Research article Special Issues

New classifications of monotonicity investigation for discrete operators with Mittag-Leffler kernel


  • Received: 07 December 2021 Revised: 26 January 2022 Accepted: 10 February 2022 Published: 15 February 2022
  • This paper deals with studying monotonicity analysis for discrete fractional operators with Mittag-Leffler in kernel. The νmonotonicity definitions, namely ν(strictly) increasing and ν(strictly) decreasing, are presented as well. By examining the basic properties of the proposed discrete fractional operators together with νmonotonicity definitions, we find that the investigated discrete fractional operators will be ν2(strictly) increasing or ν2(strictly) decreasing in certain domains of the time scale Na:={a,a+1,}. Finally, the correctness of developed theories is verified by deriving mean value theorem in discrete fractional calculus.

    Citation: Pshtiwan Othman Mohammed, Christopher S. Goodrich, Aram Bahroz Brzo, Dumitru Baleanu, Yasser S. Hamed. New classifications of monotonicity investigation for discrete operators with Mittag-Leffler kernel[J]. Mathematical Biosciences and Engineering, 2022, 19(4): 4062-4074. doi: 10.3934/mbe.2022186

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  • This paper deals with studying monotonicity analysis for discrete fractional operators with Mittag-Leffler in kernel. The νmonotonicity definitions, namely ν(strictly) increasing and ν(strictly) decreasing, are presented as well. By examining the basic properties of the proposed discrete fractional operators together with νmonotonicity definitions, we find that the investigated discrete fractional operators will be ν2(strictly) increasing or ν2(strictly) decreasing in certain domains of the time scale Na:={a,a+1,}. Finally, the correctness of developed theories is verified by deriving mean value theorem in discrete fractional calculus.



    The pattern of connections between components in a parallel computer system is called the interconnection network of that system. Current massively parallel processing systems are connected by interconnection networks with tens of thousands of processors, compute nodes, storage units, etc., thus achieving spatial parallelism. As the number of processors in a parallel computer continues to increase, the overhead of communicating between processors through interconnections is also increasing. Therefore, the implementation of parallel computer system functions depends heavily on the performance of the system interconnection network.

    An interconnection network is a network in which multiple processors or functional components within a computer are interconnected by switching elements according to a certain topology and control method. The topology of an interconnection network is the main structural characteristic of an interconnection network. From a graph theory perspective, the topology of an interconnection network can be represented as a graph, with the vertices of the graph representing the processors in the system, and the edges of the graph representing the communication links between the components.

    Traditionally, the reliability of interconnection networks has been measured in terms of connectivity of a graph. In the late 1980s, in the study of the network reliability, it was found that there were some obvious shortcomings in measuring the reliability of interconnected networks in terms of edge connectivity and connectivity of a graph: first, both parameters are analyzed and applied with the implicit assumption that all neighbors of each node would fail at the same time. However, a network application practice shows that this is almost impossible. Second, when nodes fail simultaneously and the remaining network does not remain connected, no further consideration is given to whether each connected component still retains certain necessary properties.

    Inspired by the classical connectivity defect, Harary[13] proposed the conditional connectivity, which is the minimum number of vertices removed to make the graph disconnected and placing some requirements on the components. Among all kinds of conditional connectivities, the g-extra connectivity proposed by Fàbrega and Fiol [3,4] is one of the most studied conditional connectivity. Let S be a vertex set of a graph G. If S is called a g-extra cut, then GS is disconnected and each component of GS has at least g+1 vertices. The cardinality of a minimum g-extra cut of G, denoted by κg(G), is the g-extra connectivity of G. Obviously, κ0(G)=κ(G), so the g-extra connectivity can be regarded as a generalization of classical connectivity and it can more accurately measure the reliability of a network. Determining the general g-extra connectivity of a graph is not easy work, so there are many results when g is a small parameter and there are also some general results, refer to [5,9,11,12,14,19,21]. What fails in an interconnection network is often a specific structure, not just individual vertices. The concepts of the structure connectivity and the substructure connectivity were proposed by Lin et al.[10]. Let H be a connected subgraph of G and F be a set of subgraphs of a graph G such that every element in F is isomorphic to H (resp. the subgraph of H). If GV(F) is disconnected, then F is called an H-structure cut (resp. H-substructure cut). The minimum cardinality of H-structure cuts (resp. H-substructure cuts) is called the H-structure connectivity (resp. H-substructure connectivity) of G, denoted by κ(G;H) (resp. κs(G;H)). The results of the structure connectivity and substructure connectivity of many network graphs have been studied[6,7,8,17,18].

