
Let S be a polynomial ring over a field K and I be the edge ideal associated with the bristled graph of some four or five regular circulant graph. We discuss the depth, projective dimension, regularity and Stanley depth of S/I.
Citation: Ibad Ur Rehman, Mujahid Ullah Khan Afridi, Muhammad Ishaq, Asim Asiri, Aftab Hussain. Algebraic invariants of edge ideals of some bristled circulant graphs[J]. AIMS Mathematics, 2025, 10(5): 11330-11348. doi: 10.3934/math.2025515
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Let S be a polynomial ring over a field K and I be the edge ideal associated with the bristled graph of some four or five regular circulant graph. We discuss the depth, projective dimension, regularity and Stanley depth of S/I.
The classical Hermite-Hadamard (H-H) inequality, serving as a litmus test for convexity, formally establishes that if the function F:[ℑ1,ℑ2]→R is a convex function satisfying the essential containment relationship between its midpoint value and integral mean. Specifically, the following inequalities are satisfied:
F(ℑ1+ℑ22)≤1ℑ2−ℑ1∫ℑ2ℑ1F(σ)dσ≤12(F(ℑ1)+F(ℑ2)). |
Convexity inequalities have numerous applications in the study of different models from real-world applications [1,2]. Because the H-H inequality is of great significance in convex analysis, it is widely applied in various fields such as integral inequalities, information theory, and optimization theory. In recent years, it has been extended and generalized through various forms of convexity, such as log-convexity [3,4], harmonic convexity [5,6], h-convexity [7,8,9], convexity in q-calculus [10,11,12] and especially s-convexity [13]. Since 1994, s-convexity has been a significant development and widespread application and various generalizations and results regarding H-H inequalities related to s-convex mappings have been established in [14,15,16].
From another perspective, interval analysis provides an effective numerical tool for solving uncertain and nonlinear problems. Since the publication of the first monograph in 1966 [17], it has evolved into a distinct branch of mathematics, with applications spanning data mining, machine learning, and various other fields. A key focus has been interval-valued (IV) function inequalities. Recently, based on interval calculus and generalized convexity, some authors like Ali et al. [18,19,20], Budak et al. [21,22], Costa et al. [23,24,25], Du et al. [26,27], Khan et al.[28,29], Sarikaya et al. [30,31,32], and Zhao et al. [33,34,35] established the interval versions of the Chebyshev's inequality, Jensen's inequality, and H-H inequality. Furthermore, with the successive introduction of the bilateral Riemann-Liouville (R-L) fractional integral operators (left-hand and right-hand variants) for IV functions, the results related to inequalities for IV functions are more extensive and profound [36,37]. Especially, in 2023, Budak et al. [38] introduced a novel generalized integral to demonstrate the generalized H-H-type inclusion of IV convex functions.
Motivated by the above-mentioned literature, we have established some novel interval forms of H-H inequalities by using generalized fractional integral operators and combining them with IV s-convex functions. The classical convexity theory is extended to the generalized s-convex framework, some interesting theorems are proved, and the exact inequality representation of the product of two s-convex functions is given. Our findings not only extend the main conclusions in [36,38,39], but also provide new insights for the study of IV inequalities.
The organization of this work is outlined below: Section 2 introduces essential background concepts, while Section 3 establishes a series of H-H-type inequalities for IV s-convex mappings using generalized fractional integral operators. Section 4 gives the conclusion.
Let A denote a compact real interval, mathematically expressed as
[A]=[A_,¯A]={a∈R|A_≤a≤¯A}, |
where A_,¯A∈R satisfy A_≤¯A. If A_=¯A, the interval becomes degenerate. The intervals discussed in this paper are all non-degenerate intervals. An interval [A] is termed positive when A_>0, and negative if ¯A<0. The sets R−I, R+I, and RI, respectively, denote all negative, positive, and arbitrary real-valued intervals. Additionally, we adopt the partial ordering "⊇" defined by
[A_,¯A]⊇[B_,¯B]⟺A_≤B_and¯B≤¯A. |
For η∈R and A∈RI, then
ηA=η[A_,¯A]={[ηA_,η¯A],ifη≥0,[η¯A,ηA_],ifη<0. |
For arbitrary A,B∈RI, the following four arithmetic operations are given:
A+B=[A_,¯A]+[B_,¯B]=[A_+B_,¯A+¯B],A−B=[A_,¯A]−[B_,¯B]=[A_−B_,¯A−¯B],A⋅B=[min{A_B_,A_¯B,¯AB_,¯A¯B},max{A_B_,A_¯B,¯AB_,¯A¯B}],B/A=[min{B_/A_,B_/¯A,¯B/A_,¯B/¯A},max{B_/A_,B_/¯A,¯B/A_,¯B/¯A}],(0∉[A_,¯A]). |
For additional information on interval arithmetic, refer to [40].
Definition 2.1. [13] A function F:[ℑ1,ℑ2]→R is classified as s-convex if it satisfies the convexity condition:
F(ιℏ1+(1−ι)ℏ2)≤ιsF(ℏ1)+(1−ι)sF(ℏ2), | (2.1) |
for all ℏ1,ℏ2∈[ℑ1,ℑ2] and ι∈[0,1] with s∈(0,1].
In [41], Breckner introduced the IV s-convex functions.
Definition 2.2. [41] F:[ℑ1,ℑ2]→R+I is defined as an IV s-convex function if
F(ιℏ1+(1−ι)ℏ2)⊇ιsF(ℏ1)+(1−ι)sF(ℏ2), | (2.2) |
for ℏ1,ℏ2∈[ℑ1,ℑ2] and ι∈[0,1].
Consider F:[ℑ1,ℑ2]→R+I, we define ∫ℑ2ℑ1F(ι)dι by
∫ℑ2ℑ1F(ι)dι=[∫ℑ2ℑ1F_(ι)dι,∫ℑ2ℑ1¯F(ι)dι] |
and say that the function F is interval Lebesgue integrable on [ℑ1,ℑ2] (or that F∈IL[ℑ1,ℑ2]).
Definition 2.3. [42] Let F∈L[ℑ1,ℑ2]. The bilateral R-L fractional integral operators, namely the left-sided Jαℑ1+F and the right-sided Jαℑ2−F, are defined as
Jαℑ1+F(σ)=1Γ(α)∫σℑ1(σ−ι)α−1F(ι)dι, |
and
Jαℑ2−F(σ)=1Γ(α)∫ℑ2σ(ι−σ)α−1F(ι)dι, |
respectively, where α>0, ℑ1≥0, J0ℑ1+F(σ)=J0ℑ2−F(σ)=F(σ) and Γ(α) is the Gamma function.
