Related papers: The Graded Classification Conjecture holds for gra…

The Gruenberg-Kegel graph $\Gamma(G)$ associated with a finite group $G$ has as vertices the prime divisors of $|G|$, with an edge from $p$ to $q$ if and only if $G$ contains an element of order $pq$. This graph has been the subject of much…

Graded Hecke algebras can be constructed geometrically, with constructible sheaves and equivariant cohomology. The input consists of a complex reductive group G (possibly disconnected) and a cuspidal local system on a nilpotent orbit for a…

We study the first order theory of structures over graphs i.e. structures of the form ($\mathcal{G},\tau$) where $\mathcal{G}$ is the set of all (isomorphism types of) finite undirected graphs and $\tau$ some vocabulary. We define the…

We study graph products of groups from the viewpoint of measured group theory. We first establish a full measure equivalence classification of graph products of countably infinite groups over finite simple graphs with no transvection and no…

An end of a graph $G$ is an equivalence class of rays, where two rays are equivalent if there are infinitely many vertex-disjoint paths between them in $G$. The degree of an end is the maximum cardinality of a collection of pairwise…

Let G be a locally compact, Hausdorff groupoid in which s is a local homeomorphism and the unit space is totally disconnected. Assume there is a continuous cocycle c from G into a discrete group $\Gamma$. We show that the collection A(G) of…

In this paper we associate an abelian category to a finite directed graph and prove the categories arising from two graphs are equivalent if the incidence matrices of the graphs are shift equivalent. The abelian category is the quotient of…

We study the notions of continuous orbit equivalence and eventual one-sided conjugacy of finitely-aligned higher-rank graphs and two-sided conjugacy of row-finite higher-rank graphs with finitely many vertices and no sinks or sources. We…

In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints, whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain…

We provide a complete classification of the class of unital graph $C^*$-algebras - prominently containing the full family of Cuntz-Krieger algebras - showing that Morita equivalence in this case is determined by ordered, filtered…

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…

Let $\Gamma$ be the infinite cyclic group on a generator $x.$ To avoid confusion when working with $\mathbb Z$-modules which also have an additional $\mathbb Z$-action, we consider the $\mathbb Z$-action to be a $\Gamma$-action instead.…

We generalise the theory of Cuntz-Krieger families and graph algebras to the class of finitely aligned $k$-graphs. This class contains in particular all row-finite $k$-graphs. The Cuntz-Krieger relations for non-row-finite $k$-graphs look…

We formalize eight different notions of isomorphism among (unital) graph C*-algebras, and initiate the study of which of these notions may be described geometrically as generated by moves. We propose a list of seven types of moves that we…

This dissertation concerns the classification of groupoid and higher-rank graph C*-algebras and has two main components. Firstly, for a groupoid it is shown that the notions of strength of convergence in the orbit space and…

We demonstrate that when a graph exhibits a specific type of symmetry, it satisfies the Union Closed Conjecture(UCC). Additionally, we show that certain graph classes, such as Cylindrical Grid Graphs and Torus Grid Graphs also satisfy the…

A path decomposition of a graph G is a collection of edge-disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most (n+1)/2.…

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto's notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if…

In the standard category of directed graphs, graph morphisms map edges to edges. By allowing graph morphisms to map edges to finite paths (path homomorphisms of graphs), we obtain an ambient category in which we determine subcategories…

The class of cographs is one of the most well-known graph classes, which is also known to be equivalent to the class of $P_4$-free graphs. We show that Mader's conjecture is true if we restrict ourselves to cographs, that is, for any tree…