The paper studies tolerances and congruences on anticommutative conservative groupoids. These groupoids can be assigned in a one-to-one way to undirected graphs.

For each integer $k\ge 6$ and each finite graph $L$, we construct a Coxeter group $W$ and a non positively curved polygonal complex $A$ on which $W$ acts properly cocompactly, such that each polygon of $A$ has $k$ edges, and the link of each vertex of $A$ is isomorphic to $L$. If $L$ is a “generalized $m$-gon”, then $A$ is a Tits building modelled on a reflection group of the hyperbolic plane. We give a condition on $\mathrm{Aut}\left(L\right)$ for $\mathrm{Aut}\left(A\right)$ to be non enumerable (which is satisfied if $L$ is a thick classical generalized $m$-gon). On the other hand,...

A directed Cayley graph $C(\Gamma ,X)$ is specified by a group $\Gamma$ and an identity-free generating set $X$ for this group. Vertices of $C(\Gamma ,X)$ are elements of $\Gamma$ and there is a directed edge from the vertex $u$ to the vertex $v$ in $C(\Gamma ,X)$ if and only if there is a generator $x\in X$ such that $ux=v$. We study graphs $C(\Gamma ,X)$ for the direct product $Z_m\times Z_n$ of two cyclic groups $Z_m$ and $Z_n$, and the generating set $X=\left\{\right(0,1),(1,0),(2,0),\cdots ,(p,0\left)\right\}$. We present resolving sets which yield upper bounds on the metric dimension of these graphs for $p=2$ and $3$.

We analyse the spectral phase diagram of Schrödinger operators $T+\lambda V$ on regular tree graphs, with $T$ the graph adjacency operator and $V$ a random potential given by $iid$ random variables. The main result is a criterion for the emergence of absolutely continuous $\left(ac\right)$ spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials $ac$ spectrum appears at arbitrarily weak disorder $(\lambda \ll 1)$ in an energy regime which extends beyond the spectrum of$\sim T$. Incorporating...

In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand isomorphic to the field with two elements. This result is used to describe those finite rings R for which Γ(R) contains a Hamiltonian cycle where Γ(R) is the (simple) graph defined on the elements of R with an edge between vertices r and s if and only if r - s is invertible. It is also shown that for an Artinian ring R the number of connected components of the graph...

$G$-graphs are a type of graphs associated to groups, which were proposed by A. Bretto and A. Faisant (2005). In this paper, we first give some theorems regarding $G$-graphs. Then we introduce the notion of rough $G$-graphs and investigate some important properties of these graphs.

We shall show that there exist sofic groups which are not locally embeddable into finite Moufang loops. These groups serve as counterexamples to a problem and two conjectures formulated in the paper by M. Vodička, P. Zlatoš (2019).

Let $G$ be a group and let $X$ be a finite subset. The isoperimetric method investigates the objective function $\left|\right(XB)\setminus X|$, defined on the subsets $X$ with $\left|X\right|\ge k$ and $|G\setminus (XB\left)\right|\ge k$, where $XB$ is the product of $X$ by $B$.In this paper we present all the basic facts about the isoperimetric method. We improve some of our previous results and obtain generalizations and short proofs for several known results. We also give some new applications.Some of the results obtained here will be used in coming papers to improve Kempermann structure...