    In order to more accurately measure the reliability of a network, we combine the concepts of the g-extra connectivity and the structure connectivity and substructure connectivity to propose two novel kinds of connectivity: g-extra H-structure connectivity and g-extra H-substructure connectivity. The two novel connectivities not only retain each connected component property after deleting the faulty vertices, but also take into account the structure of the deleted vertices, which is a more general conditional connectivity.

    In this paper, we use GF to represent the subgraph obtained from G by deleting all vertices in F. The following are definitions of the g-extra H-structure connectivity and g-extra H-substructure connectivity.

    Definition 1.1. Let H be a connected subgraph of G and g be a nonnegative number. If F is a set of subgraphs of G such that every element in F is isomorphic to H, then F is called a g-extra H-structure cut satisfying that GF is disconnected and each component of GF has at least g+1 vertices. The g-extra H-structure connectivity of G, denoted by κg(G;H), is defined as κg(G;H) = min{|F||F is a g-extra H-structure cut of G}, where |F| is the number of elements in F.

    Definition 1.2. Let H be a connected subgraph of G and g be a nonnegative number. If F is a set of subgraphs of G such that every element in F is isomorphic to a connected subgraph H, then F is called a g-extra H-substructure cut satisfying that GF is disconnected and each component of GF has at least g+1 vertices. The g-extra H-substructure connectivity of G, denoted by κsg(G;H), is defined as κsg(G;H) = min{|F||F is a g-extra H-substructure cut of G}, where |F| is the number of elements in F.

    Obviously, κsg(G;H)κg(G;H).

    Table 1 gives some of the notations that will be used in this paper. For relevant concepts and notations which are not mentioned in this Table 1 below may refer to [1].

    Table 1.  Notations in this paper.
    Notations Meaning
    Pn a path of length n1, denoted by v1v2vn
    Cn a cycle of length n, denoted by v1v2vnv1
    V(G) the vertex set of a graph G
    E(G) the edge set of a graph G
    N(v) the set of vertices adjacent to the vertex v in Qn
    κ(G) the connectivity of a graph G
    κg(G) the g-extra connectivity of a graph G
    S a set of vertices
    N(S) the neighbors in V(Qn)S of vertices in S
    |S| the number of vertices in S
    G[S] the subgraph induced by S
    k-regular every vertex of a graph has exactly k neighbors
    (u,v) an edge whose end vertices are u and v
    K1,h a star where one vertex has h neighbors and h vertices have a common neighbor
    GH a graph G is isomorphic to a graph H

     | Show Table
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    The hypercube is the most dominant interconnection network topology used by computer systems today, for example, the famous parallel computing systems Cray T3D, Cray T3E, IBM Blue Gene, etc., are built with it as the interconnection network topology. The hypercube Qn has n2n1 edges and 2n vertices. For every vertex v in Qn, it is represented as v=x1x2xn for xi{0,1} and 1in. If two vertices differ in only one position, then they are adjacent. Noting that vi=x1x2¯xixn as the neighbor of v with position i different from v. Denoting vi,j as the vertex that position i and position j are different from v and the other positions are the same. Similarly, v1,2,,k denotes a vertex that is not identical to vertex v at position from 1 to k, for 1kn. We set Qin be the subgraph of Qn induced by the bit n being i, where i{0,1}. Obviously, Qin is isomorphic to Qn1 for i{0,1} (See Figure 1). Note that Qn is a bipartite graph, so there is no odd cycles in Qn.

