Polyhedra of small order and their Hamiltonian properties

We describe the results of an enumeration of several classes of polyhedra. The enumerated classes include polyhedra with up to 12 vertices and up to 26 edges, simplical polyhedra with up to 16 vertices, 4-connected polyhedra with up to 15 vertices, and bipartite polyhedra with up to 22 vertices.The results of the enumeration were used to systematically search for certain minimal non-Hamiltonian polyhedra. In particular, the smallest polyhedra satisfying certain toughness-like properties are presented here, as are the smallest non-Hamiltonian, 3-connected, Delaunay tessellations and triangulations. Improved upper and lower bounds on the size of the smallest non-Hamiltonian, inscribable polyhedra are also given

On Some Combinatorial Problems in Cographs

The family of graphs that can be constructed from isolated vertices by disjoint union and graph join operations are called cographs. These graphs can be represented in a tree-like representation termed parse tree or cotree. In this paper, we study some popular combinatorial problems restricted to cographs. We first present a structural characterization of minimal vertex separators in cographs. Further, we show that listing all minimal vertex separators and the complexity of some constrained vertex separators are polynomial-time solvable in cographs. We propose polynomial-time algorithms for connectivity augmentation problems and its variants in cographs, preserving the cograph property. Finally, using the dynamic programming paradigm, we present a generic framework to solve classical optimization problems such as the longest path, the Steiner path and the minimum leaf spanning tree problems restricted to cographs, our framework yields polynomial-time algorithms for all three problems.Comment: 21 pages, 4 figure

Sparse Stable Matrices

In the design of decentralized networked systems, it is useful to know whether a given network topology can sustain stable dynamics. We consider a basic version of this problem here: given a vector space of sparse real matrices, does it contain a stable (Hurwitz) matrix? Said differently, is a feedback channel (corresponding to a non-zero entry) necessary for stabilization or can it be done without. We provide in this paper a set of necessary and a set of sufficient conditions for the existence of stable matrices in a vector space of sparse matrices. We further prove some properties of the set of sparse matrix spaces that contain Hurwitz matrices. The conditions we exhibit are most easily stated in the language of graph theory, which we thus adopt in this paper.Comment: 19 page

A Note on Using the Resistance-Distance Matrix to solve Hamiltonian Cycle Problem

An instance of Hamiltonian cycle problem can be solved by converting it to an instance of Travelling salesman problem, assigning any choice of weights to edges of the underlying graph. In this note we demonstrate that, for difficult instances, choosing the edge weights to be the resistance distance between its two incident vertices is often a good choice. We also demonstrate that arguably stronger performance arises from using the inverse of the resistance distance. Examples are provided demonstrating benefits gained from these choices

Circle actions on symplectic four-manifolds

We complete the classification of Hamiltonian torus and circle actions on symplectic four-dimensional manifolds. Following work of Delzant and Karshon, Hamiltonian circle and 2-torus actions on any fixed simply connected symplectic four-manifold were characterized by Karshon, Kessler and Pinsonnault. What remains is to study the case of Hamiltonian actions on blowups of S^2-bundles over a Riemann surface of positive genus. These do not admit 2-torus actions. In this paper, we characterize Hamiltonian circle actions on them. We then derive combinatorial results on the existence and counting of these actions. As a by-product, we provide an algorithm that determines the g-reduced form of a blowup form. Our work is a combination of "soft" equivariant and combinatorial techniques, using the momentum map and related data, with "hard" holomorphic techniques, including Gromov-Witten invariants.Comment: 24 pages, 8 figures; two appendices, one of which is authored by Tair Pnini; in version 3, the definition of blowup form is adjuste

Hydras: Directed Hypergraphs and Horn Formulas

We introduce a new graph parameter, the hydra number, arising from the minimization problem for Horn formulas in propositional logic. The hydra number of a graph $G=(V,E)$ is the minimal number of hyperarcs of the form $u,v\rightarrow w$ required in a directed hypergraph $H=(V,F)$, such that for every pair $(u, v)$, the set of vertices reachable in $H$ from $\{u, v\}$ is the entire vertex set $V$ if $(u, v) \in E$, and it is $\{u, v\}$ otherwise. Here reachability is defined by forward chaining, a standard marking algorithm. Various bounds are given for the hydra number. We show that the hydra number of a graph can be upper bounded by the number of edges plus the path cover number of the line graph of a spanning subgraph, which is a sharp bound in several cases. On the other hand, we construct single-headed graphs for which that bound is off by a constant factor. Furthermore, we characterize trees with low hydra number, and give a lower bound for the hydra number of trees based on the number of vertices that are leaves in the tree obtained from $T$ by deleting its leaves. This bound is sharp for some families of trees. We give bounds for the hydra number of complete binary trees and also discuss a related minimization problem.Comment: 17 pages, 4 figure

Cycles and paths in Jacobson graphs

All finite Jacobson graphs with a Hamiltonian cycle or path, or Eulerian tour or trail are determined, and it is shown that a finite Jacobson graph is Hamiltonian if and only if it is pancyclic. Also, the length of the longest induced cycles and paths in finite Jacobson graphs are obtained.Comment: 11 pages, 10 figure

A computer-assisted proof of Barnette-Goodey conjecture: Not only fullerene graphs are Hamiltonian

Fullerene graphs, i.e., 3-connected planar cubic graphs with pentagonal and hexagonal faces, are conjectured to be Hamiltonian. This is a special case of a conjecture of Barnette and Goodey, stating that 3-connected planar graphs with faces of size at most 6 are Hamiltonian. We prove the conjecture

Spectral Continuity for Aperiodic Quantum Systems II. Periodic Approximations in 1D

The existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schr\"odinger operators. In a forthcoming work [9] (arXiv:1709.00975) this task was boiled down to the existence and construction of periodic approximations of the underlying dynamical systems in the Hausdorff topology. As a result the one-dimensional systems admitting such approximations are completely classified in the present work. In addition explicit constructions are provided for dynamical systems defined by primitive substitutions covering all studied examples such as the Fibonacci sequence or the Golay-Rudin-Shapiro sequence. One main tool is the description of the Hausdorff topology by the local pattern topology on the dictionaries as well as the GAP-graphs describing the local structure. The connection of branching vertices in the GAP-graphs and defects is discussed.Comment: 30 pages, 5 figure

Linking covariant and canonical LQG II: Spin foam projector

In a seminal paper, Kaminski, Kisielowski an Lewandowski for the first time extended the definition of spin foam models to arbitrary boundary graphs. This is a prerequisite in order to make contact to the canonical formulation of Loop Quantum Gravity (LQG) whose Hilbert space contains all these graphs. This makes it finally possible to investigate the question whether any of the presently considered spin foam models yields a rigging map for any of the presently defined Hamiltonian constraint operators. In the analysis of this would-be spin foam rigging map we are able to identify an elementary spin foam transfer matrix that allows to generate any finite foam as a finite power of the transfer matrix. However, it transpires that the resulting object, as written, does not define a projector on the physical Hilbert space. This statement is independent of the concrete spin foam model and Hamiltonian constraint. Nevertehless, the transfer matrix potentially contains the necessary ingredient in order to construct a proper rigging map in terms of a modified transfer matrix.Comment: 62 pages, 14 figures Abstract changed, slightly reorganized, minor errors correcte