The H-H inequality in the form of R-L fractional integrals was proved by Sarikaya et al. in [32] as follows:
Theorem 2.4. Let F∈L[ℑ1,ℑ2] be a mapping from [ℑ1,ℑ2] to R+. If F satisfies the convexity condition, then
F(ℑ1+ℑ22)≤Γ(α+1)2(ℑ2−ℑ1)α(Jαℑ1+F(ℑ2)+Jαℑ2−F(ℑ1))≤12(F(ℑ1)+F(ℑ2)), |
with α>0.
Definition 2.5. [36,37] Let F:[ℑ1,ℑ2]→R+I. The IV R-L fractional integral operators, including both left-sided and right-sided variants, are defined as follows:
Iαℑ1+F(σ)=1Γ(α)∫σℑ1(σ−ι)α−1F(ι)dι,σ>ℑ1, |
and
Iαℑ2−F(σ)=1Γ(α)∫ℑ2σ(ι−σ)α−1F(ι)dι,σ<ℑ2, |
respectively. Here, Iαℑ1+F(σ)=[Jαℑ1+F_(σ),Jαℑ1+¯F(σ)], and α>0.
In [36], Budak et al. gave the fractional H-H inequality for IV convex functions as follows:
Theorem 2.6. Let F:[ℑ1,ℑ2]→R+I, and α>0. If F is an IV convex function, then
F(ℑ1+ℑ22)⊇Γ(α+1)2(ℑ2−ℑ1)α(Iαℑ1+F(ℑ2)+Iαℑ2−F(ℑ1))⊇F(ℑ1)+F(ℑ2)2. | (2.3) |
Definition 2.7. [31] Let φ:[ℑ1,ℑ2]→R+ be a monotonically increasing mapping, and suppose F,φ∈L[ℑ1,ℑ2]. The generalized R-L fractional integral operators Jα,kℑ1+,φF and Jα,kℑ2−,φF are defined by
Jα,kℑ1+,φ(F)(σ)=1Γ(α)∫σℑ1(σ−ι)α−1(φ(σ)−φ(ι))kF(ι)dι,σ>ℑ1, |
and
Jα,kℑ2−,φ(F)(σ)=1Γ(α)∫ℑ2σ(ι−σ)α−1(φ(ι)−φ(σ))kF(ι)dι,σ<ℑ2, |
respectively, where k∈N∪{0}, α>0, and ℑ1≥0.
In 2023, Budak et al. introduced the generalized fractional integral for IV function as follows:
Definition 2.8. [38] Let φ:[ℑ1,ℑ2]→R+ be a monotonically increasing function, and F:[ℑ1,ℑ2]→R+I. The generalized R-L fractional integrals Iα,kℑ1+,φF and Iα,kℑ2−,φF of interval-valued functions are defined by
Iα,kℑ1+,φ(F)(σ)=1Γ(α)∫σℑ1(σ−ι)α−1(φ(σ)−φ(ι))kF(ι)dι,σ>ℑ1, |
and
Iα,kℑ2−,φ(F)(σ)=1Γ(α)∫ℑ2σ(ι−σ)α−1(φ(ι)−φ(σ))kF(ι)dι,σ<ℑ2, |
respectively, where α>0, ℑ1≥0 and k∈N∪{0}.
The families of all functions that are Riemann integrable and interval Riemann integrable over the closed interval [ℑ1,ℑ2] are respectively represented by the notations R[ℑ1,ℑ2] and IR[ℑ1,ℑ2].
Theorem 2.9. [38] Suppose F∈IR[ℑ1,ℑ2], φ:[ℑ1,ℑ2]→R is a monotonically increasing function on (ℑ1,ℑ2) with φ∈L[ℑ1,ℑ2]. Letting Φ(ι)=F(ι)+F(ℑ1+ℑ2−ι), then Φ∈IR[ℑ1,ℑ2], and we have
F(12(ℑ1+ℑ2))(Iα,kℑ1+,φ(1)(ℑ2)+Iα,kℑ2−,φ(1)(ℑ1))⊇12(Iα,kℑ1+,φ(Φ)(ℑ2)+Iα,kℑ2−,φ(Φ)(ℑ1))⊇12(F(ℑ1)+F(ℑ2))(Iα,kℑ1+,φ(1)(ℑ2)+Iα,kℑ2−,φ(1)(ℑ1)), |
where α>0 and k∈N∪{0}.
Let Φ(ϖ)=F(ϖ)+F(ℑ1+ℑ2−ϖ) and Λ(ϖ)=G(ϖ)+G(ℑ1+ℑ2−ϖ) for ϖ∈[ℑ1,ℑ2]. It is straightforward to demonstrate that if F,G∈IR[ℑ1,ℑ2], then Φ(ϖ),Λ(ϖ)∈IR[ℑ1,ℑ2].
Theorem 3.1. Let F:[ℑ1,ℑ2]→R+I, F∈IR[ℑ1,ℑ2], and φ:[ℑ1,ℑ2]→R+ be a monotonically increasing function. If F is an IV s-convex function, then
F(ℑ1+ℑ22)(Iα,kℑ1+,φ(1)(ℑ2)+Iα,kℑ2−,φ(1)(ℑ1))⊇12s(Iα,kℑ1+,φ(Φ)(ℑ2)+Iα,kℑ2−,φ(Φ)(ℑ1))⊇12s(F(ℑ1)+F(ℑ2))(Iα,kℑ1+,φ(1)(ℑ2)+Iα,kℑ2−,φ(1)(ℑ1)), | (3.1) |
where α>0 and k∈N∪{0}.
