Research article

Two new generalized iteration methods for solving absolute value equations using M-matrix

  • Received: 19 October 2021 Revised: 24 January 2022 Accepted: 06 February 2022 Published: 25 February 2022
  • MSC : 90C30, 65F10

  • In this paper, we present two new generalized Gauss-Seidel iteration methods for solving absolute value equations Ax|x|=b, where A is an M-matrix. Furthermore, we demonstrate their convergence under specific assumptions. Numerical tests indicate the efficiency of the suggested methods with suitable parameters.

    Citation: Rashid Ali, Ilyas Khan, Asad Ali, Abdullah Mohamed. Two new generalized iteration methods for solving absolute value equations using M-matrix[J]. AIMS Mathematics, 2022, 7(5): 8176-8187. doi: 10.3934/math.2022455

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  • In this paper, we present two new generalized Gauss-Seidel iteration methods for solving absolute value equations Ax|x|=b, where A is an M-matrix. Furthermore, we demonstrate their convergence under specific assumptions. Numerical tests indicate the efficiency of the suggested methods with suitable parameters.



    Associating graphs to algebraic structures and studying their algebraic properties employing the methods in graph theory has been an interesting topic for mathematicians in the last decades and continuously arousing people's wide attention. For instance, Cayley graphs of groups [14], power graphs of groups ([2,15,24]), zero-divisor graphs of semigroups [17], zero-divisor graphs of commutative rings [5], total graphs of rings [1], unit graphs of rings [6], zero-divisor graphs of modules [9]. Bosák initiated the study of the intersection graphs of semigroups in [10]. Later, Csákány and Pollák defined the intersection graph of subgroups of a finite group and studied the graphic properties in [16]. In the recent years, several interesting properties of the intersection graphs of subgroups of groups have been obtained in the literature. In 1975, Zelinka investigated the intersection graphs of subgroups of finite abelian groups [33]. Akbari, Heydari and Maghasedi obtained some properties of the intersection graphs of subgroups of solvable groups in [7]. Shen classified finite groups with disconnected intersection graphs of subgroups in [29]. Ma gave an upper bound for the diameter of intersection graphs of finite simple groups in [23]. Rajkumar and Devi defined the intersection graph of cyclic subgroups of groups and obtained the clique number of intersection graph of subgroups of some non-abelian groups [26,27]. Recall that, for a finite group G, the intersection graph of subgroups of G, denoted by Γ(G), is a graph with all non-trivial subgroups of G as its vertices and two distinct vertices H and K are adjacent in Γ(G) if and only if HK1.

    The thickness of a graph, as a topological invariant of a graph, was first defined by Tutte in [30]. The problem is the same as determining whether K9 is biplanar or not, that is, a union of two planar graphs and it was showed in [11]. Tutte generalized the problem by defining the concept of the thickness of a graph. It is an important research object in topological graph theory, and it also has important applications to VLSI design and network design (see [4,28]). Outerthickness seems to be studied first in Geller's unpublished manuscript, where it was shown that θ0(K7) is 3 by similar exhaustive search as in the case of the thickness of K9. It is known that determining the thickness of a graph is NP-hard [22]. Until now, there are only a few results on the thickness of graphs. The attention has been focused on finding upper bounds of thickness and outerthickness. In 1964, Beineke, Frank and Moon obtained the thickness of the complete bipartite graph [13]. Beineke and Harary obtained the thickness of the complete graph ([3,12]). Chartrand, Geller and Hedetniemi conjectured that planar graph has an edge partition into two outerplanar graphs in [21]. Guy and Nowwkowski obtained the outerthickness of the complete graph and complete bipartite graph in [19,20]. In [18], Ding et al. proved that every planar graph has an edge partition into two outerplanar graphs.

    For the planarity of intersection graph of subgroups of a finite group, Ahmadi and Taeri, Kayacan and Yaraneri independently classified all finite groups with planar intersection graphs in [8,25], respectively. By their main theorem, it is easy to obtain the planarity of intersection graph of subgroups of a finite abelian group.

    Theorem 1.1. [8,25] Let G be a finite abelian group. Then Γ(G) is planar if and only if G is one of the following groups: Zpα(α=2,3,4,5),Zpαq(α=1,2),Zpqr,ZpZp,Z2Z2p, where p,q,r are distinct primes.

    Motivated by the above theorem, the aim of this paper is to classify finite abelian groups whose thickness and outerthickness of intersection graphs of subgroups are 1 and 2, respectively. We also investigate the thickness and outerthickness of intersection graphs of subgroups of some finite non-abelian groups in the last section.

    Let us recall some basic definitions and notations in graph theory. Let Γ be a graph with vertex set V(Γ) and edge set E(Γ), and all graphs in this paper are simple graphs, that are, undirect, no loops and no multiple edges. We use Kn to denote the complete graph on n vertices and Km,n to denote the complete bipartite graph with one partition consisting of m vertices and another partition consisting of n vertices. The degree of a vertex v in Γ, denoted by degΓ(v), is the number of vertices incident to v. If degΓ(v)=k, then the vertex is denoted by vk. A subgraph Γ of Γ is provided V(Γ)V(Γ) and E(Γ)E(Γ). A clique of a graph Γ is a complete subgraph of Γ. A maximal clique of a graph is a clique such that adding any vertex into the clique cannot be a clique. The clique number of a graph Γ is the number of vertices in a maximal clique in Γ with maximum size and it is denoted by ω(Γ). A subgraph H of Γ is a spanning subgraph of Γ if V(H)=V(Γ). The union of graphs Γ1 and Γ2 is the graph Γ1Γ2 with vertex set V(Γ1)V(Γ2) and edge set E(Γ1)E(Γ2). Let Γ1+Γ2 be a graph obtained from the union Γ1Γ2 by adding edges between each vertex in Γ1 and each vertex in Γ2. If Γ2 has exactly one vertex v, we shall use Γ1+v instead of Γ1+Γ2. A graph is planar if it can be drawn in a plane such that no two edges intersect expect at their end vertices. An outerplanar graph is a planar graph that can be embedded in the plane without crossings in such a way that all the vertices lie in the boundary of the unbounded face of the embedding. Recall that the thickness of a graph Γ, denoted by θ(Γ), is the minimum number of planar subgraphs whose union is Γ. Similarly, the outerthickness θ0(Γ) is obtained where the planar subgraphs are replaced by outerplanar subgraphs in the previous definition. Obviously, the thickness of a planar graph is 1 and the outerthickness of an outerplanar graph is 1.

    The following lemmas are obvious and are used frequently later.

    Lemma 2.1. [32,Lemma 2.1] If Γ1 is a subgraph of Γ2, then θ(Γ1)θ(Γ2) and θ0(Γ1)θ0(Γ2).

    Lemma 2.2. [12,19,3] (1) θ(Kn)={n+76n9,103n=9,10.

    (2) θ0(Kn)={n+14n73n=7.

    where x denotes the least integer greater than or equal to x, x denotes the largest integer greater than or equal to x

    By Lemma 2.2, one can see that θ(Kn)3 for n9, θ0(Kn)3 for n7.

    From Euler's formula, a planar graph Γ has at most 3|V|6 edges and an outerpalnar graph Γ has at most 2|V|3 edges, furthermore, one can get the lower bound for the thickness and outerthickness of a graph as follows.

    Lemma 2.3. [13] If Γ=(V,E) is a graph with |V|=n and |E|=m, then θ(Γ)m3n6 and θ0(Γ)m2n3.

    Lemma 2.4. Let Γ be a graph. If θ(Γ)=k, then θ(Γ+vk)=k. In particular, if Γ is a complete graph, then θ(Γ+vt)=k(1t2k).

    Proof. If Γ is a planar graph, then Γ+v1 is clearly a planar graph. If Γ is not a planar graph, then we may assume that {Γ1,Γ2,,Γk} is a planar decomposition of Γ, where each Γi is a spanning subgraph of Γ, then Γi+v1 are planar graphs. Hence {Γ1+v1,Γ2+v1,,Γk+v1} is a decomposition of Γ+vk. Thus, θ(Γ+vk)k. By Lemma 2.1, θ(Γ+vk)θ(Γ)=k and thus θ(Γ+vk)=k. In particular, if Γ is a complete graph, then θ(Γ+vt)=k(1t2k).

    Lemma 2.5. [31] A graph is outerplanar if and only if it has no subgraph homeomorphic to K4 or K2,3.

    The aim of this section is to classify finite abelian groups whose intersection graphs have thickness and outerthickness 1 and 2. We first consider the case of finite cyclic groups.

    Theorem 3.1. Let G=Zn be a cyclic group of order n and Γ(G) is not an empty graph. Then the following statements are hold.

    (1) θ(Γ(G))=1 if and only if G is one following groups:

    Zpα(α=2,3,4,5),Zp1pα2(α=1,2),Zp1p2p3,

    where p1, p2, p3 and p are distinct primes.

    (2) θ(Γ(G))=2 if and only if G is one following groups:

    Zpα(α=6,7,8,9),Zp1pα2(α=3,4),Zp21pα2(α=2,3),Zp1p2p23,Zp1p2p3p4,

    where p1, p2, p3, p4 and p are distinct primes.

    Proof. Suppose that n=pα11pα22pαss with α1α2αs. Then we have V(Γ(Zn))=si=1(αi+1)2. We complete our proofs by discussing the number of prime divisors of n.

    Case 1. s5. It is clear that

    p1,p2,p3,p4,p1p2,p1p3,p1p4,p1p2p3,p1p2p3p4

    are nine non-trivial subgroups of G. Note that any pair of these nine subgroups has non-trivial intersection. So they form a complete graph K9, which is a subgraph of Γ(G). Thus, ω(Γ(G))9. By Lemma 2.1, θ(Γ(G))θ(K9)=3.

    Case 2. s=4. If α42, then G has at least nine non-trivial subgroups, namely,

    p1,p2,p3,p4,p1p2,p1p3,p1p4,p1p2p3,p1p2p3p4.

    Note that any pair of these nine subgroups has non-trivial intersection. So they form a complete graph K9, which is a subgraph of Γ(G). Thus, θ(Γ(G))θ(K9)=3 by Lemma 2.1. If α4=1, then GZp1p2p3p4. So, G has totally 14 non-trivial subgroups, namely,

    p1,p2,p3,p4,p1p2,p1p3,p1p4,p2p3,p2p4,
    p3p4,p1p2p3,p1p2p4,p1p3p4,p2p3p4.

    Note that p1,p2,p3, p1p2, p1p3, p2p3, p1p2p3 form K7 as a subgraph of Γ(G). So ω(Γ(Zp1p2p3p4))7 and V(Γ(Zp1p2p3p4))=14. By Lemma 2.1 and Lemma 2.2, 2=θ(K7)θ(Γ(Zp1p2p3p4))θ(K14)=3. A planar decomposition of Γ(Zp1p2p3p4) shown in Figure 1 deduces that θ(Γ(Zp1p2p3p4))=2.

    Figure 1.  A planar decomposition of Γ(Zp1p2p3p4).

    Case 3. s=3. If α33, then

    p1,p2,p3,p23,p1p2,p1p3,p2p3,p1p2p3,p1p23

    are nine non-trivial subgroups of G. Any pair of these nine subgroups has non-trivial intersection. So they form K9, which is a subgraph of Γ(G). By Lemma 2.1, θ(Γ(G))θ(K9)=3. If α2=α3=2, then

    p1,p2,p3,p23,p1p2,p1p3,p2p3,p1p2p3,p1p23

    are nine non-trivial subgroups of G. Any pair of these nine subgroups has non-trivial intersection. They also form K9 as a subgraph of Γ(G). So θ(Γ(G))θ(K9)=3 by Lemma 2.1.

    If α1=α2=1,α3=2, then G has totally ten non-trivial subgroups, namely,

    p1,p2,p3,p23,p1p2,p1p3,p1p23,p2p3,p2p23,p1p2p3.

    Note that p1, p2, p3, p1p2, p1p3, p2p3, p1p2p3 form K7 as a subgraph of Γ(G). Then ω(Γ(G))7 and V(Γ(G))=10. By Lemma 2.1 and Lemma 2.2, 2=θ(K7)θ(Γ(G))θ(K10)=3. We can give a planar decomposition of Γ(Zp1p2p23) as shown in Figure 2. Thus, θ(Γ(G)=2.

    Figure 2.  A planar decomposition of Γ(Zp1p2p23).

    If α1=α2=α3=1, by Theorem 1.1, Γ(Zp1p2p3) is a planar graph. So θ(Γ(Zp1p2p3))=1.

    Case 4. s=2. In this case, it easy to see that Γ(G)Kα1α21+(Kα1Kα2). Then ω(Γ(Zpα11pα22))=α1α21+α2. If α1α21+α29, then θ(Γ(G))θ(K9)=3. So α1α21+α28 and implies that α24.

    If α2=4, then α1=1. So, Γ(G)K3+(K4K1). By Lemma 2.1 and Lemma 2.2, 2=θ(K7)θ(Γ(Zp1p42))θ(K8)=2. This deduces that θ(Γ(Zp1p42))=2.

    If α2=3, then α12. If α1=2, then Γ(G)K5+(K3K2), By Lemma 2.1 and Lemma 2.2, 2=θ(K8)θ(Γ(Zp21p32))θ(K10)=3. A planar decomposition of Γ(Zp21p32) as shown in Figure 3 deduces that θ(Γ(Zp21p32))=2. If α1=1, then Γ(G)K2+(K3K1). By Lemma 2.1 and Lemma 2.2, 2=θ(K5)θ(Γ(Zp1p32))θ(K6)=2. So, θ(Γ(Zp1p32))=2.

    Figure 3.  A planar decomposition of Γ(Zp21p32).

    If α2=α1=2, then Γ(G)K3+(K2K2). By Lemma 2.1 and Lemma 2.2, 2=θ(K5)θ(Γ(Zp21p22))θ(K7)=2. Thus, θ(Γ(Zp21p22))=2.

    If α2=2 and α1=1 or α1=α2=1, then by Theorem 1.1, Γ(Zp1p2) and Γ(Zp1p22) are planar graphs. So, θ(Γ(Zp1p2))=θ(Γ(Zp1p22))=1.

    Case 5. s=1. In this case, Γ(Zpα)Kα1. By Lemma 2.2, it deduces that if α=2,3,4,5, then θ(Γ(Zpα))=1; and if α=6,7,8,9, then θ(Γ(Zpα))=2.