    Figure 1.  Qn for n=1,2,3.

    The following are some useful lemmas in this paper.

    Lemma 2.1. [20] There is no 3-cycle in Qn and the cardinality of cycle in Qn is at least four.

    Lemma 2.2. [15] Any two vertices in Qn (n3) have exactly two common neighbors, if they have any.

    Lemma 2.3. [16] Let C be a subgraph of Qn with |V(C)|=g+1 for n4. Then |NQn(C)|(g+1)n2g(g2).

    Lemma 2.4. [16] For n4,

    κg(Qn)={(g+1)n2g(g2),if0gn4;n(n1)2,ifn3gn.

    Lemma 2.5. [16] For HV(Qn) and QnH is disconnected, if |H|2n2 for n3, then QnH has an isolated vertex (or an isolated edge) and a large component. Moreover, when |H|=2n2, we have that QnH has an isolated edge.

    Lemma 2.6. Let Pk(k3) be a path with k vertices in Qn. For any edge (u,v)E(Qn) and {u,v}V(QnPk), |N({u,v})V(Pk)|2k3.

    Proof. For any three consecutive vertices, denoted by x,y and z, on a path Pk in Qn with {(x,y),(y,z)}E(Qn). We claim that |N({u,v}){x,y,z}|2. We prove this result by contradiction. Suppose that |N({u,v}){x,y,z}|=3. If there exist two adjacent vertices of x,y,z such that both are adjacent to either u or v, then there is a 3-cycle, a contradiction to Lemma 2.1 (See Figure 2(a, b)). Otherwise, by symmetry, we may assume that x and z are adjacent only to u and y is adjacent only to v. Then, N(u)N(y)={x,z,v}, contradicting with Lemma 2.2 (See Figure 2(c)).

    Figure 2.  Illustration of the graph for Lemma 2.6.

    Lemma 2.7. Let Pk(k4) be a path with k vertices in Qn. For any P3 in Qn, denoted by uvw and {u,v,w}V(QnPk), then |N({u,v,w})V(Pk)|3k4.

    Proof. For any four consecutive vertices, denoted by t,x,y,z, on a path Pk in Qn with {(t,x),(x,y), (y,z)}E(Qn). We claim that |N({u,v,w}){t,x,y,z}|3. Prove this result with a contradiction. Suppose that |N({u,v,w}){t,x,y,z}|=4. By Lemma 2.6, at most two of the three consecutive vertices on a path Pk are neighbors of {u,v} and |N({u,v}){t,x,y,z}|3. By symmetry, two cases need to be considered: (1) u is adjacent to t and v is adjacent to x and z; (2) u is adjacent to x and z and v is adjacent to t. If |N({u,v,w}){t,x,y,z}|=4, then w is adjacent to y. In Case (1), N(v)N(y)={x,z,w}, contradicting with Lemma 2.2. In Case (2), there is a 5-cycle, a contradiction (See Figure 3).

    Figure 3.  Illustration of the graph for Lemma 2.7.

    In the following section 3, we will give the main results of this paper.

    Lemma 3.1. κ1(Qn;P2)n1 for n4.

    Proof. For any two vertices u and v in Qn and (u,v)E(Qn), without loss generality, suppose that u=000 and v=100. By the structure of Qn, we know that ui and vi are adjacent for 2in. Hence, (ui,vi)E(Qn) and uiviP2. It follows that there are (n1) P2s to be adjacent to uv (See Figure 4). Let F={uivi|2in}. Then |F|=n1 and |V(F)|=2n2. By Lemma 2.5, QnF has an isolated edge (u,v) and a large component. Thus, κ1(Qn;P2)n1.

    Figure 4.  Illustration of the graph for Lemma 3.1.

    Lemma 3.2. κs1(Qn;P2)n1 for n4.