Proof. By the assumption, we have
F(ℏ1+ℏ22)⊇F(ℏ1)+F(ℏ2)2s. | (3.2) |
Letting ℏ1=ιℑ1+(1−ι)ℑ2, ℏ2=(1−ι)ℑ1+ιℑ2 for ι∈[0,1], we obtain
F(ℑ1+ℑ22)⊇12s(F(ιℑ1+(1−ι)ℑ2)+F((1−ι)ℑ1+ιℑ2)). | (3.3) |
Then, multiplying the two sides of (3.3) by ια−1(φ(ℑ2)−φ(ιℑ1+(1−ι)ℑ2))k and integrating on [0, 1], we obtain
F(ℑ1+ℑ22)∫10ια−1(φ(ℑ2)−φ(ιℑ1+(1−ι)ℑ2))kdι⊇12s(∫10ια−1(φ(ℑ2)−φ(ιℑ1+(1−ι)ℑ2))kF(ιℑ1+(1−ι)ℑ2)dι+∫10ια−1(φ(ℑ2)−φ(ιℑ1+(1−ι)ℑ2))kF((1−ι)ℑ1+ιℑ2)dι). |
Letting y=ιℑ1+(1−ιℑ2), we have
F(ℑ1+ℑ22)∫ℑ2ℑ1(ℑ2−y)α−1(φ(ℑ2)−φ(y))kdy⊇12s(∫ℑ2ℑ1(ℑ2−y)α−1(φ(ℑ2)−φ(y))kF(y)dy+∫ℑ2ℑ1(ℑ2−y)α−1(φ(ℑ2)−φ(y))kF(ℑ1+ℑ2−y)dy). |
That is,
F(ℑ1+ℑ22)Iα,kℑ1+,φ(1)(ℑ2)⊇12sIα,kℑ1+,φ(Φ)(ℑ2). | (3.4) |
By multiplying the two sides of (3.3) by ια−1(φ((1−ι)ℑ1+ιℑ2)−φ(ℑ1))k and integrating on [0, 1], we obtain
F(ℑ1+ℑ22)∫10τα−1(φ((1−ι)ℑ1+ιℑ2)−φ(ℑ1))kdι⊇12s(∫10ια−1(φ((1−ι)ℑ1+ιℑ2)−φ(ℑ1))kF(ιℑ1+(1−ι)ℑ2)dι+∫10ια−1(φ((1−ι)ℑ1+ιℑ2)−φ(ℑ1))kF((1−ι)ℑ1+ιℑ2)dι). |
Letting y=(1−ι)ℑ1+ιℑ2, we have
F(ℑ1+ℑ22)Iα,kℑ2−,φ(1)(ℑ1)⊇12sIα,kℑ2−,φ(Φ)(ℑ2). | (3.5) |
Combining with conclusions (3.4) and (3.5), we obtain
F(ℑ1+ℑ22)(Iα,kℑ1+,φ(1)(ℑ2)+Iα,kℑ2−,φ(1)(ℑ1))⊇12s(Iα,kℑ1+,φ(Φ)(ℑ2)+Iα,kℑ2−,φ(Φ)(ℑ1)). |
According to (2.2), we have
F(ιℑ1+(1−ι)ℑ2)⊇ιsF(ℑ1)+(1−ι)sF(ℑ2),F((1−ι)ℑ1+ιℑ2)⊇ιsF(ℑ2)+(1−ι)sF(ℑ1). |
That is,
F(ιℑ1+(1−ι)ℑ2)+F((1−ι)ℑ1+ιℑ2)⊇(F(ℑ1)+F(ℑ2)). | (3.6) |
Multiplying the two sides of (3.6) by ια−1(φ(ℑ2)−φ(ιℑ1+(1−ι)ℑ2))k and integrating on [0, 1], we obtain
∫10ια−1(φ(ℑ2)−φ(ιℑ1+(1−ι)ℑ2))kF(ιℑ1+(1−ι)ℑ2)dι+∫10ια−1(φ(ℑ2)−φ(ιℑ1+(1−ι)ℑ2))kF(ιℑ2+(1−ι)ℑ1)dι⊇(F(ℑ1)+F(ℑ2))∫10ια−1(φ(ℑ2)−φ(ιℑ1+(1−ι)ℑ2))kdι. |
Letting y=ιℑ1+(1−ι)ℑ2, we have
∫ℑ2ℑ1(ℑ2−yℑ2−ℑ1)α−1(φ(ℑ2)−φ(y))kF(y)dy+∫ℑ2ℑ1(ℑ2−yℑ2−ℑ1)α−1(φ(ℑ2)−φ(y))kF(ℑ1+ℑ2−y)dy⊇(F(ℑ1)+F(ℑ2))∫ℑ2ℑ1(ℑ2−yℑ2−ℑ1)α−1(φ(ℑ2)−φ(y))kdy. |
That is,
Iα,kℑ1+,φ(Φ)(ℑ2)⊇(F(ℑ1)+F(ℑ2))Iα,kℑ1+,φ(1)(ℑ2). | (3.7) |
Similarly, multiplying the two sides of (3.6) by ια−1(φ((1−ι)ℑ1+ιℑ2)−φ(ℑ1))k and integrating on [0, 1], let y=ιℑ2+(1−ι)ℑ1, we have
∫ℑ2ℑ1(y−ℑ1ℑ2−ℑ1)α−1(φ(y)−φ(ℑ1))kF(ℑ1+ℑ2−y)dy+∫ℑ2ℑ1(y−ℑ1ℑ2−ℑ1)α−1(φ(y)−φ(ℑ1))kF(y)dy⊇(F(ℑ1)+F(ℑ2))∫ℑ2ℑ1(y−ℑ1ℑ2−ℑ1)α−1(φ(y)−φ(ℑ1))kdy. |
Then, we obtain
Iα,kℑ2−,φ(Φ)(ℑ1)⊇(F(ℑ1)+F(ℑ2))Iα,kℑ2−,φ(1)(ℑ1). | (3.8) |
By (3.7) and (3.8), we have
Iα,kℑ1+,φ(Φ)(ℑ2)+Iα,kℑ2−,φ(Φ)(ℑ1)⊇(F(ℑ1)+F(ℑ2))(Iα,kℑ1+,φ(1)(ℑ2)+Iα,kℑ2−,φ(1)(ℑ1)). |
Hence, Theorem 3.1 is verified.
Corollary 3.2. If φ(ι)=ι−ω for ω∈R and s=1, then
F(ℑ1+ℑ22)⊇Γ(α+k+1)2(ℑ2−ℑ1)α+k(Iα+kℑ1+F(ℑ2)+Iα+kℑ2−F(ℑ1))⊇12(F(ℑ1)+F(ℑ2)). |
Remark 3.3. If s=1, Theorem 3.1 is reduced to Theorem 4.1 obtained by Budak et al. in[38].
Remark 3.4. If k=0, then
F(ℑ1+ℑ22)⊇Γ(α+1)2s(ℑ2−ℑ1)α(Iαℑ1+F(ℑ2)+Iαℑ2−F(ℑ1))⊇12s(F(ℑ1)+F(ℑ2)). |
Remark 3.5. If s=1 and k=0, then Theorem 3.1 simplifies to Theorem 3.2 as presented by Budak et al. in [36].
