In this paper we study a random walk on an affine building of type Ãr, whose radial part, when suitably normalized, converges toward the brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane (Probab. Theory Related Fields89 (1991) 117–129). This extends also the link at the probabilistic level between riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol...

Let $G$ be a finite group. The main supergraph $𝒮\left(G\right)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o\left(x\right)\mid o\left(y\right)$ or $o\left(y\right)\mid o\left(x\right)$. In this paper, we will show that $G\cong Sz\left(q\right)$ if and only if $𝒮\left(G\right)\cong 𝒮\left(Sz\right(q\left)\right)$, where $q=2^{2m+1}\ge 8$.

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.