# Stanley Depth of the Edge Ideal of Extended Gear Networks and Application in Circuit Analysis

Authors: Guiling Zeng; Muhammad Mobeen Munir; Raheel Farooki; Muhammad Athar; Jia Bao Liu
Journal: Journal of Mathematics
 (2022)

---

## 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 extendedm−level gear graph and then establish general closed formulas for the sharp bounds of Stanley depth of quotient of edge ideals associated to extended m-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.


---

## Body

## 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 M, where M is of the form M=I/J, where J⊂I⊂S is a monomial ideal of S, 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 IPn denote the edge ideal associated to path on n vertices. Then, bounds and values for Stanley depth and depth for module S/IPn 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 n Zn-cyclic graph along with some precise formulas for Stanley depth when n=0,3 (mod 4) and tight bounds when n=1,2 (mod 4). In [13], the authors found an upper bound for the Stanley depth of S/Q∩Q′, where S is polynomial ring and Q,Q′ are monomial primary ideals, and also, showed that this conjecture holds for Q1∩Q2 and S/Q1∩Q2∩Q3, where Q1,Q2,Q3 are monomial irreducible ideals. In [14], authors found upper bound for sdepth of k-partite complete graph and s-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 k-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 setSm=Kv1,v2,v3,…,vm, suppu=i:vi|u, and suppJ=i:vi|tfor somet∈GJ, where GJ denotes the unique minimal set of monomial generators of the monomial ideal J for the rest of this study.


## 2. Definitions and Notations

LetG=VG,EG be a simple connected graph with edge set EG and vertex set VG. Let VG=1,2,…,n be the set of vertices; then, the edge ideal associated to G is the square free monomial ideal of S, that is,(1)IG=vlvk:l,k∈EG.LetS=Kv1,v2,…,vn be a polynomial ring in n variables, where K is a field and M be finitely generated S-module, which is also ℤn-graded. For a homogeneous element q∈M and a subset Z⊆v1,v2,…,vn, qKZ de is called Stanley depth of M.Stanley [1] in his article “Linear Diophantine equation and local cohomology” stated this conjecture, in a generalized way, as follows.LetR be a finitely generated ℕn-graded K-algebra R0=K, and let M be a finitely generated ℤn-graded R-module. Then, there exist finitely many subalgebras S1,…,St of R, each generated by algebraically independent ℕn-homogeneous elements of R, and there exist ℤn-homogeneous elements η1,…,ηt of M such that(2)M=⊕i=1tηiSi,where dim Si≥ depth(M), for all i, and ηiSi is a free Si-module.Definition 1 (m-level gear graph).
Gear graphs are extensions of wheel graphs. We may describe then−gear, Gn, as Wn with an additional vertex inserted between each pair of vertices on the cycle. In other words, the n−gear consists of a cycle on 2n vertices, with every other vertex on the cycle adjacent to an additional 2n+1 st vertex called the center. The m-level gear graph, Gn1,n2,…,nm, consists of m cycles of lengths n1,n2,…,nm, respectively, where ni≥4 are even, for all 1≤i≤m.
The edge idealIGn1,n2,…,nm is given as(3)IGn1,n2,…,nm=x1x2,x2x3,x3x4,…,xn1−1xn1,xn1x1,xn1+1xn1+2,xn1+2xn1+3,…,xn1+n2−1,xn1+n2,xn1+n2xn1+1,xn1+n2+1xn1+n2+2,xn1+n2+2xn1+n2+3,…,xn1+n2+n3−1xn1+n2+n3,xn1+n2+n3xn1+n2+1,…,xn1+n2+n3+⋯+nm−1+1xn1+n2+n3+⋯+nm−1+2,xn1+n2+n3+⋯+nm−1+2xn1+n2+n3+⋯+nm−1+3,…,xn1+n2+n3+⋯+nm−1xn1+n2+n3+⋯+nm,xn1+n2+n3+⋯+nmxn1+n2+n3+⋯+nm−1+1,x1xn1+n2+n3+⋯+nm+1,x3xn1+n2+n3+⋯+nm+1,x5xn1+n2+n3+⋯+nm+1,…,xn1−1xn1+n2+n3+⋯+nm+1,xn1+1xn1+n2+n3+⋯+nm+1,…,xn1+n2−1xn1+n2+n3+⋯+nm+1,xn1+n2+1xn1+n2+n3+⋯+nm+1,…,xn1+n2+n3+⋯+nm−1xn1+n2+n3+⋯+nm+1.
The graph ofIG6,8,10 is given in Figure 1.
Note: theK-subspace of M is generated by qv, where v is a monomial in KZ. The space qKZ is ℤn-graded K-subspace and is called Stanley space of dimension Z if it is a free KZ-module, where Z denotes the cardinality of Z. A decomposition of M as a finite direct sum of ℤn-graded K-vector spaces is called Stanley decomposition:(4)D:M=⊕j=1sqjKZj,where each qjKZj is a Stanley space.
The number,(5)sdepthD=minZj:j=1,2,3,…,s,is called Stanley depth of D, where D is a decomposition of M and the quantity is(6)sdepthM=maxsdepthD|,Dis a Stanley decomposition ofM.
The following main lemmas will be crucial for the proof of main results.Figure 1
Graph ofIG6,8,10.Lemma 1 (see [2]).
Let(7)0⟶U⟶V⟶W⟶0be a short exact sequence of ℤn-graded S-modules. Then,(8)sdepthV≥minsdepthU,sdepthW.Lemma 2 (see [16]).
LetJ⊂S be a square-free monomial ideal such that suppJ=n; let w=xj1xj2xj3…xjt∈S/J, such that xiw∈I, for all i∈n/suppw. Then, sdepthS/J≤t.