Example 3.6. Consider F:[0,2]→R+I, where F(x)=[2x2−4,−8x2+20x−3], φ(ι)=ι−ω for ω∈R, α=2,s=1, and k=1. Then we have
Δ1=F(ℑ1+ℑ22)(Iα,kℑ1+,φ(1)(ℑ2)+Iα,kℑ2−,φ(1)(ℑ1))=F(1)1Γ(2)∫20((2−x)2+x2)dx=163[−2,9]=[−323,48], |
Δ2=12s(Iα,kℑ1+,φ(Φ)(ℑ2)+Iα,kℑ2−,φ(Φ)(ℑ1))=121Γ(2)∫20((2−x)2+x2)(F(x)+F(2−x))dx=12[−645,163+1285]=[−325,46415], |
and
Δ3=12s(F(ℑ1)+F(ℑ2))(Iα,kℑ1+,φ(1)(ℑ2)+Iα,kℑ2−,φ(1)(ℑ1))=12(F(0)+F(2))1Γ(2)∫20((2−x)2+x2)dx=83([−4,−3]+[4,5])=[0,163]. |
Then, we obtain
Δ1⊇Δ2⊇Δ3. |
The graphical representation (Figure 1) confirms the results. $
Theorem 3.7. Assuming that the conditions of Theorem 3.1 are met, then
F(ℑ1+ℑ22)(Iα,k(12(ℑ1+ℑ2))+,φ(1)(ℑ2)+Iα,k(12(ℑ1+ℑ2))−,φ(1)(ℑ1))⊇12s(Iα,k(12(ℑ1+ℑ2))+,φ(Φ)(ℑ2)+Iα,k(12(ℑ1+ℑ2))−,φ(Φ)(ℑ1))⊇12s(F(ℑ1)+F(ℑ2))(Iα,k(12(ℑ1+ℑ2))+,φ(1)(ℑ2)+Iα,k(12(ℑ1+ℑ2))−,φ(1)(ℑ1)), | (3.9) |
where α>0 and k∈N∪{0}.
Proof. First, consider ℏ1=12ιℑ1+12(2−ι)ℑ2 and ℏ2=12ιℑ2+12(2−ι)ℑ1 for ι∈[0,1] in the inclusion (3.2). Then,
F(ℑ1+ℑ22)⊇12s(F(12ιℑ1+12(2−ι)ℑ2)+F(12ιℑ2+12(2−ι)ℑ1))⊇12s(F(ℑ1)+F(ℑ2)). | (3.10) |
Then, multiplying the two sides of (3.10) by ια−1(φ(ℑ2)−φ(12ιℑ1+12(2−ι)ℑ2))k and integrating on [0, 1], we obtain
F(ℑ1+ℑ22)∫10ια−1(φ(ℑ2)−φ(12ιℑ1+12(2−ι)ℑ2))kdι⊇12s∫10ια−1(φ(ℑ2)−φ(12ιℑ1+12(2−ι)ℑ2))k⋅(F(12ιℑ1+12(2−ι)ℑ2)+F(12ιℑ2+12(2−ι)ℑ1))dι⊇12s∫10ια−1(φ(ℑ2)−φ(12ιℑ1+12(2−ι)ℑ2))k(F(ℑ1)+F(ℑ2))dι. |
Letting y=12ιℑ1+12(2−ι)ℑ2, we have
F(ℑ1+ℑ22)(2ℑ2−ℑ1)α∫ℑ2ℑ1+ℑ22(ℑ2−y)α−1(φ(ℑ2)−φ(y))kdy⊇12s(2ℑ2−ℑ1)α∫ℑ2ℑ1+ℑ22(ℑ2−y)α−1(φ(ℑ2)−φ(y))k(F(y)+F(ℑ1+ℑ2−y))dy⊇12s(F(ℑ1)+F(ℑ2))(2ℑ2−ℑ1)α∫ℑ2ℑ1+ℑ22(ℑ2−y)α−1(φ(ℑ2)−φ(y))kdy, |
that is,
F(ℑ1+ℑ22)Iα,k(12(ℑ1+ℑ2))+,φ(1)(ℑ2)⊇12sIα,k(12(ℑ1+ℑ2))+,φ(Φ)(ℑ2)⊇12s(F(ℑ1)+F(ℑ2))Iα,k(12(ℑ1+ℑ2))+,φ(1)(ℑ2). | (3.11) |
Similarly, multiplying the two sides of (3.10) by ια−1(φ(12(2−ι)ℑ1+12ιℑ2)−φ(ℑ1))k and integrating on [0, 1], we have
F(ℑ1+ℑ22)∫10ια−1(φ(12(2−ι)ℑ1+12ιℑ2)−φ(ℑ1))kdι⊇12s∫10ια−1(φ(12(2−ι)ℑ1+12ιℑ2)−φ(ℑ1))k×(F(12ιℑ1+12(2−ι)ℑ2)+F(12ιℑ2+12(2−ι)ℑ1))dι⊇12s(F(ℑ1)+F(ℑ2))∫10ια−1(φ(12(2−ι)ℑ1+12ιℑ2)−φ(ℑ1))kdι. |
Letting y=12(2−ι)ℑ1+12ιℑ2, we have
F(ℑ1+ℑ22)(2ℑ2−ℑ1)α∫ℑ1+ℑ22ℑ1(y−ℑ1)α−1(φ(y)−φ(ℑ1))kdy⊇12s(2ℑ2−ℑ1)α∫ℑ1+ℑ22ℑ1(y−ℑ1)α−1(φ(y)−φ(ℑ1))k(F(y)+F(ℑ1+ℑ2−y))dy⊇12s(F(ℑ1)+F(ℑ2))(2ℑ2−ℑ1)α∫ℑ1+ℑ22ℑ1(y−ℑ1)α−1(φ(y)−φ(ℑ1))kdy, |
that is,
F(ℑ1+ℑ22)Iα,k(12(ℑ1+ℑ2))−,φ(1)(ℑ1))⊇12sIα,k(12(ℑ1+ℑ2))−,φ(Φ)(ℑ1)⊇12s(F(ℑ1)+F(ℑ2))Iα+s,k(12(ℑ1+ℑ2))−,φ(1)(ℑ1). | (3.12) |
By (3.11) and (3.12), we obtain the result.
