5,661 research outputs found
Classifying Higher Rank Toeplitz Operators.
To a higher rank directed graph (Λ, d), in the sense of Kumjian and Pask, 2000, one can associate natural noncommutative analytic Toeplitz algebras, both weakly closed and norm closed. We introduce methods for the classification of these algebras in the case of single vertex graphs
Inverse monoids of partial graph automorphisms
A partial automorphism of a finite graph is an isomorphism between its vertex
induced subgraphs. The set of all partial automorphisms of a given finite graph
forms an inverse monoid under composition (of partial maps). We describe the
algebraic structure of such inverse monoids by the means of the standard tools
of inverse semigroup theory, namely Green's relations and some properties of
the natural partial order, and give a characterization of inverse monoids which
arise as inverse monoids of partial graph automorphisms. We extend our results
to digraphs and edge-colored digraphs as well
Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs
Building on work of Cai, F\"urer, and Immerman \cite{CFI92}, we show two
hardness results for the Graph Isomorphism problem. First, we show that there
are pairs of nonisomorphic -vertex graphs and such that any
sum-of-squares (SOS) proof of nonisomorphism requires degree . In
other words, we show an -round integrality gap for the Lasserre SDP
relaxation. In fact, we show this for pairs and which are not even
-isomorphic. (Here we say that two -vertex, -edge graphs
and are -isomorphic if there is a bijection between their
vertices which preserves at least edges.) Our second result is that
under the {\sc R3XOR} Hypothesis \cite{Fei02} (and also any of a class of
hypotheses which generalize the {\sc R3XOR} Hypothesis), the \emph{robust}
Graph Isomorphism problem is hard. I.e.\ for every , there is no
efficient algorithm which can distinguish graph pairs which are
-isomorphic from pairs which are not even
-isomorphic for some universal constant . Along the
way we prove a robust asymmetry result for random graphs and hypergraphs which
may be of independent interest
The graded Grothendieck group and the classification of Leavitt path algebras
This paper is an attempt to show that, parallel to Elliott's classification
of AF -algebras by means of -theory, the graded -group classifies
Leavitt path algebras completely. In this direction, we prove this claim at two
extremes, namely, for the class of acyclic graphs (graphs with no cycles) and
comet and polycephaly graphs (graphs which each head is connected to a cycle or
a collection of loops).Comment: 30 pages more polishe
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