## 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].LetS=Kv1,v2 and I=v12,v22 be an ideal. It is well known that a module can have many Stanley decompositions, see Figure 2.Figure 2
DecompositionD1.We usex-axis for variable v1 and y-axis for v2. Then, the decomposition D1 of I given in Figure 2 is given as(9)D1:I=v12Kv1⊕v1v22Kv2⊕v23Kv2⊕v23K⊕v12v2Kv1,v2.So, it is clear thatsdepthD1:I=0, as it points to a green point in Figure 2. As a result, we see that sdepth≥0. 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 S=Kv1,v2,v3 and I=v1,v2,v3 be an ideal. Then, the Stanley decomposition of I is given as(10)D:I=v1v2Kv1,v2⊕v1v3Kv1,v3⊕v2v3Kv2,v3⊕v1v2v3Kv1,v2,v3.The next two figures show two different views of decomposition of the above ideal. These Figures3 and 4 are the decomposition of ideals in 3−D.Figure 3
DecompositionD.Figure 4
DecompositionD.Figure4 is still another version of the same decomposition of Ideal I. Here, colored part corresponds to the ideal I.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 sdepthI≥2. 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.
LetIGn1,n2,…,nm be the edge ideal of m-level gear graph, where each ni≥4 and is even, for all 1≤i≤m. Then,(11)sdepthSIGn1,n2,…,nm≥∑k=1mnk−13.Proof.
Using equation (8), along with the short exact sequence given by(12)0⟶S/I′⟶S/IGn1,n2,…,nm⟶S/I″⟶0,where I′=IGn1,n2,…,nm:xn1+n2+n3+⋯+nm+1 and I′′=IGn1,n2,…,nm,xn1+n2+n3+⋯+nm+1. We obtain(13)I′=x1,x3,x5,…,xn1−1,xn1+1,…,xn1+n2−1,xn1+n2+1,…,xn1+n2+n3+⋯+nm−1,S/I′≅Kx2,x4,x6,…,xn1,xn1+2,…,xn1+n2,xn1+n2+2,…,xn1+n2+n3+⋯+nm,xn1+n2+n3+⋯+nm+1. So, sdepth S/I′≥n1+n2+n3+⋯+nm+1/2 and I′′=Jn1,Jn2,Jn3,…,Jnm,xn1+n2+n3+⋯+nm+1, where Jn1=x1x2,x2x3,x3x4,…,xn1−1xn1,xn1x1, Jn2=xn1+1xn1+2,xn1+2xn1+3,…,xn1+n2−1xn1+n2,xn1+n2xn1+1, and Jni=x∑k=1i−1nk+1x∑k=1i−1nk+2,x∑k=1i−1nk+2x∑k=1i−1nk+3,…,x∑k=1i−1nk+ni−1x∑k=1i−1nk+ni,x∑k=1i−1nk+nix∑k=1i−1nk+1, for 3≤i≤m. So, we get S/IGn1,n2,…,nm,xn1+n2+n3+⋯+nm+1≅Sn1+n2+⋯+nm/Jn1,Jn2,Jn3,…,Jnm. Now, by using Theorem 3.1 of [2], sdepth S/IGn1,n2,…,nm,xn1+n2+n3+⋯+nm+1≥∑k=1m sdepthSnk/Jnk, where Sn1=Kx1,x2,x3,x4,…,xn1,Sn2=Kxn1+1,xn1+2,xn1+3,xn1+4,…,xn1+n2, and Sni=Kx∑k=1i−1nk+1,x∑k=1i−1nk+2,x∑k=1i−1nk+3,…,x∑k=1i−1nk+ni, for 3≤i≤m. So, by using Proposition 1.8 of [4], sdepth Sni/Jni≥ni−1/3, 1≤i≤m.
We get sdepthS/IGn1,n2,…,nm,xn1+n2+n3+⋯+nm+1≥∑k=1mnk−1/3. Now, by applying Lemma 1 on the above short exact sequence, we obtain(14)sdepthS/IGn1,n2,…,nm≥minsdepthS/IGn1,n2,…,nm:xn1+n2+n3+⋯+nm+1,sdepthS/IGn1,n2,…,nm,xn1+n2+n3+⋯+nm+1.
