405 research outputs found
Uniform random colored complexes
We present here random distributions on -edge-colored, bipartite
graphs with a fixed number of vertices . These graphs are dual to
-dimensional orientable colored complexes. We investigate the behavior of
quantities related to those random graphs, such as their number of connected
components or the number of vertices of their dual complexes, as . The techniques involved in the study of these quantities also yield a
Central Limit Theorem for the genus of a uniform map of order , as .Comment: 36 pages, 9 figures, minor additions and correction
Stratified graphs for imbedding systems
AbstractTwo imbeddings of a graph G are considered to be adjacent if the second can be obtained from the first by moving one or both ends of a single edge within its or their respective rotations. Thus, a collection of imbeddings S of G, called a ‘system’, may be represented as a ‘stratified graph’, and denoted SG; the focus here is the case in which S is the collection of all orientable imbeddings. The induced subgraph of SG on the set of imbeddings into the surface of genus k is called the ‘kth stratum’, and the cardinality of that set of imbeddings is called the ‘stratum size’; one may observe that the sequence of stratum sizes is precisely the genus distribution for the graph G. It is known that the genus distribution is not a complete invariant, even when the category of graphs is restricted to be simplicial and 3-connected. However, it is proved herein that the link of each point — that is, the subgraph induced by its neighbors — of SG is a complete isomorphism invariant for the category of graphs whose minimum valence is at least three. This supports the plausibility of a probabilistic approach to graph isomorphism testing by sampling higher-order imbedding distribution data. A detailed structural analysis of stratified graphs is presented
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On the Average Genus of a Graph
Not all rational numbers are possibilities for the average genus of an individual graph. The smallest such numbers are determined, and varied examples are constructed to demonstrate that a single value of average genus can be shared by arbitrarily many different graphs. It is proved that the number one is a limit point of the set of possible values for average genus and that the complete graph K4 is the only 3-connected graph whose average genus is less than one. Several problems for future study are suggested
-Colored Graphs - a Review of Sundry Properties
We review the combinatorial, topological, algebraic and metric properties
supported by -colored graphs, with a focus on those that are pertinent
to the study of tensor model theories. We show how to extract a limiting
continuum metric space from this set of graphs and detail properties of this
limit through the calculation of exponents at criticality
Unimodular lattice triangulations as small-world and scale-free random graphs
Real-world networks, e.g. the social relations or world-wide-web graphs,
exhibit both small-world and scale-free behaviour. We interpret lattice
triangulations as planar graphs by identifying triangulation vertices with
graph nodes and one-dimensional simplices with edges. Since these
triangulations are ergodic with respect to a certain Pachner flip, applying
different Monte-Carlo simulations enables us to calculate average properties of
random triangulations, as well as canonical ensemble averages using an energy
functional that is approximately the variance of the degree distribution. All
considered triangulations have clustering coefficients comparable with real
world graphs, for the canonical ensemble there are inverse temperatures with
small shortest path length independent of system size. Tuning the inverse
temperature to a quasi-critical value leads to an indication of scale-free
behaviour for degrees . Using triangulations as a random graph model
can improve the understanding of real-world networks, especially if the actual
distance of the embedded nodes becomes important.Comment: 17 pages, 6 figures, will appear in New J. Phy
Quantum Deformation of Lattice Gauge Theory
A quantum deformation of 3-dimensional lattice gauge theory is defined by
applying the Reshetikhin-Turaev functor to a Heegaard diagram associated to a
given cell complex. In the root-of-unity case, the construction is carried out
with a modular Hopf algebra. In the topological (weak-coupling) limit, the
gauge theory partition function gives a 3-fold invariant, coinciding in the
simplicial case with the Turaev-Viro one. We discuss bounded manifolds as well
as links in manifolds. By a dimensional reduction, we obtain a q-deformed gauge
theory on Riemann surfaces and find a connection with the algebraic
Alekseev-Grosse-Schomerus approach.Comment: 31 pp.; uses epic.sty and eepic.st
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