809 research outputs found
Flows on the join of two graphs
summary:The join of two graphs and is a graph formed from disjoint copies of and by connecting each vertex of to each vertex of . We determine the flow number of the resulting graph. More precisely, we prove that the join of two graphs admits a nowhere-zero -flow except for a few classes of graphs: a single vertex joined with a graph containing an isolated vertex or an odd circuit tree component, a single edge joined with a graph containing only isolated edges, a single edge plus an isolated vertex joined with a graph containing only isolated vertices, and two isolated vertices joined with exactly one isolated vertex plus some number of isolated edges
Algebraic Structures in the Coupling of Gravity to Gauge Theories
This article is an extension of the author's second master thesis [1]. It
aims to introduce to the theory of perturbatively quantized General Relativity
coupled to Spinor Electrodynamics, provide the results thereof and set the
notation to serve as a starting point for further research in this direction.
It includes the differential geometric and Hopf algebraic background, as well
as the corresponding Lagrange density and some renormalization theory. Then, a
particular problem in the renormalization of Quantum General Relativity coupled
to Quantum Electrodynamics is addressed and solved by a generalization of
Furry's Theorem. Next, the restricted combinatorial Green's functions for all
two-loop propagators and all one-loop divergent subgraphs thereof are
presented. Finally, relations between these one-loop restricted combinatorial
Green's functions necessary for multiplicative renormalization are discussed.
Keywords: Quantum Field Theory; Quantum Gravity; Quantum General Relativity;
Quantum Electrodynamics; Perturbative Quantization; Hopf Algebraic
RenormalizationComment: 57 pages, 259 Feynman diagrams, article; minor revisions; version to
appear in Annals of Physic
Topological Analogues of the Tutte polynomial and their Decompositions
The Tutte polynomial is one of the most influential graph polynomials, whose importance stems from all of the combinatorial and structural information it captures about a graph. Not only that, but the polynomial itself has been subject to much study due to its inherent well-behaved structure and universal nature. In recent years, much interest has been shown in finding an analogue of the Tutte polynomial for graphs embedded in surfaces, resulting in a plethora of topological graph polynomials. There is now an active body of research into these polynomials, the combinatorial information of the embedded graph that they capture, and the properties and structure they exhibit.This thesis consists of three parts. The first part, Chapters 1–3, paints a picture of what is already known. It provides an overview of topological graphs, how to represent them, and their polynomials, along with notable results across their development in the literature. In the second part, Chapter 4, we unify a recent divergence in the development of topological Tutte polynomials to obtain a new polynomial for embedded graphs that encapsulates the properties of its predecessors. Notably, this new topological graph polynomial possesses a Universality Property, meaning that any graph parameter satisfying a specified set of deletion-contraction relations is a specialisation of this polynomial. A natural consequence of this is a recursive method for counting the number of flows and tensions (i.e., proper colourings) in an embedded graph. The final part, Chapters 5 and 6, provides several tensor product formulae that split well-known topological Tutte polynomials by decomposing a larger graph into smaller graphs that are “multiplied” together. These formulae mirror a result of Brylawski for the Tutte polynomial, which is known for its applications as a computational aid and in understanding the complexity of the Tutte polynomial
Turbulence, representations, and trace-preserving actions
We establish criteria for turbulence in certain spaces of C*-algebra
representations and apply this to the problem of nonclassifiability by
countable structures for group actions on a standard atomless probability space
(X,\mu) and on the hyperfinite II_1 factor R. We also prove that the conjugacy
action on the space of free actions of a countably infinite amenable group on R
is turbulent, and that the conjugacy action on the space of ergodic
measure-preserving flows on (X,\mu) is generically turbulent.Comment: 27 page
Generalized Killing spinors and Lagrangian graphs
We study generalized Killing spinors on the standard sphere ,
which turn out to be related to Lagrangian embeddings in the nearly K\"ahler
manifold and to great circle flows on . Using our
methods we generalize a well known result of Gluck and Gu concerning
divergence-free geodesic vector fields on the sphere and we show that the space
of Lagrangian submanifolds of has at least three connected
components.Comment: 11 page
Integer flows and Modulo Orientations
Tutte\u27s 3-flow conjecture (1970\u27s) states that every 4-edge-connected graph admits a nowhere-zero 3-flow. A graph G admits a nowhere-zero 3-flow if and only if G has an orientation such that the out-degree equals the in-degree modulo 3 for every vertex. In the 1980ies Jaeger suggested some related conjectures. The generalized conjecture to modulo k-orientations, called circular flow conjecture, says that, for every odd natural number k, every (2k-2)-edge-connected graph has an orientation such that the out-degree equals the in-degree modulo k for every vertex. And the weaker conjecture he made, known as the weak 3-flow conjecture where he suggests that the constant 4 is replaced by any larger constant.;The weak version of the circular flow conjecture and the weak 3-flow conjecture are verified by Thomassen (JCTB 2012) recently. He proved that, for every odd natural number k, every (2k 2 + k)-edge-connected graph has an orientation such that the out-degree equals the in-degree modulo k for every vertex and for k = 3 the edge-connectivity 8 suffices. Those proofs are refined in this paper to give the same conclusions for 9 k-edge-connected graphs and for 6-edge-connected graphs when k = 3 respectively
Of McKay Correspondence, Non-linear Sigma-model and Conformal Field Theory
The ubiquitous ADE classification has induced many proposals of often
mysterious correspondences both in mathematics and physics. The mathematics
side includes quiver theory and the McKay Correspondence which relates finite
group representation theory to Lie algebras as well as crepant resolutions of
Gorenstein singularities. On the physics side, we have the graph-theoretic
classification of the modular invariants of WZW models, as well as the relation
between the string theory nonlinear -models and Landau-Ginzburg
orbifolds. We here propose a unification scheme which naturally incorporates
all these correspondences of the ADE type in two complex dimensions. An
intricate web of inter-relations is constructed, providing a possible guideline
to establish new directions of research or alternate pathways to the standing
problems in higher dimensions.Comment: 35 pages, 4 figures; minor corrections, comments on toric geometry
and references adde
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