Abstract
Graph theory is widely used in power network analysis, complex network, and engineering calculation. Stanley depth is a geometric invariant of the module which is closely related to an algebraic invariant called depth of the module. At first, we propose a generalization of classical gear graph and extended level gear graph and then establish general closed formulas for the sharp bounds of Stanley depth of quotient of edge ideals associated to extended -level gear graph. We establish general closed formulas for the sharp bounds of Stanley depth of quotient of edge ideals associated to extended gear graph. We recover these bounds for Stanley depth of the quotient of edge ideals associated to classical gear graphs.
1. Introduction
Stanley is popular for the contribution he made to combinatorial aspects of algebra and geometry. His important work on simplicial complexes occupies central place in combinatorial commutative algebra. Roughly speaking, the two of the most important such complexes are partitionable and Cohen–Macauly complexes. Stanley’s conjecture that “whether all Cohen–Macaulay complexes are partitionable” attracted the attentions of algebraists as well as people working in geometry and combinatorics. He introduced the notion of Stanley depth in 1982 in [1]. The Stanley conjecture is interesting in the sense that it compares a combinatorial invariant with a homological invariant of module. Herzog and Popescu studied this conjecture in 2006 and then a number of articles have been published in which this conjecture has been discussed for different cases. In general, Dual proved that this conjecture does not hold, [2]. Herzog et al. gave the first simple method to compute Stanley depth for the module , where is of the form , where is a monomial ideal of , in [3]. Rinaldo transformed this into an algorithm in [4]. This algorithm has been effectively implemented in CoCoA [5]. In [6], Rauf proved some important basic results regarding depth and Stanley depth of multigraded module.
Stanley depth of the edge ideal associated to some graphs has been subject matter of recent results. Iqbal et al. computed depth and Stanley depth of square paths and cycles in [7]. Stanley depth of general powers of path ideals is computed in [8]. In [9], Cimpoeas proved several inequalities regarding computation of Stanely depth. Let denote the edge ideal associated to path on vertices. Then, bounds and values for Stanley depth and depth for module are given in [7, 8, 10]. The Stanley depth of the quotient of wheel graph and square wheel graph is computed in [11]. Cimpoeas, in [12], discussed the Stanley depth of modules having small number of generators. Authors, in [10, 12], computed depth and sdepth for the quotient of path ideal of length 3 of Z-cyclic graph along with some precise formulas for Stanley depth when (mod 4) and tight bounds when (mod 4). In [13], the authors found an upper bound for the Stanley depth of , where is polynomial ring and are monomial primary ideals, and also, showed that this conjecture holds for and , where are monomial irreducible ideals. In [14], authors found upper bound for sdepth of -partite complete graph and -uniform complete bipartite hypergraph. In [15], authors found upper bounds for the sdepth of edge ideals of complete graph and complete bipartite graph when some variables are added as generators. In [16], authors gave the lower and upper bound for the cyclic module associated with the complete -partite hypergraph. Stanley depth of the quotient of wheel graph and square wheel graph is computed in [11]. In [17], Biro et al. discussed interval partition associated with partially ordered set which is required to find Stanley depth and also how this interval partition is used to find out Stanley depth. Recently, in [18], Liu et al. computed Stanley depth of edge ideals of some wheel-related graphs. For the friendly introduction to commutative algebra, we refer the reader to [19].
In this study, at first, we propose a natural extension to classical gear graphs. Then, we obtain bounds for the Stanley depth of the quotient of edge ideal of this extension. In the end, we also recover these bounds for classical gear graph. We set , , and , where denotes the unique minimal set of monomial generators of the monomial ideal for the rest of this study.
2. Definitions and Notations
Let be a simple connected graph with edge set and vertex set . Let be the set of vertices; then, the edge ideal associated to is the square free monomial ideal of , that is,
Let be a polynomial ring in variables, where is a field and be finitely generated -module, which is also -graded. For a homogeneous element and a subset , de is called Stanley depth of .
Stanley [1] in his article “Linear Diophantine equation and local cohomology” stated this conjecture, in a generalized way, as follows.
Let be a finitely generated -graded -algebra , and let be a finitely generated -graded -module. Then, there exist finitely many subalgebras of , each generated by algebraically independent -homogeneous elements of , and there exist -homogeneous elements of such thatwhere dim depth(M), for all , and is a free -module.
Definition 1 (-level gear graph). Gear graphs are extensions of wheel graphs. We may describe the gear, , as with an additional vertex inserted between each pair of vertices on the cycle. In other words, the gear consists of a cycle on vertices, with every other vertex on the cycle adjacent to an additional st vertex called the center. The -level gear graph, , consists of cycles of lengths , respectively, where are even, for all .
The edge ideal is given asThe graph of is given in Figure 1.
Note: the -subspace of is generated by , where is a monomial in . The space is -graded -subspace and is called of dimension if it is a free -module, where denotes the cardinality of . A decomposition of as a finite direct sum of -graded -vector spaces is called Stanley decomposition:where each is a Stanley space.