Corollary 3.8. If φ(ι)=ι−ω for ω∈R and s=1, then
F(ℑ1+ℑ22)⊇2α+k−1Γ(α+k+1)(ℑ2−ℑ1)α+k(Iα+k12(ℑ1+ℑ2)+F(ℑ2)+Iα+k12(ℑ1+ℑ2)−F(ℑ1))⊇F(ℑ1)+F(ℑ2)2. |
Remark 3.9. If s=1, then Theorem 3.7 simplifies to Theorem 4.4 given by Budak et al. in[38].
Remark 3.10. If k=0, then
F(ℑ1+ℑ22)⊇2αΓ(α+1)2s(ℑ2−ℑ1)α(Iα+k12(ℑ1+ℑ2)+F(ℑ2)+Iα+k12(ℑ1+ℑ2)−F(ℑ1))⊇12s(F(ℑ1)+F(ℑ2)). |
Remark 3.11. If s=1 and k=0, then Theorem 3.7 which has been obtained by Zhao et al. in [39].
Example 3.12. Consider F:[0,1]→R+I, where F(x)=[x2,5−(12)x], φ(ι)=ι−ω for ω∈R, α=2,s=1, and k=1. Then we have
T1=F(ℑ1+ℑ22)(Iα,k(12(ℑ1+ℑ2))+,φ(1)(ℑ2)+Iα,k(12(ℑ1+ℑ2))−,φ(1)(ℑ1))=F(12)1Γ(2)(∫112(1−x)2dx+∫120x2dx)=112[14,5−(12)12]≈[0.0208,0.3750], |
T2=12s(Iα,k(12(ℑ1+ℑ2))+,φ(Φ)(ℑ2)+Iα,k(12(ℑ1+ℑ2))−,φ(Φ)(ℑ1))=121Γ(2)(∫112(1−x)2(F(x)+F(1−x))dx+∫120x2(F(x)+F(1−x))dx)≈[0.0229,0.3574], |
and
T3=12s(F(ℑ1)+F(ℑ2))(Iα,k(12(ℑ1+ℑ2))+,φ(1)(ℑ2)+Iα,k(12(ℑ1+ℑ2))−,φ(1)(ℑ1))=121Γ(2)(F(0)+F(1))(∫112(1−x)2dx+∫120x2dx)=124([0,4]+[1,92])≈[0.0417,0.3542]. |
Then, we obtain
T1⊇T2⊇T3. |
Theorem 3.13. Given that the assumptions of Theorem 3.1 hold, we can obtain
F(ℑ1+ℑ22)(Iα,kℑ1+,φ(1)(12(ℑ1+ℑ2))+Iα,kℑ2−,φ(1)(12(ℑ1+ℑ2)))⊇12s(Iα,kℑ1+,φ(Φ)(12(ℑ1+ℑ2))+Iα,kℑ2−,φ(Φ)(12(ℑ1+ℑ2)))⊇12s(F(ℑ1)+F(ℑ2))(Iα,kℑ1+,φ(1)(12(ℑ1+ℑ2))+Iα,kℑ2−,φ(1)(12(ℑ1+ℑ2))), | (3.13) |
for α>0 and k∈N∪{0}.
Proof. Considering ℏ1=12(1+ι)ℑ1+12(1−ι)ℑ2 and ℏ1=12(1−ι)ℑ1+12(1+ι)ℑ2 for ι∈[0,1] in the inclusion (3.2). Then we obtain
F(ℑ1+ℑ22)⊇12s(F(12(1+ι)ℑ1+12(1−ι)ℑ2)+F(12(1−ι)ℑ1+12(1+ι)ℑ2))⊇12s(F(ℑ1)+F(ℑ2)). | (3.14) |
Then, multiplying two sides of (3.14) by
ια−1(φ(12(ℑ1+ℑ2))−φ(12(1+ι)ℑ1+12(1−ι)ℑ2))k, |
and integrating on [0, 1], we obtain
F(ℑ1+ℑ22)∫10ια−1(φ(ℑ1+ℑ22)−φ(12(1+ι)ℑ1+12(1−ι)ℑ2)k)dι⊇12s∫10τα−1(φ(ℑ1+ℑ22)−φ(12(1+ι)ℑ1+12(1−ι)ℑ2))k⋅(F(12(1+ι)ℑ1+12(1−ι)ℑ2)+F(12(1−ι)ℑ1+12(1+ι)ℑ2))dι⊇12s(F(ℑ1)+F(ℑ2))∫10ια−1(φ(ℑ1+ℑ22)−φ(12(1+ι)ℑ1+12(1−ι)ℑ2))kdι. |
Letting z=12(1+ι)ℑ1+12(1−ι)ℑ2, we have
F(ℑ1+ℑ22)∫ℑ1+ℑ22ℑ1(ℑ1+ℑ22−z)α−1(φ(12(ℑ1+ℑ2))−φ(z))kdz⊇12s∫ℑ1+ℑ22ℑ1(ℑ1+ℑ22−z)α−1(φ(ℑ1+ℑ22)−φ(z))k(F(z)+F(ℑ1+ℑ2−z))dz⊇12s(F(ℑ1)+F(ℑ2))∫12(ℑ1+ℑ2)ℑ1(ℑ1+ℑ22−z)α−1(φ(ℑ1+ℑ22)−φ(z))kdz, |
that is,
F(ℑ1+ℑ22)Iα,kℑ1+,φ(1)(ℑ1+ℑ22)⊇12sIα,kℑ1+,φ(Φ)(ℑ1+ℑ22)⊇12s(F(ℑ1)+F(ℑ2))Iα,kℑ1+,φ(1)(ℑ1+ℑ22). | (3.15) |
Multiplying two sides of (3.14) by
ια−1(φ(12(1−ι)ℑ1+12(1+ι)ℑ2)−φ(ℑ1+ℑ22))k, |
and integrating on [0, 1], we obtain
F(ℑ1+ℑ22)∫10τα−1(φ(ℑ1+ℑ22)−φ(12(1+ι)ℑ1+12(1−ι)ℑ2))kdι⊇12s∫10ια−1(φ(ℑ1+ℑ22)−φ(12(1+ι)ℑ1+12(1−ι)ℑ2))k×(F(12(1+ι)ℑ1+12(1−ι)ℑ2)+F(12(1−ι)ℑ1+12(1+ι)ℑ2))dι⊇12s(F(ℑ1)+F(ℑ2))∫10ια−1(φ(ℑ1+ℑ22)−φ(12(1+ι)ℑ1+12(1−ι)ℑ2))kdι. |
Letting z=12(1−ι)ℑ1+12(1+ι)ℑ2, we obtain
F(ℑ1+ℑ22)Iα,kℑ2−,φ(1)(ℑ1+ℑ22)⊇12sIα,kℑ2−,φ(Φ)(ℑ1+ℑ22)⊇12s(F(ℑ1)+F(ℑ2))Iα,kℑ2−,φ(1)(ℑ1+ℑ22). | (3.16) |
Adding the inclusion (3.15) to (3.16), we complete the proof.
