Related papers: Contact Graphs, Boundaries, and a Central Limit Th…

Let $\Cal S$ be an abelian group of automorphisms of a probability space $(X, {\Cal A}, \mu)$ with a finite system of generators $(A_1, ..., A_d)$. Let $A^{\el}$ denote $A_1^{\ell_1} ... A_d^{\ell_d}$, for ${\el}= (\ell_1, ..., \ell_d)$. If…

We prove the discrete Lusternik-Schnirelmann theorem telling that tcat(G) less or equal to crit(G) for a general simple graph G=(V,E). It relates the minimal number tcat(G) of in G contractible graphs covering G, with crit(G), the minimal…

The extremal graphs $\mathrm{EX}(n,\mathcal F)$ and spectral extremal graphs $\mathrm{SPEX}(n,\mathcal F)$ are the sets of graphs on $n$ vertices with maximum number of edges and maximum spectral radius, respectively, with no subgraph in…

Two classical concepts of centrality in a graph are the median and the center. The connected notions of the status and the radius of a graph seem to be in no relation. In this paper, however, we show a clear connection of both concepts, as…

Temporal graphs are graphs where the edge set can change in each time step, and the vertex set stays the same. Exploration of temporal graphs whose snapshot in each time step is a connected graph, called connected temporal graphs, has been…

Let $G$ be a finite, connected graph and $v$ a vertex of $G$. The average distance and the eccentricity of $v$ in $G$ are defined as the arithmetic mean and the maximum, respectively, of the distances from $v$ to all other vertices of $G$.…

To a subshift over a finite alphabet, one can naturally associate an infinite family of finite graphs, called its Rauzy graphs. We show that for a subshift of subexponential complexity the Rauzy graphs converge to the line $\mathbf{Z}$ in…

The neighborhood complex $\N(G)$ of a graph $G$ were introduced by L. Lov{\'a}sz in his proof of Kneser conjecture. He proved that for any graph $G$, \begin{align} \label{abstract} \chi(G) \geq conn(\N(G))+3. \end{align} In this article we…

Let $X$ be a compact smooth manifold with boundary. The paper deals with contact $1$-forms $\beta$ on $X$, whose Reeb vector fields $v_\beta$ admit Lyapunov functions $f$. We tackle the question: how to recover $X$ and $\beta$ from the…

We study random walks on $\mathbb{Z}$ which have a linear (or almost linear) drift towards 0 in a range around 0. This drift leads to a metastable Gaussian distribution centered at zero. We give specific, fast growing, time windows where we…

Let $G$ be a finite group. The intersection graph of $G$ is a graph whose vertex set is the set of all proper non-trivial subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K \neq \{e\}$, where $e$ is…

We develop a group-theoretic approach to the Shannon capacity problem. Using this approach we extend and recover, in a structured and unified manner, various families of previously known lower bounds on the Shannon capacity. Bohman (2003)…

We derive a functional central limit theorem for the excursion of a random walk conditioned on sweeping a prescribed geometric area. We assume that the increments of the random walk are integer-valued, centered, with a third moment equal to…

Turn the set of permutations of $n$ objects into a graph $G_n$ by connecting two permutations that differ by one transposition, and let $\sigma_t$ be the simple random walk on this graph. In a previous paper, Berestycki and Durrett [In…

We show that if $X$ is an indecomposable $PD_3$-complex and $\pi_1(X) is the fundamental group of a reduced finite graph of finite groups but is not virtually cyclic then $X$ is orientable, the underlying graph is a tree, all the edge…

We show that the cop number of directed and undirected Cayley graphs on abelian groups has an upper bound of the form of $O(\sqrt{n})$, where $n$ is the number of vertices, by introducing a refined inductive method. With our method, we…

Let $G$ be a finite connected simple graph with a chosen orientation of its edges. For the edge potential $\psi(t)=\cosh t-1,$ we minimize $\sum_{e\in E^\to}\psi(z_e)$ over each affine class $\omega+dC^0(G)\subset C^1(G)$. The minimizer is…

Consider any random graph model where potential edges appear independently, with possibly different probabilities, and assume that the minimum expected degree is omega(ln n). We prove that the adjacency matrix and the Laplacian of that…

We introduce new techniques for studying boundary dynamics of CAT(0) groups. For a group $G$ acting geometrically on a CAT(0) space $X$ we show there is a flat $F\subset X$ of maximal dimension whose boundary sphere intersects every minimal…

For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set…