Narrow Results

We study the properties of the rook complex ${ℛ}$ of a polyomino ${𝒫}$ seen as independence complex of a graph $G$, and the associated Stanley–Reisner ideal $I_{{ℛ}}$. In particular, we characterize the polyominoes ${𝒫}$ having a pure rook complex, and the ones whose Stanley–Reisner ideal has linear resolution. Furthermore, we prove that for a class of polyominoes the Castelnuovo–Mumford regularity of $I_{{ℛ}}$ coincides with the induced matching number of $G$.

This paper presents a complete characterization of an edge-weighted simple graph $G_{\omega }$, whose edge ideal $I(G_{\omega })$ is integrally closed. Furthermore, it is demonstrated that if $I(G_{\omega })$ is integrally closed and $G_{\omega }$ is an edge-weighted star graph, path, or cycle, then $I(G_{\omega })$ is normal.

Let ${𝒟}$ be a weighted oriented graph and let $I({𝒟})$ be its edge ideal. Under a natural condition that the underlying (undirected) graph of ${𝒟}$ contains a perfect matching consisting of leaves, we provide several equivalent conditions for the Cohen–Macaulayness of $I({𝒟})$. We also completely characterize the Cohen–Macaulayness of $I({𝒟})$ when the underlying graph of ${𝒟}$ is a bipartite graph. When $I({𝒟})$ fails to be Cohen–Macaulay, we give an instance where $I({𝒟})$ is shown to be sequentially Cohen–Macaulay.

Let $D$ be a weighted oriented graph with the underlying graph $G$ when vertices with non-trivial weights are sinks and $I(D),I(G)$ be the edge ideals corresponding to $D$ and $G,$ respectively. We give an explicit description of the symbolic powers of $I(D)$ using the concept of strong vertex covers. We show that the ordinary and symbolic powers of $I(D)$ and $I(G)$ behave in a similar way. We provide a description for symbolic powers and Waldschmidt constant of $I(D)$ for certain classes of weighted oriented graphs. When $D$ is a weighted oriented odd cycle, we compute $\mathrm{\text{reg}}(I{(D)}^{(s)}/I{(D)}^s)$ and prove $\mathrm{\text{reg}}I{(D)}^{(s)}\le \mathrm{\text{reg}}I{(D)}^s$ and show that equality holds when there is only one vertex with non-trivial weight.

Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which depends on the combinatorial structure and weights of the graph. We provide a result analogous to the unweighted, unoriented graph case, to show that when the associated simple graph has only trivial even closed walks, the toric ideal is the zero ideal. Moreover, we give necessary and sufficient conditions for the toric ideal of a weighted oriented graph to be generated by a single binomial and we describe the binomial in terms of the structure of the graph.

Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. We provide one method to find all the minimal generators of $I_{\subseteq C}$, where $C$ is a maximal strong vertex cover of $D$ and $I_{\subseteq C}$ is the intersections of irreducible ideals associated to the strong vertex covers contained in $C$. If $D^{\prime }$ is an induced digraph of $D$, under a certain condition on the strong vertex covers of $D^{\prime }$ and $D$, we show that ${I(D^{\prime })}^{(s)}\ne {I(D^{\prime })}^s$ for some $s\ge 2$ implies ${I(D)}^{(s)}\ne {I(D)}^s$. We provide the necessary and sufficient condition for the equality of ordinary and symbolic powers of edge ideal of the union of two naturally oriented paths with a common sink vertex. We characterize all the maximal strong vertex covers of $D$ such that at most one edge is oriented into each of its vertices and $w(x)\ge 2$ if ${deg}_D(x)\ge 2$ for all $x\in V(D)$. Finally, if $D$ is a weighted rooted tree with the degree of root is $1$ and $w(x)\ge 2$ when ${deg}_D(x)\ge 2$ for all $x\in V(D)$, we show that ${I(D)}^{(s)}={I(D)}^s$ for all $s\ge 2$.

