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This article is mainly concerned with computations of some algebraic invariants of quotient rings of edge ideals of perfect [h,d]-ary trees and unicyclic graphs. We compute exact values of depth and Stanley depth and consequently projective dimension for above mentioned quotient rings, except for the one special case of unicyclic graph for which best possible bounds of Stanley depth are given.

In this paper, we study depth and Stanley depth of the quotient rings of the edge ideals associated to triangular and multi triangular snake and triangular and multi triangular ouroboros snake graphs. In some cases, we find exact values, otherwise, we find tight bounds. We also find lower bounds for the edge ideals of triangular and multi triangular snake and ouroboros snake graphs and prove a conjecture of Herzog for all edge ideals we considered.

Introduction ‎‎ The study of the minimal free resolution of homogenous ideals and their powers is an interesting and active area of research in commutative algebra. Two invariants which measure the complexity of the minimal free resolutions are the so-called “projective dimension” and “Castelnuovo-Mumford regularity” (or simply, regularity) of the given ideal. Projective dimension determines the length of the minimal free resolution, while regularity is defined in terms of the degree of the entries of the matrices defining the differentials of the resolution. The focus of this paper is on the regularity of powers of ideals. One of the main results in this area is obtained by Cutkosky, Herzog, Trung [7], and independently Kodiyalam [8]. They proved that for a homogenous ideal I in a polynomial ring, the regularity of powers of I is asymptotically linear. In other words, there exist integers a(I) and b(I) such that regIs=aIs+b(I) for every integer s≫0. It is known that a(I) is bounded above by the maximum degree of generators of I. Moreover, if I is generated in a single degree d, then aI=d. But in general, it is not so much known about b(I) even if I is monomial ideal. However, when I is a quadratic squarefree monomial ideal, Alilooee, Banerjee, Beyarslan and Ha [9] conjectured that bI≤regI-2. In fact, they conjectured that the inequality regIs≤2s+regI-2 holds for any integer s≥1, when I is quadratic squarefree monomial ideal. Recently, Benerjee and Nevo [10] proved this conjecture for s=2. In this paper, we provide an alternative proof for their result. While the proof in [10] is based on topological arguments and using the Hochster’s formula, our proof is purely algebraic. Material and methods To every simple graph G one associates a quadratic squarefree monomial ideal, called its edge ideal, whose generators are the quadratic squarefree monomials corresponding to the edges of G. This association is a strong tool in the study of squarefree monomial ideals, as one can use the combinatorial properties of G to obtain information about the algebraic and homological properties of it, s edge ideal. One of the main results for bounding the regularity of powers of edge ideals is obtained by Benerjee [1]. He proved that the regularity of the sth power of an edge ideal I(G) has an upper bound which is defined in terms of the regularity of its (s-1)th power and the regularity of the edge ideal of some graphs which are explicitly determined by the structure of the G. This result has an essential role in our proof. Results and discussion The main result of this paper states that for every graph G, with edge ideal I(G), we have regIG2≤reg(IG)+2. In order to prove this inequality, using the aforementioned result of Benerjee, we must prove that the regularity of certain colon ideals are at most regIG. To achieve this goal, we use a short exact sequence argument which allows us to estimate the regularity of the colon ideas in terms of the regularity of edge ideal of some graphs which are strictly smaller than G. Conclusion The following conclusions were drawn from this research. The conjectured inequality of Alilooee, Banerjee, Beyarslan and Ha [9] is true for the case of s=2. It is known that for every graph G with edge ideal I(G) and induced matching number ν(G), we have 2s+νG-1≤reg(IGs), for every integer s≥1. Thus, our result implies that if regIG=νG+1, then regIG2=νG+3. The short exact sequence argument is a common technique in the study of regularity of monomial ideals. So, it would be interesting if one can prove the above-mentioned conjecture, using this method, even in the case of s=3 .

In this paper, we find values of depth, Stanley depth, and projective dimension of the quotient rings of the edge ideals associated with r-fold bristled graphs of ladder graphs, circular ladder graphs, some king’s graphs, and circular king’s graphs.

In this paper we study depth and Stanley depth of the quotient rings of the edge ideals associated with the corona product of some classes of graphs with arbitrary non-trivial connected graph G. These classes include caterpillar, firecracker and some newly defined unicyclic graphs. We compute formulae for the values of depth that depend on the depth of the quotient ring of the edge ideal I(G). We also compute values of depth and Stanley depth of the quotient rings associated with some classes of edge ideals of caterpillar graphs and prove that both of these invariants are equal for these classes of graphs.

