Narrow Results

We compute the absorbing number of the edge ideals of special graphs such as complete graphs, path graphs, and cycle graphs. We also compute the absorbing number for other monomial ideals. In special cases, we show that the length of a prime filtration of a monomial ideal $I$ is an upper bound for its absorbing number.

The study of the edge ideal $I(D_G)$ of a weighted oriented graph $D_G$ with underlying graph $G$ started in the context of Reed–Muller type codes. We generalize some Cohen–Macaulay constructions for $I(D_G)$, which Villarreal gave for edge ideals of simple graphs. Our constructions can be used to produce large classes of Cohen–Macaulay weighted oriented edge ideals. We use these constructions to classify all the Cohen–Macaulay weighted oriented edge ideals, whose underlying graph is a cycle. We also show that $I(D_{C_n})$ is Cohen–Macaulay if and only if $I(D_{C_n})$ is unmixed and $I(C_n)$ is Cohen–Macaulay, where $C_n$ denotes the cycle of length $n$. Miller generalized the concept of Alexander dual ideals of square-free monomial ideals to arbitrary monomial ideals, and in that direction, we study the Alexander dual of $I(D_G)$ and its conditions to be Cohen–Macaulay.

For $k,n\ge 2$, a $(k,n)$-firecracker graph, denoted by $F_{k,n}$, is obtained by concatenation of $k$$n$-star graphs by connecting one leaf from each $n$-star graph. Given a finite simple graph $G$, one can associate a simplicial complex $\mathrm{\Delta }(G)$. In this paper, we compute all the graded Betti numbers of the edge ideal $I(F_{2,n})$ of the firecracker graph $F_{2,n}$ by using the combinatorial data associated with the simplicial complex $\mathrm{\Delta }(F_{2,n}).$ We also find the regularity of the ideals $I(F_{k,n})$. Further, using the domination parameters of the graphs, we explicitly compute the projective dimension of $I(F_{k,n})$.

For integers $k,n\ge 2$, the graph obtained by joining one leaf of each of $k$-copies of $n$-star graph with a single vertex distinct from all the vertices of $k$-copies of $n$-star graph is known as Banana tree graph, written as $B_{k,n}$. In this paper, we shall investigate certain homological invariants of edge rings of Banana tree graphs $B_{k,n}$ in terms of associated combinatorial data. More precisely, we shall show that the regularity and projective dimension of edge ring of $B_{k,n}$ are $k$ and $nk-k$, respectively, and deduce combinatorial formulae for computing all the nonzero graded Betti numbers of edge ring of $B_{2,n}$.

We introduce the concept of matching powers of monomial ideals. Let $I$ be a monomial ideal of $S=K[x_1,\dots ,x_n]$, where $K$ is a field. The $k$th matching power of $I$ is the monomial ideal $I^{[k]}$ generated by the products $u_1\cdots u_k$ where $u_1,\dots ,u_k$ is a sequence of support-disjoint monomials contained in $I$. This concept naturally generalizes the notion of squarefree powers of squarefree monomial ideals. We study the normalized depth function of matching powers of monomial ideals and provide bounds for the regularity and projective dimension of edge ideals of weighted oriented graphs. When $I$ is a non-quadratic edge ideal of a weighted oriented graph that contains no even cycles, we characterize when $I^{[k]}$ has a linear resolution.

In this paper, we study some algebraic invariants of t-spread ideals, $t\ge 1$, such as the projective dimension and the Castelnuovo–Mumford regularity, by means of well-known graded resolutions. We state upper bounds for these invariants and, furthermore, we identify a special class of t-spread ideals for which such bounds are optimal.

We show that for the edge ideals of the graphs consisting of one cycle or two cycles of any length connected through a vertex, the arithmetical rank equals the projective dimension of the corresponding quotient ring.

We study weighted graphs and their "edge ideals" which are ideals in polynomial rings that are defined in terms of the graphs. We provide combinatorial descriptions of m-irreducible decompositions for the edge ideal of a weighted graph in terms of the combinatorics of "weighted vertex covers". We use these, for instance, to say when these ideals are m-unmixed. We explicitly describe which weighted cycles, suspensions, and trees are unmixed and which ones are Cohen–Macaulay, and we prove that all weighted complete graphs are Cohen–Macaulay.

In this paper, we define and characterize the f-graphs. Also, we give a construction of f-graphs and importantly we show that the f-graphs obtained from this construction are Cohen–Macaulay.

We classify Cohen–Macaulay graphs of girth at least 5 and planar Gorenstein graphs of girth at least 4. Moreover, such graphs are also vertex decomposable.

We prove that, for the edge ideal of a graph whose cycles are pairwise vertex-disjoint, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes.

Let $I(G)$ be the edge ideal of a bicyclic graph $G$ with a dumbbell as the base graph. In this paper, we characterize the Castelnuovo–Mumford regularity of $I(G)$ in terms of the induced matching number of $G$. For the base case of this family of graphs, i.e. dumbbell graphs, we explicitly compute the induced matching number. Moreover, we prove that $\mathrm{\text{reg}}I{(G)}^q=2q+\mathrm{\text{reg}}I(G)-2$, for all $q\ge 1$, when $G$ is a dumbbell graph with a connecting path having no more than two vertices.

Let $G$ be a graph and $I=I(G)$ be its edge ideal. When $G$ is the clique sum of two different length odd cycles joined at single vertex then we give an explicit description of the symbolic powers of $I$ and compute the Waldschmidt constant. When $G$ is complete graph then we describe the generators of the symbolic powers of $I$ and compute the Waldschmidt constant and the resurgence of $I$. Moreover for complete graph we prove that the Castelnuovo–Mumford regularity of the symbolic powers and ordinary powers of the edge ideal coincide.

In this paper, we obtain a combinatorial formula for computing the Betti numbers in the linear strand of edge ideals of bipartite Kneser graphs. We deduce lower and upper bounds for regularity of powers of edge ideals of these graphs in terms of associated combinatorial data and show that the lower bound is attained in some cases. Also, we obtain bounds on the projective dimension of edge ideals of these graphs in terms of combinatorial data.

In this paper, we compute the projective dimension of the edge ideals of graphs consisting of some cycles and lines which are joint in a common vertex. Moreover, we show that for such graphs, the arithmetical rank equals the projective dimension. As an application, we can compute the arithmetical rank for some homogenous monomial ideals.

We prove that if G is a gap-free and chair-free simple graph, then the regularity of the edge ideal of G is no more than 3. If G is a gap-free and P₄-free graph, then it is a chair-free graph; furthermore, the complement of G is chordal, and thus the regularity of G is 2.