Narrow Results

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.

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 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.

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.

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.

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.

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.

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.