18,173 research outputs found
Invariance of KMS states on graph C*-algebras under classical and quantum symmetry
We study invariance of KMS states on graph C*-algebras coming from strongly
connected and circulant graphs under the classical and quantum symmetry of the
graphs. We show that the unique KMS state for strongly connected graphs is
invariant under quantum automorphism group of the graph. For circulant graphs,
it is shown that the action of classical and quantum automorphism group
preserves only one of the KMS states occurring at the critical inverse
temperature. We also give an example of a graph C*-algebra having more than one
KMS state such that all of them are invariant under the action of classical
automorphism group of the graph, but there is a unique KMS state which is
invariant under the action of quantum automorphism group of the graph.Comment: 15 pages, 2 figure
Approximately Counting Embeddings into Random Graphs
Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic)
copies of H contained in a graph G. We investigate the fundamental problem of
estimating C_H(G). Previous results cover only a few specific instances of this
general problem, for example, the case when H has degree at most one
(monomer-dimer problem). In this paper, we present the first general subcase of
the subgraph isomorphism counting problem which is almost always efficiently
approximable. The results rely on a new graph decomposition technique.
Informally, the decomposition is a labeling of the vertices such that every
edge is between vertices with different labels and for every vertex all
neighbors with a higher label have identical labels. The labeling implicitly
generates a sequence of bipartite graphs which permits us to break the problem
of counting embeddings of large subgraphs into that of counting embeddings of
small subgraphs. Using this method, we present a simple randomized algorithm
for the counting problem. For all decomposable graphs H and all graphs G, the
algorithm is an unbiased estimator. Furthermore, for all graphs H having a
decomposition where each of the bipartite graphs generated is small and almost
all graphs G, the algorithm is a fully polynomial randomized approximation
scheme.
We show that the graph classes of H for which we obtain a fully polynomial
randomized approximation scheme for almost all G includes graphs of degree at
most two, bounded-degree forests, bounded-length grid graphs, subdivision of
bounded-degree graphs, and major subclasses of outerplanar graphs,
series-parallel graphs and planar graphs, whereas unbounded-length grid graphs
are excluded.Comment: Earlier version appeared in Random 2008. Fixed an typo in Definition
3.
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
- …