    Proof. Let F be a 1-extra P2-substructure cut. Then the elements in F are either isolated vertices or P2s. It suffices to show that QnF is connected when |F|n2. This lemma is pvoved by contradiction. Suppose that QnF is disconnected and C is the smallest component of QnF. Since F is a 1-extra P2-substructure cut, |V(C)|2, furthermore,

    |V(F)|2(n2)=2n4<2n2=κ1(Qn), a contradiction.

    Thus, we have |F|n1. This proof is complete.

    By Lemma 3.1 and Lemma 3.2, we can easily obtain the following theorem.

    Theorem 3.3. κ1(Qn;P2)=κs1(Qn;P2)=n1 for n3.

    The following will prove the result when 3k3n4.

    Lemma 3.4. Let m, n and k be positive integers and n5, 3k3n4.

    κ1(Qn;Pk){3n4kfork=3m,3n43n42kkfork=3m+1,3n43n4kkfork=3m+2.

    Proof. For any two vertices u and v in Qn and (u,v)E(Qn), without loss generality, suppose that u=000 and v=100. By the structure of Qn, we know that ui and vi are adjacent for 2in. Furthermore, the vertex vj,j+1(resp. uj,j+1) is a common neighbor of vj(resp. uj) and vj+1(resp. uj+1) for 2jn1. It follows that (uj,vj)E(Qn), (vj,vj,j+1)E(Qn)(resp. (uj,uj,j+1)E(Qn)) and (vj,j+1,vj+1)E(Qn)(resp.(uj,j+1,uj+1)E(Qn)) (See Figure 5). In the following, we find a path formed a cut set F and every element in F is isomorphic to Pk such that QnF is disconnected and the smallest component has at least two vertices. The discussion is divided into three cases:

    Figure 5.  Illustration of the graph for Lemma 3.4.

    Case 1. k=3m.

    Let 3n4=kq+r where q and r are nonnegative integers with 0r<k. When k=3m, we have that 3n4 is not divisible by k, so r0. In the following proof, we assume that q is odd and the proof that q is even is similar.

    Case 1.1 m is odd.

    When m is odd, we can construct the following set of path cuts where each element is isomorphic to Pk:

    P1k=u2v2v2,3uk+33vk+33vk+33,k+63,

    P2k=vk+63uk+63uk+63,k+93v2k+33v2k+33u2k+33,2k+63,

    Pqk=u(q1)k+63v(q1)k+63v(q1)k+63,(q1)k+93uqk+33vqk+33vqk+33,qk+63,

    Pq+1k=vqk+63vnunun,2un,2,3un,2,3,(k(q+1)3n+4).

    Case 1.2 m is even.

    When m is even, we can construct the following set of path cuts where each element is isomorphic to Pk:

    P1k=u2v2v2,3vk+33uk+33uk+33,k+63,

    P2k=uk+63vk+63vk+63,k+93u2k+33u2k+33v2k+33,2k+63,

    Pqk=v(q1)k+63u(q1)k+63u(q1)k+63,(q1)k+93vqk+33uqk+33uqk+33,qk+63,

    Pq+1k=uqk+63unvnvn,2vn,2,3vn,2,3,(k(q+1)3n+4).

    In Case 1, whether m is odd or even, we can construct a set F={P1k,P2k,,Pq+1k} such that QnF is disconnected because {(u,v)} is a component of QnF. In this case, q+1=3n4k.

    Case 2. k=3m+1.

    Let 3n43n42k=kq+r where q and r are nonnegative integers with 0r<k.

    Case 2.1 m is odd.

    When m is odd, we can construct the following set of path cuts where each element is isomorphic to Pk:

    Case 2.1.1 r=0.

    P1k=u2v2v2,3vk+23vk+23,k+53vk+53,

    P2k=uk+53uk+53,k+63uk+63u2k+13,2k+43u2k+43v2k+43,

    P3k=v2k+73u2k+73u2k+73,2k+103u3k+33u3k+33,3k+63u3k+63,

    Pqk=v(q1)k+2q+13v(n1),nvnun. (or Pqk=u(q1)k+2q+13u(n1),nunvn.)