    This completes the proof.

    For the outerthickness of Γ(G) of a cyclic group G, we have following theorem.

    Theorem 3.2. Let G=Zn be a cyclic group of order n and Γ(G) is not an empty graph. Then the following statements are hold.

    (1) θ0(Γ(G))=1 if and only if G is one following groups: Zpα(α=2,3,4), Zp1pα2(α=1,2), Zp1p2p3, where p1, p2, p3 and p are primes.

    (2) θ0(Γ(G))=2 if and only if G is one following groups: Zpα(α=5,6,7), Zp21p22, Zp1p32, where p1, p2 and p are primes.

    Proof Suppose that n=pα11pα22pαss with α1α2αs. According to the proof of Theorem 3.1, if s4, then θ0(Γ(G))3. We proceed with three cases.

    Case 1. s=3. Again according to the proof of Theorem 3.1, if θ0(Γ(G))2, then GZp1p2p3. In this case, it is clear that Γ(G) contains no subgraph homeomorphic to K4 or K2,3. By Lemma 2.5, Γ(Zp1p2p3) is an outerplanar graph. So, θ0(Γ(Zp1p2p3))=1.

    Case 2. s=2. In this case, Γ(G)Kα1α21+(Kα1Kα2). Then ω(Γ(G))=α1α21+α2. If α1α21+α27, then θ0(Γ(G))3. So α1α21+α26. This deduces that α23.

    If α2=3, then α1=1. So Γ(G)K2+(K3K1). By Lemma 2.1 and Lemma 2.2, 2=θ0(K5)θ0(Γ(Zp1p32))θ0(K6)=2. Hence θ0(Γ(Zp1p32))=2.

    If α2=2,α1=2, then Γ(G)K3+(K2K2). By Lemma 2.1 and Lemma 2.2, 2=θ0(K5)θ0(Γ(Zp21p22))θ0(K7)=3. An outerplanar decomposition of Γ(Zp21p22) as shown in Figure 4 deduces that θ0(Γ(Zp21p22))=2.

    Figure 4.  An outerplanar decomposition of Γ(Zp21p22).

    If α2=2,α1=1, then Γ(G)K1+(K2K1). It is clear that Γ(G) contains no subgraph homeomorphic to K4 or K2,3. By Lemma 2.5, Γ(Zp1p22) is an outerplanar graph. Then, θ0(Γ(Zp1p22))=1.

    If α1=α2=1, then Γ(G)K2. So, Γ(Zp1p2) is an outerplanar graph and thus θ0(Γ(Zp1p2))=1.

    Case 3. s=1. In this case, Γ(Zpα)Kα1. By Lemma 2.2, it deduces that if α=2,3,4, then θ0(Γ(Zpα))=1; and if α=5,6,7, then θ0(Γ(Zpα))=2. The proof is complete.

    Now, we consider the case of finite non-cyclic groups. Let G be a finite Abelian group but not a cyclic group. Then by fundamental theorem of finite abelian group, we have GG1G2Gs, where Gi(i=1,2,,s) is a cyclic group of prime power order.

    Lemma 3.3. Let GG1G2Gs, where s2 and Gi(i=1,2,,s) is a cyclic group of prime power order. If s4, then θ(Γ(G))3.

    Proof. Set

    H1=(1,0,0,0,,0),H2=(1,0,0,0,,0),(0,1,0,0,,0),
    H3=(1,0,0,0,,0),(0,0,1,0,,0),H4=(1,0,0,0,,0),(0,0,0,1,,0),
    H5=(1,0,0,0,,0),(0,1,1,0,,0),H6=(1,0,0,0,,0),(0,1,0,1,,0),
    H7=(1,0,0,0,,0),(0,0,1,1,,0),H8=(1,0,0,0,,0),(0,1,1,1,,0),
    H9=(1,0,0,0,,0),(0,1,0,0,,0),(0,0,1,0,,0),(0,1,1,0,,0).

    Then they are nine non-trivial subgroups of G. Note that any pair of these nine subgroups has non-trivial intersection. So they form K9, which is a subgraph of Γ(G). Thus, θ(Γ(G))θ(K9)=3.

    Theorem 3.4. Let G be a finite non-cyclic abelian group. Then the following statements are hold.

    (1) θ(Γ(G))=1 if and only if G is one following groups: Z2Z2p, ZpZp, where p is a prime.

    (2) θ(Γ(G))=2 if and only if G is one following groups: Z3Z3q, Z5Z5q, Z4Z4, Z2Z8, Z2Z2Z2, where q is a prime.

    Proof. Let the number of distinct prime factors of order of |G| be s. By Lemma 3.3, we know that if θ(Γ(G))2, then s3. By assumption that G is not cyclic, we have three cases for our discussion.

    Case 1. GZpαZpβZpγ, p is a prime.

    Case 2. GZpαZpβZqγ, p,q are distinct primes.

    Case 3. GZpαZpβ, p is a prime.

    For the Case 1, if there is at least one in {α,β,γ} greater than 1, then we may assume γ2. Set

    A1=(0,0,1),(0,1,0),A2=(0,0,1),(1,0,0),A3=(1,1,0),(1,1,1),
    A4=(0,1,0),(1,0,0),A5=(0,1,0),(1,0,1),A6=(1,0,0),(0,1,1),
    A7=(1,1,0),(0,1,1),A8=(0,0,p),(0,1,0),A9=(0,0,p),(1,0,0).

    It is easy to see that they are nine non-trivial subgroups of G. Note that any pair of these nine subgroups has non-trivial intersection. So they form K9, which is a subgraph of Γ(G). Thus, θ(Γ(G))θ(K9)=3.

    Now, we consider the case of α=β=γ=1. If p3, we let

    B1=(0,0,1),(0,1,0),B2=(0,0,1),(1,0,0),B3=(0,0,1),(1,1,0),
    B4=(0,0,1),(1,1,0),B5=(0,1,0),(1,0,0),B6=(0,1,0),(1,0,1),
    B7=(0,1,0),(1,0,1),B8=(0,1,1),(1,0,0),B9=(0,1,1),(1,0,1).

    They also form K9 and θ(Γ(G))θ(K9)=3.

    If p=2, then GZ2Z2Z2. G has totally 14 non-trivial subgroups, namely,

    C1=(0,0,1),(0,1,0),C2=(0,0,1),(1,0,0),C3=(1,1,0),(1,1,1),
    C4=(0,1,0),(1,0,0),C5=(0,1,0),(1,0,1),C6=(1,0,0),(0,1,1),
    C7=(1,1,0),(0,1,1),C8=(0,0,1),C9=(1,0,0),C10=(1,1,0),
    C11=(0,1,0),C12=(1,0,1),C13=(0,1,1),C14=(1,1,1).

    It is easy to see that Γ(G)=K7+7v3. By Lemma 2.4, we get θ(Γ(G))=θ(K7)=2.

    For the Case 2, we may assume αβ. If β2, we let

    D1=(0,p,0),D2=(0,1,0),(0,0,1),D3=(0,1,0),(1,0,0),
    D4=(0,1,0),D5=(0,p,0),(0,0,1),D6=(0,p,0),(1,0,0),
    D7=(1,1,0),D8=(0,p,0),(0,0,1),(1,0,0),D9=(0,p,0),(0,0,1),(1,1,0).

