Related papers: Algebraically rigid simplicial complexes and graph…

We introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and we define their associated face rings. Abstract simplicial complexes and rational fans can be…

We characterize twisted right-angled Artin groups (T-RAAGs) that are subgroup separable using only their defining mixed graphs: such a group is subgroup separable if and only if the underlying simplicial graph contains neither induced paths…

A strongly regular graph is called trivial if it or its complement is a union of disjoint cliques. We prove that every infinite family of nontrivial strongly regular graphs is quasi-random in the sense of Chung, Graham and Wilson.

We examine indivisibility for classes of graphs. We show that the class of hereditarily $\alpha$-sparse graphs is indivisible if and only if $\alpha > 2$. Additionally, we show that the following classes of graphs are indivisible: perfect…

Given an adaptable separated graph, we construct an associated groupoid and explore its type semigroup. Specifically, we first attach to each adaptable separated graph a corresponding semigroup, which we prove is an $E^*$-unitary inverse…

We consider simplicial complexes admitting a free action by an abelian group. Specifically, we establish a refinement of the classic result of Hochster describing the local cohomology modules of the associated Stanley--Reisner ring,…

A graph $G$ is a non-separating planar graph if there is a drawing $D$ of $G$ on the plane such that (1) no two edges cross each other in $D$ and (2) for any cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in $D$.…

By arithmeticity and superrigidity, a commensurability class of lattices in a higher rank Lie group is defined by a unique algebraic group over a unique number subfield of $\mathbb{R}$ or $\mathbb{C}$. We prove an adelic version of…

Let $\Delta$ be simplicial complex and let $k[\Delta]$ denote the Stanley--Reisner ring corresponding to $\Delta$. Suppose that $k[\Delta]$ has a pure free resolution. Then we describe the Betti numbers and the Hilbert--Samuel multiplicity…

We introduce $k$-robust clique complexes, a family of simplicial complexes that generalizes the traditional clique complex. Here, a subset of vertices forms a simplex provided it does not contain an independent set of size $k$. We…

We classify the finite connected simple graphs whose edge rings are strongly Koszul. From the classification, it follows that if the edge ring is strongly Koszul, then its toric ideal possesses a quadratic Gr\"obner basis.

We uncover a connection between two seemingly unrelated notions: lettericity, from structural graph theory, and geometric griddability, from the world of permutation patterns. Both of these notions capture important structural properties of…

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial finite dimensional irreducible rational $KG$-module.…

We prove the stable rationality of almost simple algebraic groups, the connected components of the Dynkin diagram of anisotropic kernel of which contain at most two vertices. The (stable) rationality of many isotropic almost simple groups…

Let $\mathbb{A}$ be an annulus in the plane $\mathbb R^2$ and $g:\mathbb{A}\rightarrow \mathbb{A}$ be a boundary components preserving homeomorphism which is distal and has no periodic points. In \cite{SXY}, the authors show that there is a…

The paper studies the structure of restricted Leibniz algebras. More specifically speaking, we first give the equivalent definition of restricted Leibniz algebras, which is by far more tractable than that of a restricted Leibniz algebras in…

Let $G$ be a finite simple graph. The line graph $L(G)$ represents the adjacencies between edges of $G$. We define first the line simplicial complex $\Delta_L(G)$ of $G$ containing Gallai and anti-Gallai simplicial complexes…

A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in R^d with respect to the classical l^p norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and…

We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature…

Let $K$ be a complete non-trivially valued non-Archimedean field. Given an algebraic group over $K$ on which every regular function is constant, any rigid analytic function is shown to be constant too. It follows that an algebraic group…