Related papers: Tur\'an's theorem for Dowling geometries

Given a finitely generated subgroup $\Gamma \le \mathrm{Out}(\mathbb{F})$ of the outer automorphism group of the rank $r$ free group $\mathbb{F} = F_r$, there is a corresponding free group extension $1 \to \mathbb{F} \to E_{\Gamma} \to…

Here we study some algebraic properties of non-cyclic graphs. In this paper we show that $\overline{\Gamma}_G$ is isomorphic to $K_3\cup (n-4)K_1$ or $K_4\cup (n-5)K_1$ if and only if $G$ is isomorphic to $D_8$ or $D_{10}$, respectively. We…

For any positive integer $n$, we exhibit a cofinite subgroup $\Gamma_n$ of the mapping class group of a surface of genus at most two such that $\Gamma_n$ admits an epimorphism onto a free group of rank $n$. We conclude that…

We establish a quantum version of Frucht's Theorem, proving that every finite quantum group is the quantum automorphism group of an undirected finite quantum graph. The construction is based on first considering several quantum Cayley…

For an abelian group $\Gamma$, a $\Gamma$-labelled graph is a graph whose vertices are labelled by elements of $\Gamma$. We prove that a certain collection of edge sets of a $\Gamma$-labelled graph forms a delta-matroid, which we call a…

Given a finite group $G$ and a set $A$ of generators, the diameter diam$(\Gamma(G,A))$ of the Cayley graph $\Gamma(G,A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup…

A Mauldin-Williams graph $\mathcal{M}$ is a generalization of an iterated function system by a directed graph. Its invariant set $K$ plays the role of the self-similar set. We associate a $C^{*}$-algebra $\mathcal{O}_{\mathcal{M}}(K)$ with…

The group of automorphisms of the geometry of an integrable system is considered. The geometrical structure used to obtain it is provided by a normal form representation of integrable systems that do not depend on any additional geometrical…

Given a countable abelian group $A$, we construct a row finite directed graph $\Gamma(A)$ such that the $K_{0}$-group of the graph $\textrm{C}^{\ast}$-algebra $\textrm{C}^{\ast}(\Gamma(A))$ is canonically isomorphic to $A$. Moreover, each…

Motivated by the vast literature of quantum automorphism groups of graphs, we define and study quantum automorphism groups of matroids. A key feature of quantum groups is that there are many quantizations of a classical group, and this…

For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $r\in \mathbb{N}$, the $r$-independence number of $G$, denoted $\alpha_r(G)$ is the largest size of a…

Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of…

Leighton's graph covering theorem says that two finite graphs with a common cover have a common finite cover. We present a new proof of this using groupoids, and use this as a model to prove two generalisations of the theorem. The first…

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…

We prove a number of results to the effect that generic quantum graphs (defined via operator systems as in the work of Duan-Severini-Winter / Weaver) have few symmetries: for a Zariski-dense open set of tuples $(X_1,\cdots,X_d)$ of…

We prove that an invariant subalgebra A_n^W of the Weyl algebra A_n is a Galois order over an adequate commutative subalgebra \Gamma when W is a two-parameters irreducible unitary reflection group G(m,1,n), m\geq 1, n\geq 1, including the…

We study varieties associated to hypergraphs from the point of view of projective geometry and matroid theory. We describe their decompositions into matroid varieties, which may be reducible and can have arbitrary singularities by the…

In 1961, Dirac showed that chordal graphs are exactly the graphs that can be constructed from complete graphs by a sequence of clique-sums. In an earlier paper, by analogy with Dirac's result, we introduced the class of $GF(q)$-chordal…

Supergravity theories in more than four dimensions with grand unified gauge symmetries are an important intermediate step towards the ultraviolet completion of the Standard Model in string theory. Using toric geometry, we classify and…

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique…