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An exact solution of heat and mass transfer analysis on hydrodynamic magneto nanofluid over an infinite inclined plate using Caputo fractional derivative model

  • This paper presents the problem modeled using Caputo fractional derivatives with an accurate study of the MHD unsteady flow of Nanofluid through an inclined plate with the mass diffusion effect in association with the energy equation. H2O is thought to be a base liquid with clay nanoparticles floating in it in a uniform way. Bousinessq's approach is used in the momentum equation for pressure gradient. The nondimensional fluid temperature, species concentration, and fluid transport are derived together with Jacob Fourier sine and Laplace transforms Techniques in terms of exponential decay function, whose inverse is computed further in terms of Mittag-Leffler function. The impact of various physical quantities interpreted with fractional order of the Caputo derivatives. The obtained temperature, transport, and species concentration profiles show behaviours for 0<α<1 where α is the fractional parameter. Numerical calculations have been carried out for the rate of heat transmission and the Sherwood number is swotted to be put in the form of tables. The parameters for the magnetic field and the angle of inclination slow down the boundary layer of momentum. The distributions of velocity, temperature, and concentration expand more rapidly for higher values of the fractional parameter. Additionally, it is revealed that for the volume fraction of nanofluids, the concentration profiles behave in the opposite manner. The limiting case solutions also presented on flow field of governing model.

    Citation: J. Kayalvizhi, A. G. Vijaya Kumar, Ndolane Sene, Ali Akgül, Mustafa Inc, Hanaa Abu-Zinadah, S. Abdel-Khalek. An exact solution of heat and mass transfer analysis on hydrodynamic magneto nanofluid over an infinite inclined plate using Caputo fractional derivative model[J]. AIMS Mathematics, 2023, 8(2): 3542-3560. doi: 10.3934/math.2023180

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  • This paper presents the problem modeled using Caputo fractional derivatives with an accurate study of the MHD unsteady flow of Nanofluid through an inclined plate with the mass diffusion effect in association with the energy equation. H2O is thought to be a base liquid with clay nanoparticles floating in it in a uniform way. Bousinessq's approach is used in the momentum equation for pressure gradient. The nondimensional fluid temperature, species concentration, and fluid transport are derived together with Jacob Fourier sine and Laplace transforms Techniques in terms of exponential decay function, whose inverse is computed further in terms of Mittag-Leffler function. The impact of various physical quantities interpreted with fractional order of the Caputo derivatives. The obtained temperature, transport, and species concentration profiles show behaviours for 0<α<1 where α is the fractional parameter. Numerical calculations have been carried out for the rate of heat transmission and the Sherwood number is swotted to be put in the form of tables. The parameters for the magnetic field and the angle of inclination slow down the boundary layer of momentum. The distributions of velocity, temperature, and concentration expand more rapidly for higher values of the fractional parameter. Additionally, it is revealed that for the volume fraction of nanofluids, the concentration profiles behave in the opposite manner. The limiting case solutions also presented on flow field of governing model.



    The notion of resolving sets in general networks is introduced by Slater in 1975 and he called the minimum cardinality of a resolving set location number [1]. In next year Harary and Melter also introduced the same concept with different name and they called it the metric dimension (MD) of the connected networks. They provide a characterization of MD of the trees and they also proved that the MD of wheel W1,z and complete network Kz is 2 and z1 respectively [2]. Later on the results of the MD of W1,z, were improved by S. Khuller et al. and they also characterized the connected networks that those have MD 1 and 2 [3]. Shanmukha et al. improved the results of Harary and Melter and they computed the MD of wheel-related networks [4]. Chartrand et al. established the bounds on MD of connected networks in terms of the order and diameter of a network [5].

    The concept of MD arises in diverse areas including network discovery and verification [6], robot navigation [7], strategies for the Mastermind game [8], combinatorial optimization [9], coin weighting [10], navigation of robots in networks [11] and image processing [12]. There are some new types of MD are discovered in recent times as local MD [13], k- MD [14], edge MD [15], fault tolrent MD [16] and some interesting results of fault-tolerant MD of convex polytope networks have been derived by Raza et al [17].

