Related papers: Fundamental groupoids of k-graphs
We define cofinite graphs and cofinite groupoids in a unified way that extends the notion of cofinite groups introduced by Hartley. The common underlying structure of all these objects is that they are directed graphs endowed with a certain…
We introduce what we call `generalized higher rank $k$-graphs' as a class of categories equipped with a notion of size. They extend not only the higher rank $k$-graphs, but also the Levi categories introduced by the first author as a…
A topological groupoid G is K-pointed, if it is equipped with a homomorphism from a topological group K to G. We describe the homotopy groups of such K-pointed topological groupoids and relate these groups to the ordinary homotopy groups in…
The Gruenberg-Kegel graph of a group is the undirected graph whose vertices are those primes which occur as the order of an element of the group, and distinct vertices $p$, $q$ are joined by an edge whenever the group has an element of…
Given an adaptable separated graph, we construct an associated groupoid and explore its type semigroup. Specifically, we first attach to each adaptable separated graph a corresponding semigroup, which we prove is an $E^*$-unitary inverse…
We compute the K-groups of C^*-algebras arising from one-dimensional generalized solenoids. The results show that Ruelle algebras from one-dimensional generalized solenoids are one-dimensional generalizations of Cuntz-Krieger algebras.
We introduce a notion of a root groupoid as a replacement of the notion of Weyl group for (Kac-Moody) Lie superalgebras. The objects of the root groupoid classify certain root data, the arrows are defined by generators and relations. As an…
For a finite group $G$ and for a fixed positive integer $k$, $k\geq 2$, the $k$-power graph of $G$ is an undirected simple graph with vertex set $G$ in which two distinct vertices $x$ and $y$ are adjacent if and only if $x^k=y$ or $y^k=x$.…
We describe the fundamental groups of ordered and unordered k point sets in complex projective space of dimension n generating a projective subspace of dimension i. We apply these to study connectivity of more complicated configurations of…
We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph LG. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented…
In this paper we give a complete description of K-theory groups for Cuntz-Krieger C*-algebras associated to general locally-finite (topologically connected) graphs via Bass-Hashimoto operator. Our result generalizes the one obtained by the…
The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$, introduced by Hager in 1985, is a nice generalization of the classical connectivity. Recently, as a natural counterpart, we proposed the concept of generalized…
Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of…
We introduce a new class of C^*-algebras, which is a generalization of both graph algebras and homeomorphism C^*-algebras. This class is very large and also very tractable. We prove the so-called gauge-invariant uniqueness theorem and the…
We establish exact sequences in $KK$-theory for graded relative Cuntz-Pimsner algebras associated to nondegenerate $C^*$-correspondences. We use this to calculate the graded $K$-theory and $K$-homology of relative Cuntz-Krieger algebras of…
A new picture of Quantum Mechanics based on the theory of groupoids is presented. This picture provides the mathematical background for Schwinger's algebra of selective measurements and helps to understand its scope and eventual…
For fixed positive integers $n$ and $k$, the Kneser graph $KG_{n,k}$ has vertices labeled by $k$-element subsets of $\{1,2,\dots,n\}$ and edges between disjoint sets. Keeping $k$ fixed and allowing $n$ to grow, one obtains a family of…
Graphs with given k vertices generate an (acyclic) simplicial complex. We describe the homology of its quotient complex, formed by all connected graphs, and demonstrate its applications to the topology of braid groups, knot theory,…
In this article, we study the various fundamental groupoid schemes corresponding to Tannakian categories of certain types of vector bundles. We compute fundamental groupoid scheme of anisotropic conic, Klein bottle and abelian varieties.…
In [36, Section 8], the present author proposed the hypergraph obstruction for the existence of k-regular embeddings. In this paper, we develop the hypergraph obstruction concretely and give some homological obstructions for the k-regular…