Related papers: A strong geometric hyperbolicity property for dire…
We consider the complexity of Green's relations when the semigroup is given by transformations on a finite set. Green's relations can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes then…
We show that if $G$ is a finitely generated group hyperbolic relative to a finite collection of subgroups $\mathcal{P}$, then the natural action of $G$ on the geodesic boundary of the associated relative Cayley graph induces a hyperfinite…
We study different geometric properties on infinite graphs, related to the weak-type boundedness of the Hardy-Littlewood maximal averaging operator. In particular, we analyze the connections between the doubling condition, having finite…
This is the first of a sequence of papers devoted to studying the link between the complexity of the Word Problem for a finitely generated recursively presented group $G$ and the isoperimetric functions of the finitely presented groups in…
Many complex networks exhibit hierarchical, tree-like structures, making hyperbolic space a natural candidate wherein to learn representations of them. Based on this observation, Hyperbolic Graph Neural Networks (HGNNs) have been widely…
We show that groups presented by inverse-closed finite convergent length-reducing rewriting systems are characterised by a striking geometric property: their Cayley graphs are geodetic and side-lengths of non-degenerate triangles are…
We prove measurable analogues of Whitney's classical theorems on weak isomorphisms of finite graphs. In the setting of locally finite graphings, we introduce a notion of weak isomorphism as an edge-measure-preserving Borel bijection that…
Deciding whether a graph can be embedded in a grid using only unit-length edges is NP-complete, even when restricted to binary trees. However, it is not difficult to devise a number of graph classes for which the problem is polynomial, even…
Graph monoids arise naturally in the study of non-stable K-theory of graph C*-algebras and Leavitt path algebras. They play also an important role in the current approaches to the realization problem for von Neumann regular rings. In this…
The results of computer searches for large graphs with given (small) degree and diameter are presented. The new graphs are Cayley graphs of semidirect products of cyclic groups and related groups. One fundamental use of our ``dense graphs''…
We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result…
We establish mild conditions under which a possibly irregular, sparse graph $G$ has "many" strong orientations. Given a graph $G$ on $n$ vertices, orient each edge in either direction with probability $1/2$ independently. We show that if…
This note addresses some questions that arise in the series of works by Kyoji Saito on the growth functions of graphs. We study "hyperbolike" graphs, which include Cayley graphs of hyperbolic groups. We generalize some well-known results on…
We show that one can define and effectively compute Stallings graphs for quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or right-angled Artin groups). These Stallings graphs are finite labeled graphs, which are…
Recently, a new set of multigraph parameters was defined, called "gonalities". Gonality bears some similarity to treewidth, and is a relevant graph parameter for problems in number theory and multigraph algorithms. Multigraphs of gonality 1…
We prove a deletion-contraction formula for motivic Feynman rules given by the classes of the affine graph hypersurface complement in the Grothendieck ring of varieties. We derive explicit recursions and generating series for these motivic…
For a formation $\mathfrak{F}$ of finite groups, tight connections are established between the pro-$\mathfrak{F}$-topology of a finitely generated free group $F$ and the geometry of the Cayley graph $\Gamma(\hat{F_{\mathfrak{F}}})$ of the…
When one studies geometric properties of graphs, local finiteness is a common implicit assumption, and that of transitivity a frequent explicit one. By compactness arguments, local finiteness guarantees several regularity properties. It is…
We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion…
In graph realization problems one is given a degree sequence and the task is to decide whether there is a graph whose vertex degrees match to the given sequence. This realization problem is known to be polynomial-time solvable when the…