For a simplicial complex Δ, we introduce a simplicial complex attached to Δ, called the expansion of Δ, which is a natural generalization of the notion of expansion in graph theory. We are interested in knowing how the properties of a simplicial complex and its Stanley–Reisner ring relate to those of its expansions. It is shown that taking expansion preserves vertex decomposable and shellable properties and in some cases Cohen–Macaulayness. Also it is proved that some homological invariants of Stanley–Reisner ring of a simplicial complex relate to those invariants in the Stanley–Reisner ring of its expansions.

We give a new combinatorial characterization of the big height of a squarefree monomial ideal leading to a new bound for the projective dimension of a monomial ideal.

We graph-theoretically characterize the class of graphs $G$ such that $I{(G)}^2$ are Buchsbaum.

Given a nontrivial homogeneous ideal $I\subseteq k[x_1,x_2,\dots ,x_d]$, a problem of great recent interest has been the comparison of the $r$th ordinary power of $I$ and the $m$th symbolic power $I^{(m)}$. This comparison has been undertaken directly via an exploration of which exponents $m$ and $r$ guarantee the subset containment $I^{(m)}\subseteq I^r$ and asymptotically via a computation of the resurgence $\rho (I)$, a number for which any $m/r>\rho (I)$ guarantees $I^{(m)}\subseteq I^r$. Recently, a third quantity, the symbolic defect, was introduced; as $I^t\subseteq I^{(t)}$, the symbolic defect is the minimal number of generators required to add to $I^t$ in order to get $I^{(t)}$. We consider these various means of comparison when $I$ is the edge ideal of certain graphs by describing an ideal $J$ for which $I^{(t)}=I^t+J$. When $I$ is the edge ideal of an odd cycle, our description of the structure of $I^{(t)}$ yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.

Let $G$ be a simple graph and $I(G)$ be its edge ideal. In this paper, we study the Castelnuovo–Mumford regularity of symbolic powers of edge ideals of join of graphs. As a consequence, we prove Minh’s conjecture for wheel graphs, complete multipartite graphs, and a subclass of co-chordal graphs. We obtain a class of graphs whose edge ideals have regularity three. By constructing graphs, we prove that the multiplicity of edge ideals of graphs is independent from the depth, dimension, regularity, and degree of $h$-polynomial. Also, we demonstrate that the depth of edge ideals of graphs is independent from the regularity and degree of $h$-polynomial by constructing graphs.

We introduce the concept of minimum edge cover for an induced subgraph in a graph. Let $G$ be a unicyclic graph with a unique odd cycle and $I=I(G)$ be its edge ideal. We compute the exact values of all symbolic defects of $I$ using the concept of minimum edge cover for an induced subgraph in a graph. We describe one method to find the quasi-polynomial associated with the symbolic defects of edge ideal $I$. We classify the class of unicyclic graphs when some power of maximal ideal annihilates $I^{(s)}/I^s$ for any fixed $s$. Also for those class of graphs, we compute the Hilbert function of the module $I^{(s)}/I^s$ for all $s.$

Let $H$ be a simple undirected graph. The family of all matchings of $H$ forms a simplicial complex called the matching complex of $H$. Here, we give a classification of all graphs with a Gorenstein matching complex. Also we study when the matching complex of $H$ is Cohen–Macaulay and, in certain classes of graphs, we fully characterize those graphs which have a Cohen–Macaulay matching complex. In particular, we characterize when the matching complex of a graph with girth at least five or a complete graph is Cohen–Macaulay.

We provide some exact formulas for the projective dimension and regularity of edge ideals associated to some vertex-weighted oriented cyclic graphs with a common vertex or edge. These formulas are functions in the weight of the vertices, and the numbers of edges and cycles. Some examples show that these formulas are related to direction selection and the assumption that $w(x)\ge 2$ for any vertex $x$ cannot be dropped.