Corollary 3.14. If φ(ι)=ι−ω for ω∈R and s=1, then
F(ℑ1+ℑ22)⊇2α+k−1Γ(α+k+1)(ℑ2−ℑ1)α+k(Iα+kℑ1+F(ℑ1+ℑ22)+Iα+kℑ2−F(ℑ1+ℑ22))⊇12(F(ℑ1)+F(ℑ2)). |
Remark 3.15. If s=1, Theorem 3.13 simplifies to Theorem 4.7 presented by Budak et al. in[38].
Remark 3.16. If s=1 and k=0, then
F(ℑ1+ℑ22)⊇2α−1Γ(α+1)(ℑ2−ℑ1)α(Iαℑ1+F(ℑ1+ℑ22)+Iαℑ2−F(ℑ1+ℑ22))⊇12(F(ℑ1)+F(ℑ2)), |
which has been obtained by Zhao et al. in [39].
Example 3.17. Consider F:[0,2]→R+I, where F(x)=[x2,−x2+8], φ(ι)=ι−ω for ω∈R, α=2,s=1, and k=1. Then we have
D1=F(ℑ1+ℑ22)(Iα,kℑ1+,φ(1)(12(ℑ1+ℑ2))+Iα,kℑ2−,φ(1)(12(ℑ1+ℑ2)))=F(1)1Γ(2)(∫10(1−x)2dx+∫21(x−1)2dx)=23[1,7]=[23,143], |
D2=12s(Iα,kℑ1+,φ(Φ)(12(ℑ1+ℑ2))+Iα,kℑ2−,φ(Φ)(12(ℑ1+ℑ2)))=121Γ(2)∫20(1−x)2(F(x)+F(2−x))dx=[1615,6415], |
and
D3=12s(F(ℑ1)+F(ℑ2))(Iα,kℑ1+,φ(1)(12(ℑ1+ℑ2))+Iα,kℑ2−,φ(1)(12(ℑ1+ℑ2)))=121Γ(2)(F(0)+F(2))∫20(1−x)2dx=13([0,8]+[4,4])=[43,4]. |
Then, we obtain
D1⊇D2⊇D3. |
Theorem 3.18. Let F,G:[ℑ1,ℑ2]→R+I, s1,s2∈(0,1], and φ:[ℑ1,ℑ2]→R be a monotonically increasing function. If F and G are both IV s-convex functions, then
F(ℑ1+ℑ22)⋅G(ℑ1+ℑ22)(Iα,kℑ1+,φ(1)(ℑ2)+Iα,kℑ2−,φ(1)(ℑ1))⊇12s1+s2(Iα,kℑ1+,φ(ΦΛ)(ℑ2)+Iα,kℑ2−,φ(ΦΛ)(ℑ1))⊇12s1+s2(P(ℑ1,ℑ2)+Q(ℑ1,ℑ2))(Iα,kℑ1+,φ(1)(ℑ2)+Iα,kℑ2−,φ(1)(ℑ1)), | (3.17) |
where
P(ℑ1,ℑ2)=F(ℑ1)⋅G(ℑ1)+F(ℑ2)⋅G(ℑ2), |
and
Q(ℑ1,ℑ2)=F(ℑ1)⋅G(ℑ2)+F(ℑ2)⋅G(ℑ1). |
Proof. By hypothesis, then
F(ℏ1+ℏ22)⊇F(ℏ1)+F(ℏ2)2s1,G(ℏ1+ℏ22)⊇G(ℏ1)+G(ℏ2)2s2. |
Considering ℏ1=ιℑ1+(1−ιℑ2) and ℏ2=(1−ι)ℑ1+ιℑ2, we obtain
F(ℑ1+ℑ22)⋅G(ℑ1+ℑ22)⊇12s1+s2(F(ιℑ1+(1−ιℑ2))+F((1−ι)ℑ1+ιℑ2))⋅(G(ιℑ1+(1−ιℑ2))+G((1−ι)ℑ1+ιℑ2))⊇12s1+s2(P(ℑ1,ℑ2)+Q(ℑ1,ℑ2)). | (3.18) |
By multiplying two sides of (3.18) by ια−1(φ(ℑ2)−φ(ιℑ1+(1−ι)ℑ2))k and integrating on [0, 1], we obtain
F(ℑ1+ℑ22)⋅G(ℑ1+ℑ22)∫10ια−1(φ(ℑ2)−φ(ιℑ1+(1−ι)ℑ2))kdι⊇12s1+s2∫10ια−1(φ(ℑ2)−φ(ιℑ1+(1−ι)ℑ2))k(F(ιℑ1+(1−ιℑ2))+F((1−ι)ℑ1+ιℑ2))⋅(G(ιℑ1+(1−ιℑ2))+G((1−ι)ℑ1+ιℑ2))dτ⊇12s1+s2(P(ℑ1,ℑ2)+Q(ℑ1,ℑ2))∫10ια−1(φ(ℑ2)−φ(ιℑ1+(1−ι)ℑ2))kdι. |
Letting y=ιℑ1+(1−ι)ℑ2, then
F(ℑ1+ℑ22)⋅G(ℑ1+ℑ22)(1ℑ2−ℑ1)α∫ℑ2ℑ1(ℑ2−y)α−1(φ(ℑ2)−φ(y))kdy⊇12s1+s2(1ℑ2−ℑ1)α(∫ℑ2ℑ1(ℑ2−y)α−1(φ(ℑ2)−φ(y))k⋅(F(y)+F(ℑ1+ℑ2−y))⋅(G(y)+G(ℑ1+ℑ2−y))dy⊇12s1+s2(1ℑ2−ℑ1)α(P(ℑ1,ℑ2)+Q(ℑ1,ℑ2))⋅∫ℑ2ℑ1(ℑ2−y)α−1(φ(ℑ2)−φ(y))kdy, |
that is,
F(ℑ1+ℑ22)Iα,kℑ1+,φ(1)(ℑ2)⊇12s1+s2Iα,kℑ1+,φ(ΦΛ)(ℑ2)⊇12s1+s2(P(ℑ1,ℑ2)+Q(ℑ1,ℑ2))Iα,kℑ1+,φ(1)(ℑ2). | (3.19) |
By multiplying two sides of (3.18) by ια−1(φ((1−ι)ℑ1+ιℑ2)−φ(ℑ1))k and integrating on [0, 1], we have
F(ℑ1+ℑ22)⋅G(ℑ1+ℑ22)∫10ια−1(φ((1−ι)ℑ1+ιℑ2)−φ(ℑ1))kdι⊇12s1+s2∫10ια−1(φ((1−ι)ℑ1+ιℑ2)−φ(ℑ1))k⋅(F(ιℑ1+(1−ιℑ2))+F((1−ι)ℑ1+ιℑ2))(G(ιℑ1+(1−ιℑ2))+G((1−ι)ℑ1+ιℑ2))dτ⊇12s1+s2∫10ια−1(φ((1−ι)ℑ1+ιℑ2)−φ(ℑ1))k(P(ℑ1,ℑ2)+Q(ℑ1,ℑ2))dι. |
Letting y=(1−ι)ℑ1+ιℑ2, that is,
F(ℑ1+ℑ22)⋅G(ℑ1+ℑ22)Iα,kℑ2−,φ(1)(ℑ1)⊇12s1+s2Iα,kℑ2−,φ(ΦΛ)(ℑ1)⊇12s1+s2(P(ℑ1,ℑ2)+Q(ℑ1,ℑ2))Iα,kℑ2−,φ(1)(ℑ1). | (3.20) |
Adding the inclusion (3.19) and (3.20), we obtained the desired result.