    Case 2.1.2 r0.

    P1k=u2v2v2,3vk+23vk+23,k+53vk+53,

    P2k=uk+53uk+53,k+63uk+63u2k+13,2k+43u2k+43v2k+43,

    P3k=v2k+73u2k+73u2k+73,2k+103u3k+33u3k+33,3k+63u3k+63,

    Pqk=v(q1)k+2q+13uqk+2q3, (or Pqk=u(q1)k+2q+13vqk+2q3, )

    Pq+1k=vqk+2q3unvnvn,2vn,2,(q+1)k3n+2q1.

    (or Pq+1k=uqk+2q3vnunun,2un,2,(q+1)k3n+2q1.)

    Case 2.2 m is even.

    When m is even, we can construct the following set of path cuts where each element is isomorphic to Pk:

    Case 2.2.1 r=0.

    P1k=v2u2u2,3uk+23uk+23,k+53uk+53,

    P2k=vk+53vk+53,k+63vk+63v2k+13,2k+43v2k+43u2k+43,

    P3k=u2k+73v2k+73v2k+73,2k+103v3k+33v3k+33,3k+63v3k+63,

    Pqk=u(q1)k+2q+13u(n1),nunvn. (or Pqk=v(q1)k+2q+13v(n1),nvnun.)

    Case 2.2.2 r0.

    P1k=v2u2u2,3uk+23uk+23,k+53uk+53,

    P2k=vk+53vk+53,k+63vk+63v2k+13,2k+43v2k+43u2k+43,

    P3k=u2k+73v2k+73v2k+73,2k+103v3k+33v3k+33,3k+63v3k+63,

    Pqk=u(q1)k+2q+13vqk+2q3, (or Pqk=v(q1)k+2q+13uqk+2q3, )

    Pq+1k=uqk+2q3vnunun,2un,2,(q+1)k3n+2q1.

    (or Pq+1k=vqk+2q3unvnvn,2vn,2,(q+1)k3n+2q1.)

    In Case 2, when m is odd or even and r=0, we construct a set F={P1k,P2k,,Pqk} with q elements; when m is odd or even and r0, we construct a set F={P1k,P2k,,Pqk,Pq+1k} with q+1 elements. Then QnF is disconnected since {(u,v)} is a component of QnF. In this case, when r=0, q=3n43n42kk; when r0, q+1=3n43n42kk.

    Case 3. k=3m+2.

    Let 3n43n4k=kq+r where q and r are nonnegative integers with 0r<k.

    Case 3.1 m is odd.

    When m is odd, we can construct the following set of path cuts where each element is isomorphic to Pk:

    Case 3.1.1 r=0.

    P1k=u2v2v2,3uk+43, P2k=uk+73u2k+53, P3k=u2k+83u3k+63,

    Pqk=u(q1)k+5+q3vnun.

    Case 3.1.2 r0.

    P1k=u2v2v2,3uk+43, P2k=uk+73u2k+53, P3k=u2k+83u3k+63,

    Pqk=u(q1)k+5+q3uqk+q+33,

    Pq+1k=uqk+6+q3vnunun,2un,2,(q+1)k3n+q+4.

    Case 3.2 m is even.

    In the following proof, we assume that q is odd and the proof that q is even is similar. When m is even, we can construct the following set of path cuts where each element is isomorphic to Pk:

    Case 3.2.1 r=0.

    P1k=u2v2v2,3vk+43, P2k=vk+73u2k+53, P3k=u2k+83v3k+63,

    Pqk=u(q1)k+5+q3unvn.

    Case 3.2.2 r0.

    P1k=u2v2v2,3vk+43, P2k=vk+73u2k+53, P3k=u2k+83v3k+63,

    Pqk=u(q1)k+5+q3vkq+q+33,

    Pq+1k=v(k+1)q+63vnunun,2un,2,(q+1)k3n+q+4.