The number,is called Stanley depth of , where is a decomposition of and the quantity isThe following main lemmas will be crucial for the proof of main results.
Lemma 1 (see [2]). Letbe a short exact sequence of -graded -modules. Then,
Lemma 2 (see [16]). Let be a square-free monomial ideal such that ; let , such that , for all . Then, .
3. Example
We illustrate the above notions with the help of a simple example. We also present graphical representations used to construct a particular Stanley decomposition, explained briefly in [3, 20, 21].
Let and be an ideal. It is well known that a module can have many Stanley decompositions, see Figure 2.
We use -axis for variable and -axis for . Then, the decomposition of given in Figure 2 is given as
So, it is clear that , as it points to a green point in Figure 2. As a result, we see that . We have to consider all partitions of this ideal. It is rather hard to find this for all possible partitions as number of decompositions of ideals can be infinite. Now, we give another example of Stanley decomposition of an ideal in polynomial rings with three variables. Let and be an ideal. Then, the Stanley decomposition of is given as
The next two figures show two different views of decomposition of the above ideal. These Figures 3 and 4 are the decomposition of ideals in .
Figure 4 is still another version of the same decomposition of Ideal . Here, colored part corresponds to the ideal .
It is clear that the sdepth of this particular decomposition is 2 as minimal spaces are planes as shown in the figures. So, we can conclude that sdepth . To find it, we have to try all other decompositions of this ideal.
4. Main Results
In this section, we are going to compute our main results.
Theorem 1. Let be the edge ideal of -level gear graph, where each and is even, for all . Then,
Proof. Using equation (8), along with the short exact sequence given bywhere and . We obtain. So, sdepth and , where , , and , for . So, we get . Now, by using Theorem 3.1 of [2], sdepth sdepth, where , and , for . So, by using Proposition 1.8 of [4], sdepth , .
We get sdepth . Now, by applying Lemma 1 on the above short exact sequence, we obtainHowever, we can see that sdepth sdepth . So, we get sdepth sdepth . Hence, sdepth .
Theorem 2. Let be the edge ideal of -level gear graph, where each and is even, for all . Then,where for .
Proof. By Theorem 3.1, we show that sdepth , so we only need to show the other inequality.
We will prove this by using Lemma 2.
Let be the edge ideal of -level gear graph, where each and is even, for .
Now, let , for , where is the quotient and is the remainder of when it is divided by 3.
Now, we takeThis monomial can also be written asIn the above monomial, if , then , otherwise 1.
Then, we can see that , but , for all . So, we have by Lemma 2
the number of appearing in the product in .
Now, we have to count the number of such that belongs to .
We can easily see that there are number of in .
However, if (mod 3), then the term , which is already there in , so we do not need to count those terms again.
This means that there are total number of in , where is the number of that are multiples of 3 from . We can write .
Also, we can see thatHence, sdepth .
Applying the results obtained, we arrive at the following.
Corollary 1. Let be the edge ideal of gear graph, where each and is even, Then,
Corollary 2. Let be the edge ideal of gear graph, where each and is even. Then,where .
5. Examples
Now, we give an example of our main result.
Example 1. Let be the edge ideal of 3-level gear graph. Then,We haveConsider the short exact sequence:where and . So, we have , , and sdepth . Also, where , , and . We get . Now, by using Theorem 3.1 of [2], sdepth , where , and . So, by using Proposition 1.8 of [4], sdepth ,and sdepth . We get sdepth . Now, by applying Lemma 1 on the above short exact sequence, we obtainHowever, we can see that sdepth sdepth . So, we get sdepth sdepth . Hence, sdepth . Now, we have to find upper bound of sdepth . We will prove this by using Lemma 2.2. Now, we takeWe can easily see that , but , for all . So, we have by Lemma 2.2, the number of appearing in the product in . Hence, sdepth .
6. Conclusions and Open Problems
In this study, we have computed the Stanley depth of the quotient of edge ideals associated to some familiar families of wheel-related graphs. In particular, we establish general closed formulas of bounds for the Stanely depth of quotient of edge ideals associated to the extended Gear graphs and arrive at the following main results.
Theorem 3. Let be the edge ideal of -level gear graph, where each and is even, for all . Then,
Theorem 4. Let be the edge ideal of -level gear graph, where each and is even, for all . Then,where for .
At the same time, we pose natural open problems about the exact values of Stanely depth of the quotient of edge ideal associated to power of gear and antiweb gear graphs.
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Mobeen Munir conceived the idea, Jia Bao Liu, Athar, Guiling Zengand, and Raheel wrote the manuscript, whereas the computational part was done by Raheel Farooki. All authors read and approved the final manuscript.
Acknowledgments
This research is supported by Projects of Natural Science Research of Universities in Anhui Province (KJ2021A1327).