    The idea of MD to find the solution of specific integer programming (IPP) is introduced by Chartrand et al. [5] and Currie and Ollermann introduced the concept of fractional metric dimension (FMD) to find improved solution of IPP [18]. The concept of FMD in the field of networking theory is formally introduced by Arumugam and Mathew, they developed different combinatorial techniques to find the exact value of FMD of different connected networks. Moreover, they also found the FMD of Petersen, cycle, friendship and cartesian product of different connected networks [19,20]. Feng et al. established a computational technique to find FMD of vertex transitive networks and as an application they computed the FMD of hamming and generalized Johnson networks [21]. Javaid et al. characterize all those connected networks that attain FMD exactly 1 [22,23] and Zafar et al. computed the exact value of FMD of different connected networks [24].

    The notion of latest derived form of FMD known as a local fractional metric dimension (LFMD) is defined by Asiyah et al. and they calculated the exact values of the LFMD of the corona product of connected networks [25]. Javaid et al. purposed a unique methodology to compute the sharp bounds of LFMD for all the connected networks and they also proved that the lower bound of LFMD of non-bipartite networks is greater than 1 [26,27]. Some interesting results of LFMD of different connected networks can be seen in [28,29,30].

    In this paper, the lower and upper bounds of LFMD of generalized modified prism networks have been computed. It is also proved that all the upper bounds of all these networks is less or equal to 2, when the order of these networks approaches to . The rest of the paper is organized as follows: Section 2 deals with preliminaries, Section 3 consists of the main results of LFMD of generalized modified prism network, Section 4 represents the conclusion and comparison among all the main results.

    A network Γ is a pair (V(Γ)×E(Γ)) with V(Γ) is a vertex set and E(Γ)(V(Γ)×V(Γ)) an edge set. A walk is a sequence of edges and vertices of a network. A path is a sequence of vertices with the property that each vertex in the sequence is adjacent to the vertex next to it. For any two vertices x, y of V(Γ) then the distance d(x,y) between them is the number of edges between the shortest path connecting them. A network is called connected if there exist a path between every pair of vertices of Γ. A vertex xV(Γ) resolves a pair (a,b) if d(x,a)d(x,b). Let R={r1,r2,r3,....,rz}V(Γ) be a ordered set is considered as resolving set of Γ if each pair of vertices of Γ is resolved by some vertex in R. A resolving set with minimum cardinality is called the metric dimension of Γ and it is defined as

    dim(Γ)=min{|R|:RisresolvingsetofΓ}.

    For an edge abE(Γ) the local resolving neighbourhood set (RLN) Rx(ab) of ab is defined as Rx(ab)={cV(Γ):d(a,c)d(b,c)}. A local resolving function (LRF) is defined as η:V(Γ)[0,1] such that η(Rx(ab))1 for each Rx(ab) of Γ. A local resolving function η is called minimal if there exists a function μ:V(Γ)[0,1] such that μη and μ(a)η(a) for at least one aΓ(V) that is not a local resolving function of Γ. If |η|=aRx(ab)η(a) then LFMD of Γ is donated by dimLF(Γ) is defined as

    dimLF(Γ)=min{|η|:ηisminimallocalresolvingfunctionofΓ}.

    Throughout the paper, we have used the symbol of local resolving neighbourhood set of an edge abE(Γ) is Rx(ab). For more details about local resolving neighbourhood set and local resolving function, we refer [25].

    Lemma X. [26] Let Γ=(V(Γ)×E(Γ)) be a connected network. If |Rx(e)A|ω, eE(Γ) then

    1dimlf(Γ)|V(Γ)|ω

    where ω=min{|Rx(e)|:eE(Γ)}, where A={Rx(e):|Rx(e)=ω}.

    Lemma Y. [27] Let Γ=(V(Γ)×E(Γ)) be a connected network. Then

    dimlf(Γ)|V(Γ)|σ

    where σ=max{|Rx(e)|:eE(Γ)}.

    For z5 the modified prism network MPz,1,2 with vertex set vertex set V(MPz,1,2)={aj,aj:1jz} and edge set E(MPz,1,2)={ajaj+2:1jz2}{ajaj+1:1jz}{ajaj:1jz}{ajaj+1:1jz}, where |V(MPz,1,2)|=2z and |E(MPz,1,2)|=4z. Fore more details see Figure 1.

    Figure 1.  Modified prism network MP9,1,2.

    For z5 the modified prism network MQz,1,2 with vertex set vertex set V(MPz,1,2)={aj,aj:1jz} and edge set E(MPz,1,2)={ajaj+2:1jz2}{ajaj+1:1jz}{ajaj:1jz}{ajaj+1:1jz}{ajbj:1jz}{ajaj+1:1jz}{bjbj+1:1jz}, where |V(MPz,1,2)|=3z and |E(MPz,1,2)|=6z. Fore more details see Figure 2.