    Note that DiDj=D1 for all ij. So, they form a K9 and θ(Γ(G))θ(K9)=3.

    If γ2, then we let

    E1=(0,0,q),E2=(0,0,1),(1,0,0),E3=(0,0,1),(1,1,0),
    E4=(0,0,q),(0,1,0),E5=(0,0,q),(1,0,0),E6=(0,0,q),(1,1,0),
    E7=(0,0,1),E8=(0,0,1),(0,1,0),E9=(0,0,q),(0,1,0),(1,0,0).

    Note that EiEj=E1 for all ij. So, they form a K9 and θ(Γ(G))θ(K9)=3.

    Now, let GZpZpZq. Then all non-trivial subgroups of G are:

    I=(0,0,1),(0,1,0),
    Mi=(0,0,1),(1,i,0),where0ip1,
    T=(0,1,0),(1,0,0),K=(0,1,0),
    Nj=(1,j,0),where0jp1.

    It is not difficult to see that Γ(G)=(Kp+3e)+(p+1)v2. If p7, then θ(Γ(G))θ(K9)=3. So, p5. By Lemma 2.2, if p=2, then θ(Γ(G))=1; and if p=3,5, then θ(Γ(G))=2.

    For the Case 3, GZpαZpβ. We may assume that βα. If α2 and β3, then we set

    F1=(0,pα),F2=(0,p),F3=(0,1),F4=(0,1),(1,0),F5=(0,p),(1,0),
    F6=(0,1),(p,0),F7=(p,1),F8=(0,p),(p,0),F9=(p,p).

    They are non-trivial subgroups of G. Also, F1=FiFj for all ij. It follows that they form K9 as a subgraph of Γ(G). Thus, θ(Γ(G))θ(K9)=3.

    If α=β=2, then GZp2Zp2. Now, G has totally p+1 subgroups of order p, i.e., (0,p), (p,0), (p,p), (p2p,p); G has totally p2+p+1 subgroups of order p2, i.e., Q=(0,p),(p,0), (1,0), (0,1), (1,1), (p21,1); (1,p), (1,2p), (1,3p), (1,p2p); G has totally p+1 subgroups of order p3, i.e., (0,1),(p,0), (0,p),(1,0), (0,p),(p,0),(1,1), (0,p),(p,0),(1,2), (0,p),(p,0),(1,p1). It is not difficult to see that Γ(Zp2Zp2)Kp+2+(p+1)Kp+1. So, ω(Γ(Zp2Zp2))=2p+3,V(Γ(Zp2Zp2))=p2+3p+3. If p3, then ω(Γ(G))9 and θ(Γ(G))3. So p=2. By Lemma 2.1 and Lemma 2.2, 2=θ(K7)θ(Γ(Z4Z4)θ(K13)=3. On the other hand, a planar decomposition of Γ(Z4Z4) shown in Figure 5 gives θ(Γ(Z4Z4))=2.

    Figure 5.  A planar decomposition of Γ(Z4Z4).

    If α=1, then GZpZpβ. All subgroups of G can be listed below.

    (0,1),(1,1),(p1,1),

    (0,p),(1,p),(p1,p),

    (0,p2),(1,p2),(p1,p2),

    (0,pβ1),(1,pβ1),(p1,pβ1).

    It is clear that Γ(G)Kβ1+(Kpβp+1pK1). So Kβp+βp is a subgraph of Γ(G). If β4, then ω(Γ(G))9 and θ(Γ(G))3. So, β3.

    If β=3, then p=2 and GZ2Z8. So Γ(G)K2+(K52K1) and θ(Γ(Z2Z8)θ(K7)=2. By Lemma 2.4, θ(Γ(Z2Z8))=θ(K7)=2.

    If β=2, then p5. If p=2, then GZ2Z4 and Γ(G)K1+(K32K1). So, θ(Γ(Z2Z4))θ(K4)=1. By Lemma 2.4, θ(Γ(Z2Z4))=θ(K4)=1. If p=3, then GZ3Z9 and Γ(G)K1+(K43K1). Thus, θ(Γ(Z3Z9))θ(K5)=2. By Lemma 2.4, θ(Γ(Z3Z9))=θ(K5)=2. If p=5, then GZ5Z25 and Γ(G)K1+(K65K1). Thus, θ(Γ(Z5Z25))θ(K7)=2. By Lemma 2.4, then θ(Γ(Z5Z25))=θ(K7)=2.

    If β=1, then GZpZp. So, Γ(G)(p+1)K1. By Theorem 1.1, θ(Γ(ZpZp))=1.

    This completes our proof.

    Theorem 3.5. Let G be a finite non-cyclic abelian group. Then the following statements are hold.

    (1) θ0(Γ(G))=1 if and only if GZpZp, where p is a prime.

    (2) θ0(Γ(G))=2 if and only if GZ2Z2p or Z3Z3p, where p is a prime.

    Proof. Suppose GG1G2G3Gs, where s2 and Gi(i=1,2,3,,s) are cyclic groups. According to the proof of Theorem 3.4, if θ0(Γ(G))2, then GZpZp or GZpZpZq or GZpZp2.

    Case 1. GZpZp. In this case, Γ(G)(p+1)K1. It is clear that Γ(G) contains no subgraph homeomorphic to K4 or K2,3. By Lemma 2.5, Γ(ZpZp) is an outerplanar graph, then θ0(Γ(ZpZp))=1.

    Case 2. GZpZpZq. Again by the proof of Theorem 3.4, we know that ω(Γ(ZpZpZq))=p+2. If p5, then θ0(Γ(G))θ0(K7)=3. So p=2, 3. If p=2, then GZ2Z2q. It is easy to see that Γ(Z2Z2q)=(K5e)+3v2. By Lemma 2.1 and Lemma 2.2, 2=θ0(K4)θ0(Γ(Z2Z2q))θ0(K5)=2. Thus, θ0(Γ(Z2Z2q))=2. If p=3, then GZ3Z3q. It is easy to see that Γ(Z3Z3q)=(K6e)+4v2. By Lemma 2.1 and Lemma2.2, 2=θ0(K5)θ0(Γ(Z3Z3q))θ0(K6)=2. So θ0(Γ(Z3Z3q))=2.

    Case 3. GZpZp2. In this case, Γ(ZpZp2)K1+(Kp+1+pK1). Thus, ω(Γ(G))=p+2. If ω(Γ(G))7, then θ0(G)3. Thus, p=2,3. If p=2, then GZ2Z4 and Γ(G)K1+(K32K1). By Lemma 2.4, θ0(Γ(Z2Z4))=θ0(K4)=2. If p=3, then GZ3Z9 and Γ(G)K1+(K43K1). By Lemma 2.4, θ0(Γ(Z3Z9))=θ0(K5)=2.

    The proof is complete.

    Combine Theorems 3.1–3.5, we get the following theorem.

    Theorem 3.6. Let G be a finite abelian group and let p1,p2,p3,p4 and p be primes. Then the following statements are hold.

    (1) θ(Γ(G))=1 if and only if G is one following groups:

    Zpα(α=2,3,4,5),Zp1pα2(α=1,2),Zp1p2p3,Z2Z2p,ZpZp.