However, we can see that sdepthS/IGn1,n2,…,nm,xn1+n2+n3+⋯+nm+1≥ sdepth S/IGn1,n2,…,nm:xn1+n2+n3+⋯+nm+1. So, we get sdepth S/IGn1,n2,…,nm≥ sdepth S/IGn1,n2,…,nmxn1+n2+n3+⋯+nm+1. Hence, sdepth S/IGn1,n2,…,nm≥∑k=1mnk−1/3.Theorem 2.
LetIGn1,n2,…,nm be the edge ideal of m-level gear graph, where each ni≥4 and is even, for all 1≤i≤m. Then,(15)∑k=1mnk−13≤sdepthS/IGn1,n2,…,nm≤∑k=1mnk−sk3+∑k=1msk3,where sk≡nkmode3 for 1≤k≤m.Proof.
By Theorem 3.1, we show that sdepthS/IGn1,n2,…,nm≥∑k=1mnk−1/3, so we only need to show the other inequality.
We will prove this by using Lemma2.
LetIGn1,n2,…,nm be the edge ideal of m-level gear graph, where each ni≥4 and is even, for 1≤i≤m.
Now, letni=3qi+si, for 1≤i≤m, where qi is the quotient and si is the remainder of ni when it is divided by 3.
Now, we take(16)w=x30+1x31+1x32+1…x3q1−1+1x3q1+s1−u1,x3q1+s1+30+1x3q1+s1+31+1…x3q1+s1+3q2−1+1x3q1+s1+3q2+s2−u2,x3q1+s1+3q2+s2+30+1x3q1+s1+3q2+s2+31+1…,x3q1+s1+3q2+s2+3q3−1+1x3q1+s1+3q2+s2+3q3+s3−u3,∏k=0q4−1x∑t=133qt+st+3k+1x∑t=133qt+st+3q4+s4−u4,∏k=0q5−1x∑t=143qt+st+3k+1x∑t=143qt+st+3q5+s5−u5…,∏k=0qm−1x∑t=1m−13qt+st+3k+1x∑t=1m3qt+st+3qm+sm−um.
This monomial can also be written as(17)w=∏k=0q1−1x3k+1x3q1+s1−u1,∏k=0q2−1x3q1+s1+3k+1x3q1+s1+3q2+s2−u2,∏k=0q3−1x∑t=123qt+st+3k+1x∑t=123qt+st+3q3+s3−u3,⋮⋮⋮,∏k=0qm−1x∑t=1m−13qt+st+3k+1x∑t=1m−13qt+st+3qm+sm−um.
In the above monomial, ifni≡0mod3, then ui=2, otherwise 1.
Then, we can see thatw∈S/IGn1,n2,…,nm, but xiw∈IGn1,n2,…,nm, for all i∈n1+n2+n3+⋯+nm+1/suppw. So, we have by Lemma 2
sdepthS/IGn1,n2,…,nm≤ the number of xi appearing in the product in w.
Now, we have to count the number of suchxi that belongs to w.
We can easily see that there areq1+q2+q3+⋯+qm+m number of xi,s in w.
However, ifni≡0 (mod 3), then the term x∑t=1i−13qt+st+3qi+si−ui=x∑t=1i−13qt+st+3qi−1+1, which is already there in w, so we do not need to count those terms again.
This means that there are totalq1+q2+q3+⋯+qm+m−l number of xi,s in w, where l is the number of ni that are multiples of 3 from 1≤i≤m. We can write q1+q2+q3+⋯+qm+m−l=q1+q2+q3+⋯+qm+1+1+1+⋯+1︸m−l−times.
Also, we can see that(18)q1+q2+q3+⋯+qm+1+1+1+⋯+1︸m−l−times=∑k=1mnk−sk3+∑k=1msk3
Hence, sdepthS/IGn1,n2,…,nm≤∑k=1mnk−sk/3+∑k=1msk/3.Applying the results obtained, we arrive at the following.Corollary 1.
LetIGn be the edge ideal of gear graph, where each n≥4 and is even, Then,(19)sdepth sdepthS/IGn≥n−13Corollary 2.
LetIGn be the edge ideal of gear graph, where each n≥4 and is even. Then,(20)n−13≤sdepthS/IGn≤n−s3+s3,where s≡nmode3.


