are possibly attached to the outer boundary circle of the annulus. The genus range of a chord diagram is the genus values over all such variations of surfaces thus obtained from a given chord diagram. Genus ranges of chord diagrams for a fixed number of chords are studied. Integer intervals that can, and cannot, be realized as genus ranges are investigated. Computer calculations are presented, and play a key role in discovering and proving the properties of genus ranges.Comment: 12 pages, 8 figure

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

Unboundedness of adjacency matrices of locally finite graphs

Given a locally finite simple graph so that its degree is not bounded, every self-adjoint realization of the adjacency matrix is unbounded from above. In this note we give an optimal condition to ensure it is also unbounded from below. We also consider the case of weighted graphs. We discuss the question of self-adjoint extensions and prove an optimal criterium.Comment: Typos corrected. Examples added. Cute drawings. Simplification of the main condition. Case of the weight tending to zero more discussed

Blocking nonorientability of a surface

AbstractLet S be a nonorientable surface. A collection of pairwise noncrossing simple closed curves in S is a blockage if every one-sided simple closed curve in S crosses at least one of them. Robertson and Thomas [9] conjectured that the orientable genus of any graph G embedded in S with sufficiently large face-width is “roughly” equal to one-half of the minimum number of intersections of a blockage with the graph. The conjecture was disproved by Mohar (Discrete Math. 182 (1998) 245) and replaced by a similar one. In this paper, it is proved that the conjectures in Mohar (1998) and Robertson and Thomas (J. Graph Theory 15 (1991) 407) hold up to a constant error term: For any graph G embedded in S, the orientable genus of G differs from the conjectured value at most by O(g2), where g is the genus of S

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

The problem of deficiency indices for discrete Schr\"odinger operators on locally finite graphs

The number of self-adjoint extensions of a symmetric operator acting on a complex Hilbert space is characterized by its deficiency indices. Given a locally finite unoriented simple tree, we prove that the deficiency indices of any discrete Schr\"odinger operator are either null or infinite. We also prove that almost surely, there is a tree such that all discrete Schr\"odinger operators are essentially self-adjoint. Furthermore, we provide several criteria of essential self-adjointness. We also adress some importance to the case of the adjacency matrix and conjecture that, given a locally finite unoriented simple graph, its the deficiency indices are either null or infinite. Besides that, we consider some generalizations of trees and weighted graphs.Comment: Typos corrected. References and ToC added. Paper slightly reorganized. Section 3.2, about the diagonalization has been much improved. The older section about the stability of the deficiency indices in now in appendix. To appear in Journal of Mathematical Physic

1-factorisation of the Composition of Regular Graphs

1-factorability of the composition of graphs is studied. The followings sufficient conditions are proved: $G[H]$ is 1-factorable if $G$ and $H$ are regular and at least one of the following holds: (i) Graphs $G$ and $H$ both contain a 1-factor, (ii) $G$ is 1-factorable (iii) $H$ is 1-factorable. It is also shown that the tensor product $G\otimes H$ is 1-factorable, if at least one of two graphs is 1-factorable. This result in turn implies that the strong tensor product $G\otimes' H$ is 1-factorable, if $G$ is 1-factorable

Asymptotic behavior of the number of Eulerian orientations of graphs

We consider the class of simple graphs with large algebraic connectivity (the second-smallest eigenvalue of the Laplacian matrix). For this class of graphs we determine the asymptotic behavior of the number of Eulerian orientations. In addition, we establish some new properties of the Laplacian matrix, as well as an estimate of a conditionality of matrices with the asymptotic diagonal predominanceComment: arXiv admin note: text overlap with arXiv:1104.304

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