    (2) θ(Γ(G))=2 if and only if G is one following groups:

    Zpα(α=6,7,8,9),Zp1pα2(α=3,4),Zp21pα2(α=2,3),Zp1p2p23,Zp1p2p3p4,
    Z3Z3p,Z5Z5p,Z4Z4,Z2Z8,Z2Z2Z2.

    (3) θ0(Γ(G))=1 if and only if G is one following groups:

    Zpα(α=2,3,4),Zp1pα2(α=1,2),Zp1p2p3,ZpZp.

    (4) θ0(Γ(G))=2 if and only if G is one following groups:

    Zpα(α=5,6,7),Zp21p22,Zp1p32,Z2Z2p,Z3Z3p.

    In this section, we investigate the thickness and outerthickness of intersection graphs of subgroups of some finite non-abelian groups. For a positive integer n, τ(n) denotes the number of positive divisor of n; σ(n) denotes the sum of all the positive divisors of n. Sn and An are denoted the symmetric group and alternating group of degree n acting on {1,2,3,,n}, respectively.

    For any integer n3, the dihedral group of order 2n is defined by

    Dn={a,b|an=b2=1,ab=ba1}.

    For any integer n2, the generalized quaternion group of order 4n is defined by

    Qn={a,b|a2n=b4=1,an=b2,b1ab=a1}.

    For any prime p and integer α3, the modular group of order pα is defined by

    Mpα={a,b|apα1=bp=1,bab1=apα2+1}.

    For any integer n4, the quasi-dihedral group of order 2n is defined by

    QD2n={a,b|a2α1=b2=1,bab1=a2α21}.

    Lemma 4.1. [27] The following statements are hold.

    (1) For n3, n=pα11pα22pαkk, V(Γ(Dn))=τ(n)+σ(n)2 and ω(Γ(Dn))=σ(n)n1+ki=1αi.

    (2) Let n2 be an integer and 2n=2α1pα22pαkk, where p2,,pk are distinct odd primes and αi1. Then V(Γ(Qn))=τ(2n)+σ(n)2 and ω(Γ(Qn))=σ(n)1+α1ki=2(αi+1).

    (3) For α4, V(Γ(QD2α))=α+3(2α11) and ω(Γ(QD2α))=α+2α13.

    Theorem 4.2. Let G=Sn. Then the following statements are hold.

    (1) If n=3, then θ(Γ(G))=1.

    (2) If n4, then θ(Γ(G))3.

    Proof. If G=S3, then it is easy to see that Γ(S3)4K1, which is a planar graph. So, θ(Γ(G))=1. If n4, then we set

    S1={(1),(12),(34),(12)(34)},

    S2={(1),(12),(13),(23),(123),(132)},

    S3={(1),(12),(14),(24),(124),(142)},

    S4={(1),(13),(14),(34),(134),(143)},

    S5={(1),(23),(24),(34),(234),(243)},

    S6={(1),(13),(24),(13)(24),(14)(23),(12)(34),(1234),(1432)},

    S7={(1),(14),(23),(13)(24),(14)(23),(12)(34),(1243),(1342)},

    S8={(1),(13),(24),(13)(24),(14)(23),(12)(34),(1324),(1423)},

    S9={(1),(123),(132),(124),(142),(134),(143),(234),(243),(13)(24),(14)(23),(12)(34)}.

    Then they are nine non-trivial subgroups of Sn and form a complete graph K9 as a subgraph of Γ(Sn). Thus, θ(Γ(G))θ(K9)3.

    Theorem 4.3. Let G=An. Then the following statements are hold.

    (1) θ(Γ(G))=1 if and only if n=4.

    (2) If n5, then θ(Γ(G))3.

    Proof. If GA4, then Γ(A4)4K1K1,3, which is a planar graph. So, θ(Γ(G))=1. If n5, we let

    L1={(1),(12345),(13524),(14253),(15432),(12)(35),(13)(45),(14)(23),(15)(24),(25)(34)},

    L2={(1),(12453),(13542),(14325),(15234),(12)(34),(13)(25),(14)(35),(15)(24),(23)(45)},

    L3={(1),(12354),(13425),(14532),(15243),(12)(34),(13)(45),(14)(25),(15)(23),(24)(35)},

    L4={(1),(12435),(13254),(14523),(15342),(12)(45),(13)(24),(14)(35),(15)(23),(25)(34)},

    L5={(1),(12534),(13245),(14352),(15423),(12)(45),(13)(25),(14)(23),(15)(34),(24)(35)},

    L6={(1),(12543),(13452),(14235),(15324),(12)(35),(13)(24),(14)(25),(15)(34),(23)(45)},

    L7={(1),(123),(124),(134),(234),(132),(142),(143),(243),(13)(24),(14)(23),(12)(34)},

    L8={(1),(123),(125),(135),(235),(132),(152),(153),(253),(12)(35),(13)(25),(15)(23)},

    L9={(1),(124),(125),(145),(245),(142),(152),(154),(254),(12)(45),(14)(25),(15)(24)}.

    Then they are nine non-trivial subgroups of An and form a complete graph K9 as a subgraph of Γ(An). Thus, θ(Γ(G))θ(K9)3.

    Theorem 4.4. Let G=Dn(n3) be the dihedral group of order 2n. Then the following statements are hold.

    (1) θ(Γ(G))=1 if and only if GD4 or Dp, where p is an odd prime.

    (2) θ(Γ(G))=2 if and only if GD6,D9, D25 or D10.

    Proof. For n3, let n=pα11pα22pαkk with α1α2αk. By Lemma 4.1, we know that ω(Γ(Dn))=σ(n)n1+ki=1αi. If k2 and αk2, then ω(Γ(Dn))p1+pk+p1pk11. Thus, if θ(Γ(G))2, then n=p1p2 or n=pα.

    If n=p1p2, then by Lemma 4.1, ω(Γ(Dn))=σ(n)n1+ki=1αi=p1+p2+1. If p1+p2+19, then θ(Γ(Dn))3. Thus, if θ(Γ(Dn))2, then p1+p2+18. Thus, p1=2, p2=3 or p1=2, p2=5.

    If p1=2, p2=3, then ω(Γ(D6))=6. So, θ(Γ(D6))θ(K6)=2. Note that degΓ(D6)(aib)=τ(6)2=2 and V(aib)=6. By Lemma 2.4, θ(Γ(D6))=θ(Γ(D6)6v2). On the other hand, V(Γ(D6)6v2)=τ(6)1+σ(6)61=8, then 2=θ(K6)θ(Γ(D6)6v2)θ(K8)=2. Hence, θ(Γ(D6))=2.

    If p1=2 and p2=5, then ω(Γ(D10))=8. So, θ(Γ(D10))θ(K8)=2. Note that degΓ(D10)(aib)=τ(10)2=2 and V(aib)=n=10. By Lemma 2.4, θ(Γ(D10))=θ(Γ(D10)10v2). As V(Γ(D10)10v2)=τ(10)1+σ(10)101=10, we have 2=θ(K8)θ(Γ(D10)10v2)θ(K10)=3. On the other hand, a planar decomposition of Γ(D10)10v2 as shown in Figure 6 deduces that θ(Γ(D10))=2.

    Figure 6.  A planar decomposition of Γ(D10)10v2.