Let k be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a path on n vertices. We show that both depth and Stanley depth have the same values and can be given in terms of k and n. If n≣0, k + 1, k + 2, . . . , 2k(mod(2k + 1)), then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a cycle on n vertices and tight bounds otherwise, in terms of n and k. We also compute lower bounds for the Stanley depth of the edge ideals associated to the kth power of a path and a cycle and prove a conjecture of Herzog for these ideals.

Minimal free resolutions of a finitely generated module over a polynomial ring S=k[x], with variables x={x1,…,xn} and a field k have been extensively studied. Almost all the results in the literature about minimal free resolutions give their Betti numbers, that is, the ranks of the free modules in the resolution at each degree. Several techniques have been developed to compute Betti numbers, making this a manageable problem in many cases. However, a description of the differentials in the resolution is rarely given, as this turns out to be a more difficult problem. The main purpose of this article is to give a criterion to check when a graded free complex of an S-module is exact. Unlike previous similar criteria, this one allows us to give a description of the differentials using the combinatorics of the S-module. The criterion is given in terms of the Betti numbers of the resolutions in each degree and the set of columns of the matrix representation of the differentials. In the last section, and with the aim of illustrating how to use the criterion, we apply it to one of the first better-understood cases, the edge ideal of the complete graph. However, this criterion can be used to give an explicit description of the differentials of a resolution of several monomial ideals such as the duplication of an ideal, the edge ideal of a cograph, etc.

Let R = S/I where S = k[T_1, . . . , T_n] and I is a homogeneous ideal in S. The acyclic closure R of k over R is a DG algebra resolution obtained by means of Tate’s process of adjoining variables to kill cycles. In a similar way one can obtain the minimal model S[X], a DG algebra resolution of R over S. By a theorem of Avramov there is a tight connection between these two resolutions. In this paper we study these two resolutions when I is the edge ideal of a path or a cycle. We determine the behavior of the deviations ε_i (R), which are the number of variables in R in homological degree i. We apply our results to the study of the k-algebra structure of the Koszul homology of R.

We present an algebraic algorithm to detect the existence of and to list all indecomposable even circuits in a given graph. We also discuss an application of our work to the study of directed cycles in digraphs.

In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s conjecture, and their counterexample is a quotient of squarefree monomial ideals. On the other hand, there is evidence showing that Stanley’s inequality can be true for high powers of monomial ideals. In this survey article, we collect the recent results in this direction. More precisely, we investigate the Stanley depth of powers, integral closure of powers, and symbolic powers of monomial ideals.

We show that Cohen-Macaulay and (S 2 ) properties are equivalent for the second power of an edge ideal. We give an example of a Gorenstein squarefree monomial ideal I such that S / I 2 satisfies the Serre condition (S 2 ), but is not Cohen-Macaulay.

Stanley depth is a geometric invariant of the module and is related to an algebraic invariant called depth of the module. We compute Stanley depth of the quotient of edge ideals associated with some familiar families of wheel-related graphs. In particular, we establish general closed formulas for Stanley depth of quotient of edge ideals associated with the m t h -power of a wheel graph, for m ≥ 3 , gear graphs and anti-web gear graphs.

We compute the Betti numbers for all the powers of initial and final lexsegment edge ideals. For the powers of the edge ideal of an anti–d−path, we prove that they have linear quotients and we characterize the normally torsion–free ideals. We determine a class of non–squarefree ideals, arising from some particular graphs, which are normally torsion–free.

Cohen-Macaulayness of bipartite graphs is investigated by several mathematicians and has been characterized combinatorially. In this paper, we give some different combinatorial conditions for a bipartite graph which are equivalent to Cohen-Macaulayness of the graph. We prove that a bipartite graph is Cohen-Macaulay if and only if it is well covered and has a unique perfect matching. We also provide a fast algorithm to check Cohen-Macaulayness of a given bipartite graph. [ABSTRACT FROM AUTHOR]

Let K be a field and S = K[x1, . . .,xn] be the polynomial ring in n variables over the field K. Let G be a forest with p connected components G1, . . .,Gp and let I = I(G) be its edge ideal in S. Suppose that di is the diameter of Gi, 1 ≤ i ≤ p, and consider d = max{di | 1 ≤ i ≤ p}. Morey has shown that for every t ≥ 1, the quantity max{...} is a lower bound for depth(S/It). In this paper, we show that for every t ≥ 1, the mentioned quantity is also a lower bound for sdepth(S/It). By combining this inequality with Burch's inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem. [ABSTRACT FROM AUTHOR]

We prove that the binomial arithmetical rank of the edge ideal of a forest coincides with its big height. [ABSTRACT FROM AUTHOR]