    Figure 2.  Modified prism network MQ9,1,2.

    In this dissertation, our objective is to compute RLN Sets and LFMD of modified prism networks (MPz,1,2,MQz,1,2) in the form of sharp upper and lower bounds.

    In this section, we compute the RLN sets and LFMD of modified prism network (MPz,1,2).

    Lemma 4.1. Let MPz,1,2 be a modified prism network, where z1(mod4). Then

    (a) |Rx(ajaj+1)|=z1 and zj=1Rx(ajaj+1)=V(MPz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)| where |Rx(y)| are the other possible resolving local neighbourhood sets.

    Proof. Let aj inner, aj be the outer vertices of modified generalized Prism network, for 1jz, where z+1(1modz), we have following possibilities

    (a) Rx(ajaj+1) =V(MPz,1,2){aj+2,aj+4,aj+6.....,az+i5,az+i3,az+i1}{aj+2,aj+4,aj+6,.....,az+i5,az+i3,az+i1}{az+2i+22}{az+2i+22} and |Rx(ajaj+1)|=z1 and |zj=1Rx(ajaj+1)|=3z=|V(MPz,1,2)|.

    (b) Rx(ajaj)=V(MPz,1,2){aj+2,aj+3,az+j3,az+j4}, Rx(ajaj+2)=V(MPz,1,2){aj+1,aj+1}, Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4,aj+6,....,az+j3,ai+4,ai+6,ai+8,ai+10,...,az+i5}.

    The cardinalities among all these RLN sets are classified in Table 1.

    Table 1.  Cardinality of each RLN set.
    RLN Set Cardinality
    Rx(ajaj) 2z4>z1
    Rx(ajaj+2) 2z2>z1
    Rx(ajaj+1) z+3>z1

     | Show Table
    DownLoad: CSV

    It is clear from above Table 1 that cardinality of Rx(ajaj+1) is less then all other RLN sets.

    Theorem 4.2. Let MPz,1,2 be a modified prism network, where z1(mod4). Then

    zz1dimLF(MPz,1,2)2zz1.

    Proof. Case 1. For z=5, we have the following RLN sets

    Rx(a1a2)=Rx(a1a2)={a1,a2,a1,a2},

    Rx(a2a3)=Rx(a2a3)={a2,a3,a2,a3},

    Rx(a3a4)=Rx(a3a4)={a3,a4,a3,a4},

    Rx(a4a5)=Rx(a4a5)={a4,a5,a4,a5},

    Rx(a5a1)=Rx(a5a1)={a1,a5,a1,a5},

    Rx(a1a3)={a1,a3,a1,a3},

    Rx(a1a4)={a1,a4,a1,a4},

    Rx(a2a4)={a2,a4,a2,a4},

    Rx(a2a5)={a2,a5,a2,a5},

    Rx(a3a5)={a3,a5,a3,a5},

    Rx(a1a1)=V(MP5,1,2){a3,a4},

    Rx(a2a2)=V(MP5,1,2){a4,a5},

    Rx(a3a3)=V(MP5,1,2){a5,a1},

    Rx(a4a4)=V(MP5,1,2){a1,a2},

    Rx(a5a5)=V(MP5,1,2){a2,a3}.

    For 1j5 it is clear that |Rx(ajaj+1)|=8 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MP5,1,2. Therefore, an upper LRF η:V(MP5,1,2)[0,1] is defined as η(y)=14 for each yV(MP5,1,2). In order to show that η is a minimal LRF, we define another LRF η(y):V(MP5,1,2)[0,1] as |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MP5,1,2. Therefore, dimLF(MP5,1,2)10114=52. In the same context, for 1jz it is clear from the above RLN sets that |Rx(ajaj)|=8 and |Rx(ajaj)||Rx(e)|, where Rx(e) are the other RLN sets of MP5,1,2. Therefore, a lower LRF η:V(MP5,1,2)[0,1] is defined as η(y)=121 for all yV(MP5,1,2) hence dimLF(MP5,1,2)10118=54. Consequently,

    54dimLF(MP5,1,2)52.