Example 3.19. Assume F,G:[0,2]→R+I, where F(x)=[x2,−x2+10] and
G(x)=[12x2,−x2+8], |
respectively. Let φ(ι)=ι−ω for ω∈R, α=2,k=1, and s=1. Then we have
Θ1=F(ℑ1+ℑ22)⋅G(ℑ1+ℑ22)(Iα,kℑ1+,φ(1)(ℑ2)+Iα,kℑ2−,φ(1)(ℑ1))=F(1)⋅G(1)1Γ(2)∫20((2−x)2+x2)dx=163([1,9]⋅[12,7])=[83,336], |
and
Θ2=12s1+s2(Iα,kℑ1+,φ(ΦΛ)(ℑ2)+Iα,kℑ2−,φ(ΦΛ)(ℑ1))=141Γ(2)∫20((2−x)2+x2)(F(x)+F(2−x))⋅(G(x)+G(2−x))dx=[5.4857,303.2381], |
and
Θ3=12s1+s2(P(ℑ1,ℑ2)+Q(ℑ1,ℑ2))(Iα,kℑ1+,φ(1)(ℑ2)+Iα,kℑ2−,φ(1)(ℑ1))=14(F(0)+F(2))⋅(G(0)+G(2))1Γ(2)∫20((2−x)2+x2)dx=43([8,192])=[323,256]. |
Then, we obtain
Θ1⊇Θ2⊇Θ3. |
Analogously, we can derive the subsequent results.
Theorem 3.20. Assuming that the conditions of Theorem 3.18 hold, we consequently obtain
F(ℑ1+ℑ22)⋅G(ℑ1+ℑ22)(Iα,k(12(ℑ1+ℑ2))+,φ(1)(ℑ2)+Iα,k(12(ℑ1+ℑ2))−,φ(1)(ℑ1))⊇12s1+s2(Iα,k(12(ℑ1+ℑ2))+,φ(ΦΛ)(ℑ2)+Iα,k(12(ℑ1+ℑ2))−,φ(ΦΛ)(ℑ1))⊇12s1+s2(P(ℑ1,ℑ2)+Q(ℑ1,ℑ2))(Iα,k(12(ℑ1+ℑ2))+,φ(1)(ℑ2)+Iα+s,k(12(ℑ1+ℑ2))−,φ(1)(ℑ1)). | (3.21) |
Theorem 3.21. Assuming that the conditions of Theorem 3.18 hold, we obtain
F(ℑ1+ℑ22)⋅G(ℑ1+ℑ22)(Iα,kℑ1+,φ(1)(ℑ1+ℑ22)+Iα,kℑ2−,φ(1)(ℑ1+ℑ22))⊇12s1+s2(Iα,kℑ1+,φ(ΦΛ)(ℑ1+ℑ22)+Iα,kℑ2−,φ(ΦΛ)(ℑ1+ℑ22))⊇12s1+s2(P(ℑ1,ℑ2)+Q(ℑ1,ℑ2))(Iα,kℑ1+,φ(1)(ℑ1+ℑ22)+Iα,kℑ2−,φ(1)(ℑ1+ℑ22)). | (3.22) |
This study derives novel H-H-type inequalities through generalized fractional integrals applied to IV s-convex functions. These not only generalize but also refine the previously proposed inequalities established by Budak et al. and provide a foundation for further exploration of generalized convexity and IV estimation. The developed techniques offer new tools for uncertainty quantification in convex optimization problems with imprecise measurements. Future research directions include:
(1) Develop new interval H-H inequalities based on more general convexity.
(2) Extension to IV quasi-convex functions using variable-order fractional operators.
(3) Applications in robust portfolio optimization with IV risk measures.
G. Deng and D. Zhao: Conceptualization, Methodology, Formal analysis; S. Etemad: Conceptualization, Software, Formal analysis, Investigation, Writing–original draft preparation; G. Deng: Writing–original draft preparation; D. Zhao: Validation, Investigation, Supervision, Project administration, Writing–review and editing; J. Tariboon: Validation, Formal analysis, Project administration, Writing–review and editing. All authors read and approved the final manuscript.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This research was funded by the National Science, Research and Innovation Fund (NSRF) and King Mongkut's University of Technology North Bangkok, under Contract No. KMUTNB-FF-68-B-04, and the Foundation of Hubei Normal University (2024Z022).
The authors declare no conflict of interest.