## 5. Examples

Now, we give an example of our main result.Example 1.
LetIG6,8,10 be the edge ideal of 3-level gear graph. Then,(21)7≤sdepthSIG6,8,10≤9.
We have(22)IG6,8,10=x1x2,x2x3,x3x4,…,x5x6,x6x1,x7x8,x8x9,…,x13x14,x14x7,x15x16,x16x17,…,x23x24,x24x15,x1x25,x3x25,x5x25,x7x25,…,x13x25,x15x25,…,x23x25.
Consider the short exact sequence:(23)0⟶SI′⟶SIG6,8,10⟶SI″⟶0,where I′=IG6,8,10:x25 and I′′=IG6,8,10,x25. So, we have I′=x1,x3,x5,x7,…,x13,x15,…,x23, S/I′≅Kx2,x4,x6,x8,…,x14,x16,x24,x25, and sdepth S/I′≥2/52=13. Also, I′′=Jn1,Jn2,Jn3,x25, where Jn1=x1x2,x2x3,x3x4,…,x5x6,x6x1, Jn2=x7x8,x8x9,…,x13x14,x14x7, and Jn3=x15x16,x16x17,…,x23x24,x24x15. We get S/IG6,8,10,x25≅S24/Jn1,Jn2,Jn3. Now, by using Theorem 3.1 of [2], sdepth S/IG6,8,10,x25≥∑k=13sdepthSnk/Jnk, where Sn1=Kx1,x2,x3,x4,…,x6,Sn2=Kx7,x8,x9,…,x14, and Sn3=Kx15,x16,x17,…,x24. So, by using Proposition 1.8 of [4], sdepth Sn1/Jn1≥6−1/3=2,(24)sdepthSn2/Jn2≥8−13=2,and sdepth Sn3/Jn3≥10−1/3=3. We get sdepth S/IG6,8,10,x25≥2+2+3=7. Now, by applying Lemma 1 on the above short exact sequence, we obtain(25)sdepthS/IG6,8,10≥minsdepthS/IG6,8,10:x25,sdepthS/IG6,8,10,x25.
However, we can see that sdepthS/IG6,8,10,x25≥ sdepth S/IG6,8,10:x25. So, we get sdepth S/IG6,8,10≥ sdepth S/IG6,8,10,x25. Hence, sdepth S/IG6,8,10≥2+2+3=7. Now, we have to find upper bound of sdepth S/IG6,8,10. We will prove this by using Lemma 2.2. Now, we take(26)w=x1x4x7x10x13x15x18x21x23.
We can easily see thatw∈S/IG6,8,10, but xiw∈IG6,8,10, for all i∈25/suppw. So, we have by Lemma 2.2, sdepthS/IG6,8,10≤ the number of xi appearing in the product in w. Hence, sdepth S/IG6,8,10≤2+3+4=9.


## 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.
LetIGn1,n2,…,nm be the edge ideal of m-level gear graph, where each ni≥4 and is even, for all 1≤i≤m. Then,(27)sdepthSIGn1,n2,…,nm≥∑k=1mnk−13.Theorem 4.
LetIGn1,n2,…,nm be the edge ideal of m-level gear graph, where each ni≥4 and is even, for all 1≤i≤m. Then,(28)∑k=1mnk−13≤sdepthSIGn1,n2,…,nm≤∑k=1mnk−sk3+∑k=1msk3,where sk≡nkmode3 for 1≤k≤m.At the same time, we pose natural open problems about the exact values of Stanely depth of the quotient of edge ideal associated tomth−power of gear and antiweb gear graphs.



---

*Source: 9706112-2022-06-06.xml*