Related papers: Stanley-Reisner ideals of higher independence comp…

Let $G$ be a finite simple graph on the vertex set $V(G)$ and let $r$ be a positive integer. We consider the hypergraph $\mathrm{Con}_r(G)$ whose vertices are the vertices of $G$ and the (hyper)edges are all $A\subseteq V(G)$ such that…

We describe the Betti numbers of the edge ideals $I(G)$ of uniform hypergraphs $G$ such that $I(G)$ has linear graded free resolution. We give an algebraic equation system and some inequalities for the components of the $f$--vector of the…

Given a simple undirected graph $G$ there is a simplicial complex $\mathrm{Ind}(G)$, called the independence complex, whose faces correspond to the independent sets of $G$. This is a well studied concept because it provides a fertile ground…

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 $G$ be a finite simple graph on $n$ vertices and set $R=\Bbbk[x_1,\dots,x_n]$, with edge ideal $I(G)$ and cover ideal $J(G)$. We give an explicit description of the $h$-polynomial of $R/J(G)$, in a form that extends to the Alexander…

For $r\geq 1$, the $r$-independence complex of a graph $G$, denoted Ind$_r(G)$, is a simplicial complex whose faces are subsets $A \subseteq V(G)$ such that each component of the induced subgraph $G[A]$ has at most $r$ vertices. In this…

Using the concept of $d$-collapsibility from combinatorial topology, we define chordal simplicial complexes and show that their Stanley-Reisner ideals are componentwise linear. Our construction is inspired by and an extension of "chordal…

As a higher analogue of the edge ideal of a graph, we study the $t$-connected ideal $\operatorname{J}_{t}$. This is the monomial ideal generated by the connected subsets of size $t$. For chordal graphs, we show that $\operatorname{J}_{t}$…

The independence complex $\mathrm{Ind}(G)$ of a graph $G$ is the simplicial complex formed by its independent sets. This article introduces a deformation of the simplicial boundary map of $\mathrm{Ind}(G)$ that gives rise to a double…

Let $i_t(G)$ denote the number of independent sets of size $t$ in a graph $G$. Levit and Mandrescu have conjectured that for all bipartite $G$ the sequence $(i_t(G))_{t \geq 0}$ (the {\em independent set sequence} of $G$) is unimodal. We…

For a finite simple graph $G$ and an integer $r \ge 1$, the $r$-connected ideal $I_r(G)$ is the squarefree monomial ideal generated by the vertex sets of connected induced subgraphs of size $r+1$, extending the classical edge ideal. We…

We introduce a new family of pure simplicial complexes, called the $r$-co-connected complex of $G$ with respect to $A$, $\Sigma_r(A,G)$, where $r\geq 1$ is a natural number, $G$ is a simple graph, and $A$ is a subset of vertices.…

Several algebraic properties of a binomial edge ideal $J_G$ can be interpreted in terms of combinatorial properties of its associated graph $G$. In particular, the so-called cut sets of a graph $G$, special sets of vertices that disconnect…

Let $G$ be a graph and $r \ge 1$. A vertex subset is $r$-independent if every connected component of its induced subgraph has size at most $r$. The family of all such subsets forms a simplicial complex, the $r$-independence complex…

An independent set in a simple graph $G$ is a set of pairwise non-adjacent vertices in $G$. The independence polynomial of $G$, denoted by $I_G$ is defined as $1 + a_1 z + a_2 z^2+\cdots+a_d z^{d}$, where $a_i$ denotes the number of…

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex not in $S$ is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The independent domination number…

Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Alavi, Erd\H{o}s, Malde and Schwenk made the conjecture that if $G$ is a tree then the independent set sequence $\{i_t(G)\}_{t\geq 0}$ of $G$ is unimodal; Levit and…

The deficiency of a graph $G$, denoted by $\kd(G)$, is the number of vertices not saturated by a maximum matching. A bone $B_i$ is the tree obtained by attaching two pendent edges to each of the end vertices of a path $P_{i}$. The local…

An independent set of a graph $G$ is a vertex subset $I$ such that there is no edge joining any two vertices in $I$. Imagine that a token is placed on each vertex of an independent set of $G$. The $\mathsf{TS}$- ($\mathsf{TS}_k$-)…

When $I$ is the edge ideal of a graph $G$, we use combinatorial properities, particularly Property $P$ on connectivity of neighbors of an edge, to classify when a binomial sum of vertices is a regular element on $R/I(G)$. Under a mild…