    In Case 3, when m is odd or even and r=0, we construct a set F={P1k,P2k,,Pqk} with q elements; when m is odd or even and r0, we construct a set F={P1k,P2k,,Pqk,Pq+1k} with q+1 elements. Then QnF is disconnected since {(u,v)} is a component of QnF. In this case, when r=0, q=3n43n4kk; when r0, q+1=3n43n4kk.

    Lemma 3.5. Let m, n and k be positive integers and n5.

    κs1(Qn;Pk){3n4kfork=3mand3k3n4,3n43n42kkfork=3m+1and4k3n4,3n43n4kkfork=3m+2and5k3n4.

    Proof. Let F={Pjii|1jini,1ik} be a 1-extra Pk-substructure cut, and such that each element Pjii is isomorphic to Pi, where ni indicates the number of pi.

    Case 1. k=3m.

    In this case, it suffices to prove that if |F|3n4k1, then QnF is connected. This proof is by contradiction. Assume that QnF is disconnected, then

    |V(F)|=ki=1ni|V(Pi)|kki=1ni=k|F|k(3n4k1)k(3n4+k2k1)=3n6<3n5=κ2(Qn).

    Since F is a 1-extra Pk-substructure cut, the smallest component S of QnF with |V(S)|2. Hence, it follows that |V(S)|=2. Let V(S)={u,v}, and (u,v)E(Qn). By Lemma 2.6, we have

    |NQn({u,v})V(F)|2k3ki=1ni=2k3|F|2k3(3n4k1)2k3(3n4+k2k1)=23(3n6)=2n4<2n2=κ1(Qn),

    a contradiction.

    Case 2. k=3m+1.

    In this case, it suffices to prove that if |F|3n43n42kk1, then QnF is connected. This proof is by contradiction. Assume that QnF is disconnected, then

    |V(F)|=ki=1ni|V(Pi)|kki=1ni=k|F|k(3n43n42kk1)k(3n4(3n42k1)k1)=k((3n4)2k(3n4)+2k2k21)k((3n4)2k(3n4)+2k+2k222k21)=k((3n4)2k(3n4)+2k+2k222k22k2)=2k12k(3n4)+k1k<3n44n9=κ3(Qn)forn5.

    Since F is a 1-extra Pk-substructure cut, the smallest component S of QnF with |V(S)|2. Hence, divided into two subcases.

    Case 2.1. |V(S)|=2.

    Let V(S)={u,v}, and (u,v)E(Qn). By Lemma 2.6, we have

    |NQn({u,v})V(F)|2k3ki=1ni=2k3|F|2k3(3n43n42kk1)2k+13((3n4)2k(3n4)+2k+2k222k21)=2k+13(2k12k2(3n4)+2k22k2)=4k216k2(3n4)+2k+132k22k2<2k23k2(3n4)+2k+232k22k2=23(3n4)+4k246k2<23(3n4)+23=2n2=κ1(Qn),

    a contradiction.

    Case 2.2. |V(S)|=3.

    Let V(S)={u,v,w}. There is no 3-cycle in Qn, so G[S] is a P3. By Lemma 2.7, we have

    |NQn({u,v,w})V(F)|3k4ki=1ni=3k4|F|3k4(3n43n42kk1)3k+34((3n4)2k(3n4)+2k+2k222k21)=3k+34(2k12k2(3n4)+k1k2)=6k2+3k38k2(3n4)+3k234k2<7k28k2(3n4)+3k24k2=218n184<3n5=κ2(Qn),

    a contradiction.

    Case 3. k=3m+2.

    In this case, it suffices to prove that if |F|(3n43n4kk1), then QnF is connected. This proof is by contradiction. Assume that QnF is disconnected, then

    |V(F)|=ki=1ni|V(Pi)|kki=1ni=k|F|k(3n43n4kk1)k(3n4(3n4k1)k1)=k((3n4)k(3n4)+kk21)k((3n4)k(3n4)+k+k22k21)=k((3n4)k(3n4)+k+k22k2k2)=k1k(3n4)+k2k<3n44n9=κ3(Qn)forn5.