[1] |
B. Alspach, T. D. Parsons, Isomorphism of circulant graphs and digraphs, Discrete Math., 25 (1979), 97–108. https://doi.org/10.1016/0012-365X(79)90011-6 doi: 10.1016/0012-365X(79)90011-6
![]() |
[2] |
F. K. Hwang, A survey on multi-loop networks, Theor. Comput. Sci., 299 (2003), 107–121. https://doi.org/10.1016/S0304-3975(01)00341-3 doi: 10.1016/S0304-3975(01)00341-3
![]() |
[3] |
J. C. Bermond, F. Comellas, D. F. Hsu, Distributed loop computer-networks: A survey, J. Parallel Distr. Com., 24 (1995), 2–10. https://doi.org/10.1006/jpdc.1995.1002 doi: 10.1006/jpdc.1995.1002
![]() |
[4] |
B. Shaukat, M. Ishaq, A. U. Haq, Algebraic invariants of edge ideals of some circulant graphs, AIMS Mathematics, 9 (2023), 868–895. https://doi.org/10.3934/math.2024044 doi: 10.3934/math.2024044
![]() |
[5] | CoCoATeam, CoCoA: A system for computations in commutative algebra. Available from: http://cocoa.dima.unige.it/. |
[6] | D. R. Grayson, M. E. Stillman, Macaulay2: A software system for research in algebraic geometry, 2002. Available from: https://www.unimelb-macaulay2.cloud.edu.au/home. |
[7] |
R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math., 68 (1982), 175–193. https://doi.org/10.1007/BF01394054 doi: 10.1007/BF01394054
![]() |
[8] |
A. M. Duval, B. Goeckner, C. J. Klivans, J. L. Martin, A non-partitionable Cohen–Macaulay simplicial complex, Adv. Math., 299 (2016), 381–395. https://doi.org/10.1016/j.aim.2016.05.011 doi: 10.1016/j.aim.2016.05.011
![]() |
[9] |
J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, 322 (2009), 3151–3169. https://doi.org/10.1016/j.jalgebra.2008.01.006 doi: 10.1016/j.jalgebra.2008.01.006
![]() |
[10] | J. Herzog, A survey on Stanley depth, In: Monomial ideals, computations and applications, Berlin, Heidelberg: Springer, 2013, 3–45. https://doi.org/10.1007/978-3-642-38742-5_1 |
[11] |
M. U. K. Afridi, I. Ur Rehman, M. Ishaq, Algebraic invariants of the edge ideals of whisker graphs of cubic circulant graphs. Heliyon, 11 (2025), e41783. https://doi.org/10.1016/j.heliyon.2025.e41783 doi: 10.1016/j.heliyon.2025.e41783
![]() |
[12] |
M. M. S. Shahid, M. Ishaq, A. Jirawattanapanit, K. Subkrajang, Depth and Stanley depth of the edge ideals of multi triangular snake and multi triangular ouroboros snake graphs, AIMS Mathematics, 7 (2022), 16449–16463. https://doi.org/10.3934/math.2022900 doi: 10.3934/math.2022900
![]() |
[13] |
Z. Iqbal, M. Ishaq, M. Aamir, Depth and Stanley depth of the edge ideals of square paths and square cycles, Commun. Algebra, 46 (2018), 1188–1198. https://doi.org/10.1080/00927872.2017.1339068 doi: 10.1080/00927872.2017.1339068
![]() |
[14] | W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge university press, 1998. https://doi.org/10.1017/CBO9780511608681 |
[15] |
A. Rauf, Depth and Stanley depth of multigraded modules Commun. Algebra, 38 (2010), 773–784. https://doi.org/10.1080/00927870902829056 doi: 10.1080/00927870902829056
![]() |
[16] |
G. Caviglia, H. T. Hà, J. Herzog, M. Kummini, N. Terai, N. V. Trung, Depth and regularity modulo a principal ideal, J. Algebr. Comb., 49 (2019), 1–20. https://doi.org/10.1007/s10801-018-0811-9 doi: 10.1007/s10801-018-0811-9
![]() |
[17] | R. H. Villarreal, Monomial algebras, monographs and textbooks in pure and applied mathematics, New York: Marcel Dekker, Inc., 2001. |
[18] | A. Alipour, A. Tehranian, Depth and Stanley depth of edge ideals of star graphs, Int. J. Appl. Math. Stat., 56 (2017), 63–69. |
[19] | B. Shaukat, M. Ishaq, A. U. Haq, Z. Iqbal, Algebraic properties of edge ideals of corona product of certain graphs, 2022, arXiv: 2211.05721. https://doi.org/10.48550/arXiv.2211.05721 |
[20] | M. Cimpoeas, Several inequalities regarding Stanley depth, Rom. J. Math. Comput. Sci., 2 (2012), 28–40. |
[21] |
N. U. Din, M. Ishaq, Z. Sajid, Values and bounds for depth and Stanley depth of some classes of edge ideals, AIMS Mathematics, 6 (2021), 8544–8566. https://doi.org/10.3934/math.2021496 doi: 10.3934/math.2021496
![]() |
[22] | S. Morey, R. H. Villarreal, Edge ideals: Algebraic and combinatorial properties, Progress in Commut. Algebr., 1 (2012), 85–126. |
[23] |
G. Kalai, R. Meshulam, Intersections of Leray complexes and regularity of monomial ideals, J. Comb. Theory A, 113 (2006), 1586–1592. https://doi.org/10.1016/j.jcta.2006.01.005 doi: 10.1016/j.jcta.2006.01.005
![]() |
[24] |
J. Herzog, A generalization of the Taylor complex construction, Commun. Algebra, 35 (2007), 1747–1756. https://doi.org/10.1080/00927870601139500 doi: 10.1080/00927870601139500
![]() |
[25] |
M. Katzman, Characteristic-independence of Betti numbers of graph ideals, J. Comb. Theory A, 113 (2006), 435–454. https://doi.org/10.1016/j.jcta.2005.04.005 doi: 10.1016/j.jcta.2005.04.005
![]() |
[26] |
H. T. Hà, A. V. Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebr. Comb., 27 (2008), 215–245. https://doi.org/10.1007/s10801-007-0079-y doi: 10.1007/s10801-007-0079-y
![]() |
[27] | R. Woodroofe, Matchings, coverings, and Castelnuovo-Mumford regularity, J. Commut. Algebra, 6 (2014), 287–304. |
[28] |
L. T. Hoa, N. D. Tam, On some invariants of a mixed product of ideals, Arch. Math., 94 (2010), 327–337. https://doi.org/10.1007/s00013-010-0112-6 doi: 10.1007/s00013-010-0112-6
![]() |
[29] |
Y. Muta, M. R. Pournaki, N. Terai, A local cohomological viewpoint on edge rings associated with multi-whisker graphs, Commun. Algebra, 53 (2024), 1856–1865. https://doi.org/10.1080/00927872.2024.2423888 doi: 10.1080/00927872.2024.2423888
![]() |