    If n=pα, then by Lemma 4.1, ω(Γ(Dn))=pαpp1+α. If ω(Γ(Dn))9, then θ(Γ(Dn))3. So, α2.

    If α=1, then Γ(Dp)(p+1)K1 and thus θ(Γ(Dp))=1. If α=2, then ω(Γ(Dp2))=p+2. Note that degΓ(Dp2)(aib)=τ(p2)2=1, V(aib)=n=p2, and V(Γ(Dp2))=τ(n)+σ(n)2=1+p+p2+32=p2+p+2. So, Γ(Dp2)=Kp+2+p2v1. By Lemma 2.4, θ(Γ(Dp2))=θ(Kp+2). Since p+29 deduces θ(Γ(Dp2))3, we have p+28, that is, p=2,3,5. By Lemma 2.2, we get if p=2, then θ(Γ(D4))=1; and if p=3,5, then θ(Γ(D9))=θ(Γ(D25))=2.

    Theorem 4.5. Let G=Qn. Then the following statements are hold.

    (1) θ(Γ(G))=1 if and only if G=Q2.

    (2) θ(Γ(G))=2 if and only if GQ3 or Q5.

    Proof. Let n2 be an integer and 2n=2α1pα22pαkk, where pi are distinct primes and αi1. By Lemma 4.1, ω(Γ(Qn))=σ(n)1+α1ki=2(αi+1). If k2, then ω(Γ(Qn))1+p1+p2+p1p21+1×2×2=4+p1+p2+p1p2>4+2+3+2×3>9 and θ(Γ(Qn))>3. So, if θ(Γ(Qn))2, then we have n=pα.

    If n=2α, ω(Γ(Q2α))=σ(n)1+α1ki=2(αi+1)=2α+11+α,V(Γ(Q2α))=τ(2n)+σ(n)2=2α+11+α, then Γ(Q2α) is a complete graph. If 2α+11+α9, then θ(Γ(Qn))3. So, if θ(Γ(Qn))2, then 2α+11+α8 and thus α=1. In this case, θ(Γ(Q2))=1.

    If n=pα and p2, then ω(Γ(Qn))=σ(n)1+α1ki=2(αi+1)=1+p+p2++pα1+pα+1×(α+1)1=pα+11p1+α9, then θ(Γ(Qn))3. So, if θ(Γ(Qn))2, then pα+11p1+α8, then α=1.

    For α=1, ω(Γ(Qn))=σ(n)1+α1ki=2(αi+1)=p+2 and V(Γ(Qn))=τ(2n)+σ(n)2=p+3. So, θ(Kp+2)θ(Γ(Qn))θ(Kp+3). If ω(Γ(Qn))=p+29, then θ(Γ(Qn))3. So, if θ(Γ(Qn))2, then p+28, then p=3,5.

    By Lemma 2.1 and Lemma 2.2, 2=θ(K5)θ(Γ(Q3))θ(K6)=2 and 2=θ(K7)θ(Γ(Q5))θ(K8)=2, then θ(Γ(Q3))=θ(Γ(Q5))=2.

    Theorem 4.6. Let G=QD2α. Then θ(Γ(G))3.

    Proof. For α4, by Lemma 4.1, we have ω(Γ(QD2α))=α+2α139. Thus, θ(Γ(QD2α))θ(K9)=3.

    Theorem 4.7. Let GMpα. Then the following statements are hold.

    (1) θ(Γ(G))=1 if and only if G=M8.

    (2) θ(Γ(G))=2 if and only if GM27,M125 or M16.

    Proof. If pα=8, then Γ(M8)Γ(D4). So θ(Γ(M8))=1. If pα8, then Γ(Mpα)Γ(ZpZpα1). If θ(Γ(G))2, then by the proof of Theorem 3.5, we know α=3,4. If α=3, then Γ(Mp3)K1+(Kp+1pK1). By θ(Γ(G))2, we get p=3,5. By Lemma 2.4, θ(Γ(M27))=θ(K5)=2 and θ(Γ(M125))=θ(K7)=2. If α=4, then p=2. By Theorem 3.5, θ(Γ(M16))=θ(Γ(Z2Z8))=2.

    Theorem 4.8. Let G=Sn. Then the following statements are hold.

    (1) If n=3, then θ0(Γ(G))=1.

    (2) If n4, then θ0(Γ(G))3.

    Proof. If G=S3, then Γ(S3)4K1, which is an outerplanar graph. So θ0(Γ(S3))=1. By the proof of Theorem 4.2, if n4, then θ0(Γ(G))θ0(Γ(K9))=3.

    Theorem 4.9. Let GAn. Then the following statements are hold.

    (1) If n=4, then θ0(Γ(G))=1.

    (2) If n5, then θ0(Γ(G))3.

    Proof. Note that Γ(A4)4K1K1,3. It is clear that Γ(A4) contains no subgraph homeomorphic to K4 or K2,3. By Lemma 2.5, Γ(A4) is an outerplanar graph. So θ0(Γ(A4))=1. By the proof of Theorem 4.3, if n5, then θ0(Γ(G))θ0(K9)=3.

    Theorem 4.10. Let GDn. Then the following statements are hold.

    (1) θ0(Γ(G))=1 if and only if GDp, where p is a prime.

    (2) θ0(Γ(G))=2 if and only if G is one following groups: D4, D6, D9.

    Proof. For n3, let n=pα11pα22pαkk with α1α2αk. According to the proof of Theorem 4.4, if θ0(Γ(Dn))2, then n=p1p2 or pα.

    If n=p1p2, then ω(Γ(Dn))=σ(n)n1+ki=1αi=p1+p2+1. If p1+p2+17, then θ0(Γ(Dn))3. So, if θ0(Γ(Dn))2, then p1+p2+16, that is, p1=2, p2=3. As ω(Γ(D6))=2+3+1=6, we know θ0(Γ(D6))θ0(K6)=2. An outerplanar decomposition of Γ(D6)6v2 as shown in Figure 7 gives θ0(Γ(D6))=2.

    Figure 7.  An outerplanar decomposition of Γ(D66v2).

    If n=pα, then ω(Γ(Dn))=σ(n)n1+ki=1αi=p+p2++pα1+α=pαpp1+α7, then θ0(Γ(Dn))3. So, if θ0(Γ(Dn))2, then pαpp1+α6, so α=1,2.

    If α=1, then Γ(Dp)(p+1)K1. So, θ0(Γ(Dp))=1.

    If α=2, Γ(Dp2)=Kp+2+p2v1. By Lemma 2.4, θ0(Γ(Dp2))=θ0(Kp+2). As p+27, then θ0(Γ(Dp2))3. So, if θ0(Γ(Dp2))2, then p+26, then p=2,3. By Lemma 2.2, we get if p=2, then θ0(Γ(D4))=θ0(K4)=2; and if p=3, then θ0(Γ(D9))=θ0(K5)=2.

    Theorem 4.11. Let GQn. Then the following statements are hold.

    (1) θ0(Γ(G))2.

    (2) θ0(Γ(G))=2 if and only if GQ2 or Q3.

    Proof. According the proof of Theorem 4.5, if θ0(Γ(Qn))2, then n=pα.