    Since F is a 1-extra Pk-substructure cut, the smallest component S of QnF with |V(S)|2. Hence, divided into two subcases.

    Case 3.1. |V(S)|=2.

    Let V(S)={u,v}, and (u,v)E(Qn). By Lemma 2.6, we have

    |NQn({u,v})V(F)|2k3ki=1ni=2k3|F|2k3(3n43n4kk1)2k+13(3n4(3n4k1)k1)2k+13((3n4)k(3n4)+k+k22k2k2)=2k+13((k1)(3n4)+k2k2)=2k2k13k2(3n4)+2k2k23k2<2k23k2(3n4)+2k23k2=23(3n4)+23=2n2=κ1(Qn),

    a contradiction.

    Case 3.2. |V(S)|=3.

    Let V(S)={u,v,w}. There is no 3-cycle in Qn, so G[S] is a P3. By Lemma 2.7, we have

    |NQn({u,v,w})V(F)|3k4ki=1ni=3k4|F|3k4(3n43n4kk1)3k+24((3n4)k(3n4)+k+k22k21)=3k+24(k1k2(3n4)+k2k2)=3k2k24k2(3n4)+3k2k44k2<3k24k2(3n4)+3k24k2=94n3+34=94n94<3n5=κ2(Qn),

    a contradiction.

    By Lemma 3.4 and Lemma 3.5, we can easily obtain the following theorem.

    Theorem 3.6. Let m, n and k be positive integers and n4.

    κ1(Qn;Pk)=κs1(Qn;Pk)={3n4kfork=3mand3k3n4,3n43n42kkfork=3m+1and4k3n4,3n43n4kkfork=3m+2and5k3n4.

    In this section, we do two sets of comparison to compare the structure connectivity results of the hypercube with the 1-extra structure connectivity results of the hypercube. In [2], the authors determined the structure connectivity and substructure connectivity of the hypercube: Let 3kn. Then κ(Qn,Pk)=κs(Qn,Pk)=2nk+1 if k is odd and κ(Qn,Pk)=κs(Qn,Pk)=2nk if k is even. The value of the structure connectivity of the hypercube is equal to the value of the substructure connectivity and the value of the 1-extra-structure connectivity of the hypercube is equal to the value of the 1-extra substructure connectivity, here we compare only the structure connectivity of the hypercube with the 1-extra structure connectivity. In the first set of comparisons, we obtain the results for the number of Pks when the dimension of the hypercube is 30 and the length of the Pks ranges from 4 to 25. In Figure 6(a), it is clear that the results for 1-extra structure connectivity are better than the results for structure connectivity. In comparison 2, we obtain the results for the number of Pks when the hypercube has dimension n={10,15,20,25} and the length of the Pks goes from 4 to 10. From the comparative results (Figure 6(b)), it is seen that the 1-extra structure connectivity is better than the structure connectivity when the dimensions n is the same and the lengths of the Pks are the same.

    In this paper, we propose two new parameters for measuring the network reliability: g-extra H-structure connectivity and g-extra H-substructure connectivity, and obtain some results for Qn:

    κ1(Qn;Pk)=κs1(Qn;Pk)={n1fork=2,3n4kfork=3mand3k3n4,3n43n42kkfork=3m+1and4k3n4,3n43n4kkfork=3m+2and5k3n4.

    The experiments show that our results are better than those of structure connectivity and substructure connectivity. Therefore, the proposed two new parameters are meaningful. One can further consider the results for the hypercube when g is larger. Of course, the results of some other well-known networks can be considered.

    The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.

    This work is supported by the Science Foundation of Qinghai Province (No. 2021-ZJ-703), the National Science Foundation of China (Nos.11661068, 12261074 and 12201335).

    The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.



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