Related papers: On outer automorphisms of certain graph $C^{*}$-al…

To a large class of graphs of groups we associate a C*-algebra universal for generators and relations. We show that this C*-algebra is stably isomorphic to the crossed product induced from the action of the fundamental group of the graph of…

Given a short exact sequence of locally compact abelian groups $0 \to A \to B \to C \to 0$ and a continuous $C$-valued $1$-cocycle $\phi$ on a locally compact Hausdorff groupoid $\Gamma$ we construct a twist of $\Gamma$ by $A$ that is…

We give an accessible and modern description of the automorphisms of a finite abelian group $G$. Included is an explicit formula for the cardinality of $Aut(G)$.

A graph $\Gamma$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of arcs of $\Gamma$, where an arc is an ordered pair of adjacent vertices. Let $\Gamma$ be a $G$-symmetric graph such that its…

We construct a 2-generated group $\Gamma $ such that its Cayley graph possesses finite connected subsets with arbitrarily big finite Heesch number.

Let $G$ be a finite, non-abelian group of the form $G = A N$, where $A \leq G$ is abelian, and $N \trianglelefteq G$ is cyclic. We prove that the commuting graph $\Gamma(G)$ of $G$ is either a connected graph of diameter at most four, or…

A graph $\Gamma$ labelled by a set $S$ defines a group $G(\Gamma)$ whose generators are the set of labels $S$ and whose relations are all words which can be read on closed paths of this graph. We introduce the notion of aspherical graph and…

For a graph $\Gamma$ and group $G$, $G^\Gamma$ is the subgroup of $G^{|\Gamma|}$ generated by elements with $g$ in the coordinates corresponding to $v$ and its neighbors in $\Gamma$. There is a natural epimorphism $G^\Gamma \to…

Let $P$ be a finitely generated cancellative abelian monoid. A $P$-graph $\Lambda$ is a natural generalization of a higher rank graph. A pullback of $\Lambda$ is constructed by pulling it back over a given monoid morphism to $P$, while a…

We present an infinite sequence of finite graphs with trivial automorphism group and non-trivial quantum automorphism group. These are the first known examples of graphs with this property. Moreover, to the best of our knowledge, these are…

We show that the automorphism group of a graph product of finite groups $Aut(G_\Gamma)$ has Kazhdan's property (T) if and only if $\Gamma$ is a complete graph.

Let $\Gamma=\mathrm{Cay}(G,S)$ be a Cayley digraph on a group $G$ and let $A=\mathrm{Aut}(\Gamma)$. The Cayley index of $\Gamma$ is $|A:G|$. It has previously been shown that, if $p$ is a prime, $G$ is a cyclic $p$-group and $A$ contains a…

For every finite-to-one map $\lambda:\Gamma\to\Gamma$ and for every abelian group $K$, the generalized shift $\sigma_\lambda$ of the direct sum $\bigoplus_\Gamma K$ is the endomorphism defined by…

We geometrically describe the relation induced on a set of graphs by isomorphism of their associated graph C*-algebras as the smallest equivalence relation generated by five types of moves. The graphs studied have finitely many vertices and…

We consider the problem of identifying exactly which AF-algebras are isomorphic to a graph C*-algebra. We prove that any separable, unital, Type I C*-algebra with finitely many ideals is isomorphic to a graph C*-algebra. This result allows…

For a finite group $G$, denote by $\alpha(G)$ the minimum number of vertices of any graph $\Gamma$ having $\text{Aut}(\Gamma)\cong G$. In this paper, we prove that $\alpha(G)\leq |G|$, with specified exceptions. The exceptions include four…

Assume that $G$ is a finite group. For every $a, b \in\mathbb N,$ we define a graph $\Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if…

Topological quivers generalize the notion of directed graphs in which the sets of vertices and edges are locally compact (second countable) Hausdorff spaces. Associated to a topological quiver $Q$ is a $C^*$-correspondence, and in turn, a…

We determine the structure of automorphism groups of finite graphs of bounded Hadwiger number. Our proof includes a structural analysis of finite edge-transitive graphs. In particular, we show that for connected, $K_{h+1}$-minor-free,…

Let $R=\oplus_{\Gamma\in\Gamma}R_{\gamma}$ be a $\Gamma$-graded $K$-algebra over a field $K$, where $\Gamma$ is a totally ordered semigroup, and let $I$ be an ideal of $R$. Considering the $\Gamma$-grading filtration $FR$ of $R$ and the…