    If n=2α, again by the proof of Theorem 4.5, we have Γ(Q2α)=K2α+11+α. If 2α+11+α7, then θ0(Γ(Qn))3. So θ0(Γ(Qn))2, then 2α+11+α6, so α=1. In this case, θ0(Γ(Q2))=2.

    If n=pα and p2, then ω(Γ(Qn))=pα+11p1+α7 deduces θ0(Γ(Qn))3. So, if θ0(Γ(Qn))2, then pα+11p1+α6 and α=1.

    In this case, ω(Γ(Qn))=σ(n)1+α1ki=2(αi+1)=p+2 and V(Γ(Qn))=τ(2n)+σ(n)2=p+3. So, θ0(Kp+2)θ0(Γ(Qn))θ0(Kp+3). If ω(Γ(Qn))=p+27, then θ0(Γ(Qn))3. So, if θ0(Γ(Qn))2, then p+26, so p=3. By Lemma 2.1 and Lemma 2.2, 2=θ0(K5)θ0(Γ(Q3))θ0(K6)=2, then θ0(Γ(Q3))=2.

    Theorem 4.12. Let GQD2α. Then θ0(Γ(G)3.

    Proof. For α4, by Lemma 4.1, we have ω(Γ(QD2α))=α+2α139. Thus, θ0(Γ(QD2α))θ0(K9)=3.

    Theorem 4.13. Let G=Mpα. Then the following statements are hold.

    (1) θ0(Γ(G))2.

    (2) θ0(Γ(G))=2 if and only if GM8 or M27.

    Proof. If pα=8, then Γ(M8)Γ(D4). By Theorem 4.4, θ0(Γ(M8))=2. If pα8, then Γ(Mpα)Γ(ZpZpα1). By the proof of Theorem 4.7, we get α=3,p=3. In this case, we have θ0(Γ(M27))=2.

    In this paper, we classify all finite abelian groups whose thickness and outerthickness of subgroup intersection graphs are 1 and 2, respectively. We also investigate the thickness and outerthickness of subgroup intersection graphs for some finite non-abelian groups. The method is finding a forbidden subgraph of Γ(G), like K9 and then determining the remaining cases. The key point is how to separate Γ(G) into two planar graphs. Next, one can consider to determine all finite abelian groups G whose Γ(G) has thickness or outerthickness 3. For a general finite group G, how to determine the thickness and outerthickness of Γ(G) is a challenge. Of course, investigating other properties of Γ(G) is a valuable exploration.

    We would like to thank the referees for a careful reading of the paper and for their valuable comments. This work was supported by the National Natural Science Foundation of China (Grant No. 11661013, 11961050), the Guangxi Natural Science Foundation (Grant No. 2020GXNSFAA159103, 2020GXNSFAA159053) and High-level talents for scientific research of Beibu Gulf University (2020KYQD07). The second author is supported by the Science and Technology Research Project of Jiangxi Education Department.

    The authors declared that they have no conflicts of interest to this work.



    [1] J. Feng, S. Liu, An improved generalized Newton method for absolute value equations, SpringerPlus, 5 (2016), 1042. https://doi.org/10.1186/s40064-016-2720-5 doi: 10.1186/s40064-016-2720-5
    [2] J. Feng, S. Liu, A new two-step iterative method for solving absolute value equations, J. Inequal. Appl., 2019 (2019), 39. https://doi.org/10.1186/s13660-019-1969-y doi: 10.1186/s13660-019-1969-y
    [3] L. Abdallah, M. Haddou, T. Migot, Solving absolute value equation using complementarity and smoothing functions, J. Comput. Appl. Math., 327 (2018), 196–207. https://doi.org/10.1016/j.cam.2017.06.019 doi: 10.1016/j.cam.2017.06.019
    [4] F. Mezzadri, On the solution of general absolute value equations, Appl. Math. Lett., 107 (2020), 106462. https://doi.org/10.1016/j.aml.2020.106462 doi: 10.1016/j.aml.2020.106462
    [5] I. Ullah, R. Ali, H. Nawab, Abdussatar, I. Uddin, T. Muhammad, et al., Theoretical analysis of activation energy effect on prandtl–eyring nanoliquid flow subject to melting condition, J. Non-Equil. Thermody., 47 (2022), 1–12. https://doi.org/10.1515/jnet-2020-0092 doi: 10.1515/jnet-2020-0092
    [6] M. Amin, M. Erfanian, A dynamic model to solve the absolute value equations, J. Comput. Appl. Math., 333 (2018), 28–35. https://doi.org/10.1016/j.cam.2017.09.032 doi: 10.1016/j.cam.2017.09.032
    [7] L. Caccetta, B. Qu, G. L. Zhou, A globally and quadratically convergent method for absolute value equations, Comput. Optim. Appl., 48 (2011), 45–58. https://doi.org/10.1007/s10589-009-9242-9 doi: 10.1007/s10589-009-9242-9
    [8] C. Chen, D. Yu, D. Han, Optimal parameter for the SOR-like iteration method for solving the system of absolute value equations, arXiv. Available from: https://arXiv.org/abs/2001.05781.
    [9] M. Dehghan, A. Shirilord, Matrix multisplitting Picard-iterative method for solving generalized absolute value matrix equation, Appl. Numer. Math., 158 (2020), 425–438. https://doi.org/10.1016/j.apnum.2020.08.001 doi: 10.1016/j.apnum.2020.08.001
    [10] X. Dong, X. H. Shao, H. L. Shen, A new SOR-like method for solving absolute value equations, Appl. Numer. Math., 156 (2020), 410–421. https://doi.org/10.1016/j.apnum.2020.05.013 doi: 10.1016/j.apnum.2020.05.013
    [11] V. Edalatpour, D. Hezari, D. K. Salkuyeh, A generalization of the Gauss-Seidel iteration method for solving absolute value equations, Appl. Math. Comput., 293 (2017), 156–167. https://doi.org/10.1016/j.amc.2016.08.020 doi: 10.1016/j.amc.2016.08.020
    [12] A. J. Fakharzadeh, N. N. Shams, An efficient algorithm for solving absolute value equations, J. Math. Ext., 15 (2021), 1–23. https://doi.org/10.30495/JME.2021.1393 doi: 10.30495/JME.2021.1393
    [13] X. M. Gu, T. Z. Huang, H. B. Li, S. F. Wang, L. Li, Two-CSCS based iteration methods for solving absolute value equations, J. Appl. Math. Comput., 7 (2017), 1336–1356. https://doi.org/10.11948/2017082 doi: 10.11948/2017082
    [14] P. Guo, S. L. Wu, C. X. Li, On the SOR-like iteration method for solving absolute value equations, Appl. Math. Lett., 97 (2019), 107–113. https://doi.org/10.1016/j.aml.2019.03.033 doi: 10.1016/j.aml.2019.03.033
    [15] F. Hashemi, S. Ketabchi, Numerical comparisons of smoothing functions for optimal correction of an infeasible system of absolute value equations, Numer. Algebra Control Optim., 10 (2020), 13–21. https://doi.org/10.3934/naco.2019029 doi: 10.3934/naco.2019029
    [16] I. Uddin, I. Ullah, R. Ali, I. Khan, K. S. Nisar, Numerical analysis of nonlinear mixed convective MHD chemically reacting flow of Prandtl-Eyring nanofluids in the presence of activation energy and Joule heating, J. Therm. Anal. Calorim., 145 (2021), 495–505. https://doi.org/10.1007/s10973-020-09574-2 doi: 10.1007/s10973-020-09574-2
    [17] S. L. Hu, Z. H. Huang, A note on absolute value equations, Optim. Lett., 4 (2010), 417–424. https://doi.org/10.1007/s11590-009-0169-y doi: 10.1007/s11590-009-0169-y
    [18] S. Ketabchi, H. Moosaei, Minimum norm solution to the absolute value equation in the convex case, J. Optim. Theory Appl., 154 (2012), 1080–1087. https://doi.org/10.1007/s10957-012-0044-3 doi: 10.1007/s10957-012-0044-3
    [19] Y. F. Ke, C. F. Ma, SOR-like iteration method for solving absolute value equations, Appl. Math. Comput., 311 (2017), 195–202. https://doi.org/10.1016/j.amc.2017.05.035 doi: 10.1016/j.amc.2017.05.035
    [20] Y. F. Ke, The new iteration algorithm for absolute value equation, Appl. Math. Lett., 99 (2020), 105990. https://doi.org/10.1016/j.aml.2019.07.021 doi: 10.1016/j.aml.2019.07.021
    [21] C. X. Li, A preconditioned AOR iterative method for the absolute value equations, Int. J. Comput. Methods, 14 (2017), 1750016. https://doi.org/10.1142/S0219876217500165 doi: 10.1142/S0219876217500165
    [22] H. Moosaei, S. Ketabchi, M. A. Noor, J. Iqbal, V. Hooshyarbakhsh, Some techniques for solving absolute value equations, Appl. Math. Comput., 268 (2015), 696–705. https://doi.org/10.1016/j.amc.2015.06.072 doi: 10.1016/j.amc.2015.06.072
    [23] O. L. Mangasarian, R. R. Meyer, Absolute value equation, Linear Algebra Appl., 419 (2006), 359–367. https://doi.org/10.1016/j.laa.2006.05.004 doi: 10.1016/j.laa.2006.05.004
    [24] O. L. Mangasarian, Absolute value programming, Comput. Optim. Applic., 36 (2007), 43–53. https://doi.org/10.1007/s10589-006-0395-5 doi: 10.1007/s10589-006-0395-5
    [25] O. L. Mangasarian, Absolute value equation solution via concave minimization, Optim. Lett., 1 (2007), 3–8. https://doi.org/10.1007/s11590-006-0005-6 doi: 10.1007/s11590-006-0005-6
    [26] O. L. Mangasarian, Linear complementarity as absolute value equation solution, Optim. Lett., 8 (2014), 1529–1534. https://doi.org/10.1007/s11590-013-0656-z doi: 10.1007/s11590-013-0656-z
    [27] X. H. Miao, J. T. Yang, B. Saheya, J. S. Chen, A smoothing Newton method for absolute value equation associated with second-order cone, Appl. Numer. Math., 120 (2017), 82–96. https://doi.org/10.1016/j.apnum.2017.04.012 doi: 10.1016/j.apnum.2017.04.012
    [28] C. T. Nguyen, B. Saheya, Y. L. Chang, J. S. Chen, Unified smoothing functions for absolute value equation associated with second-order cone, Appl. Numer. Math., 135 (2019), 206–227. https://doi.org/10.1016/j.apnum.2018.08.019 doi: 10.1016/j.apnum.2018.08.019
    [29] O. A. Prokopyev, On equivalent reformulations for absolute value equations, Comput. Optim. Appl., 44 (2009), 363. https://doi.org/10.1007/s10589-007-9158-1 doi: 10.1007/s10589-007-9158-1
    [30] J. Rohn, A theorem of the alternatives for the equation Ax+B|x|=b, Linear Multilinear Algebra, 52 (2004), 421–426. https://doi.org/10.1080/0308108042000220686 doi: 10.1080/0308108042000220686
    [31] J. Rohn, V. Hooshyarbakhsh, R. Farhadsefat, An iterative method for solving absolute value equations and sufficient conditions for unique solvability, Optim. Lett., 8 (2014), 35–44. https://doi.org/10.1007/s11590-012-0560-y doi: 10.1007/s11590-012-0560-y
    [32] B. Saheya, C. H. Yu, J. S. Chen, Numerical comparisons based on four smoothing functions for absolute value equation, J. Appl. Math. Comput., 56 (2018), 131–149. https://doi.org/10.1007/s12190-016-1065-0 doi: 10.1007/s12190-016-1065-0
    [33] R. S. Varga, Matrix iterative analysis, New Jersey: Prentice-Hall, Englewood Cliffs, 1962.
    [34] S. L. Wu, C. X. Li, A special shift splitting iteration method for absolute value equation, AIMS Math., 5 (2020), 5171–5183. https://doi.org/10.3934/math.2020332 doi: 10.3934/math.2020332
    [35] S. L. Wu, The unique solution of a class of the new generalized absolute value equation, Appl. Math. Lett., 116 (2021), 107029. https://doi.org/10.1016/j.aml.2021.107029 doi: 10.1016/j.aml.2021.107029
    [36] R. Ali, M. R. Khan, A. Abidi, S. Rasheed, A. M. Galal, Application of PEST and PEHF in magneto-Williamson nanofluid depending on the suction/injection, Case Stud. Therm. Eng., 27 (2021). https://doi.org/10.1016/j.csite.2021.101329 doi: 10.1016/j.csite.2021.101329
    [37] C. X. Li, S. L. Wu, Modified SOR-like iteration method for absolute value equations, Math. Probl. Eng., 2020 (2020), 9231639. https://doi.org/10.1155/2020/9231639 doi: 10.1155/2020/9231639
    [38] M. R. Khan, M. X. Li, S. P. Mao, R. Ali, S. Khan, Comparative study on heat transfer and friction drag in the flow of various hybrid nanofluids efiected by aligned magnetic fleld and nonlinear radiation, Sci. Rep., 11 (2021), 3691.
    [39] J. Y. Bello Cruz, O. P. Ferreira, L. F. Prudente, On the global convergence of the inexact semi-smooth Newton method for absolute value equation, Comput. Optim. Appl., 65 (2016), 93–108. https://doi.org/10.1007/s10589-016-9837-x doi: 10.1007/s10589-016-9837-x
    [40] G. Ning, Y. Zhou, An improved differential evolution algorithm for solving absolute value equations, In: J. Xie, Z. Chen, C. Douglas, W. Zhang, Y. Chen, High performance computing and applications, Lecture Notes in Computer Science, Springer, 9576 (2016), 38–47. https://doi.org/10.1007/978-3-319-32557-6
    [41] D. K. Salkuyeh, The Picard-HSS iteration method for absolute value equations, Optim. Lett., 8 (2014), 2191–2202. https://doi.org/10.1007/s11590-014-0727-9 doi: 10.1007/s11590-014-0727-9
    [42] Z. Z. Bai, Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl., 17 (2010), 917–933. https://doi.org/10.1002/nla.680 doi: 10.1002/nla.680
    [43] R. Ali, A. Ali, S. Iqbal, Iterative methods for solving absolute value equations, J. Math. Comput. Sci., 26 (2022), 322–329. https://doi.org/10.22436/jmcs.026.04.01 doi: 10.22436/jmcs.026.04.01
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