    Case 2. For 1jz from Lemma 4.1 it is clear that |Rx(ajaj+1)|=z+1 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MPz,1,2. Therefore, an upper LRF η:V(MPz,1,2)[0,1] is defined as η(y)=1z1 for each yV(MPz,1,2). In order to show that η is a minimal RLF, we define another RLF η:V(MPz,1,2)[0,1] as |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a RLF of (MPz,1,2). Therefore, by Lemma X dimLF(MPz,1,2)2zj=11z1=2zz1. In the same way, for 1jz it is clear from Lemma 4.1 |Rx(ajaj+1)|=2z2 and |Rx(ajaj+2)||Rx(e)|, where Rx(e) are the other LRN sets of MPz,1,2. Therefore, a lower RLF η:V(MPz,1,2)[0,1] is defined as η(y)=12z4 for each yV(MPz,1,2) hence by Lemma Y dimLF(MPz,1,2)2zj=112z2=zz2. Consequently,

    zz2dimLF(MPz,1,2)2zz1.

    Lemma 4.3. Let MPz,1,2 be a modified prism network, where z3(mod4). Then

    (a) |Rx(ajaj+1)|=z+1 and zj=1Rx(ajaj+1)=V(MPz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)| where |Rx(e)| are the other possible RLN sets.

    Proof. Let aj inner, aj be the outer vertices of modified prism network, for 1jz, where z+11(modz), we have following possibilities,

    (a)Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4,aj+6,.....,az+i5,az+i3,az}{aj+2,aj+4,aj+6,.....,az+i5,az+i3,az} and |Rx(ajaj+1)|=z+1 and |zj=1Rx(ajaj+1)|=3z=|V(MPz,1,2)|.

    (b) Rx(ajaj)=V(MPz,1,2){aj+2,aj+3,az+j3,az+j4}, Rx(ajaj+2)=V(MPz,1,2){aj+1,aj+1}, Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4,aj+6,....,az+j3,ai+4,ai+6,ai+8,ai+10,...,az+i5}.

    The RLN sets are classified in Table 2 and it is clear that |Rx(ajaj+1)| is less then the all other RLN sets of MPz,1,2.

    Table 2.  Cardinality of each RLN set.
    RLN Set Cardinality
    Rx(ajaj) 2z4>z+1
    Rx(ajaj+2) 2z2>z+1
    Rx(ajaj+1) z+3>z+1

     | Show Table
    DownLoad: CSV

    Theorem 4.4. Let MPz,1,2 be a modified prism network, where z3(mod4). Then

    zz1dimLF(MPz,1,2)2zz+1.

    Proof. Case 1. For z=7, we have the following RLN sets

    Rx(a1a2)=V(MP7,1,2){a3,a5,a7,a3,a5,a7},

    Rx(a2a3)=V(MP7,1,2){a4,a6,a1,a4,a6,a1},

    Rx(a3a4)=V(MP7,1,2){a5,a7,a2,a5,a7,a2},

    Rx(a4a5)=V(MP7,1,2){a6,a1,a3,a6,a1,a3},

    Rx(a5a6)=V(MP7,1,2){a7,a2,a4,a7,a2,a4},

    Rx(a6a7)=V(MP7,1,2){a1,a3,a5,a1,a3,a5},

    Rx(a7a1)=V(MP7,1,2){a2,a4,a6,a2,a4,a6},

    Rx(a1a1)=V(MP7,1,2){a3,a4,a5,a6},

    Rx(a2a2)=V(MP7,1,2){a4,a5,a6,a7},

    Rx(a3a3)=V(MP7,1,2){a5,a6,a7,a1},

    Rx(a4a4)=V(MP7,1,2){a6,a7,a1,a2},

    Rx(a5a5)=V(MP7,1,2){a7,a1,a2,a3},

    Rx(a6a6)=V(MP7,1,2){a1,a2,a3,a4},

    Rx(a7a7)=V(MP7,1,2){a2,a3,a4,a5,},

    Rx(a1a3)=V(MP7,1,2){a2,a2},

    Rx(a2a4)=V(MP7,1,2){a3,a3},

    Rx(a3a5)=V(MP7,1,2){a4,a4},

    Rx(a4a6)=V(MP7,1,2){a5,a5},

    Rx(a5a7)=V(MP7,1,2){a6,a6},

    Rx(a6a1)=V(MP7,1,2){a7,a7},

    Rx(a7a2)=V(MP7,1,2){a1,a1},

    Rx(a1a2)=V(MP7,1,2){a3,a5,a7,a5},

    Rx(a2a3)=V(MP7,1,2){a4,a6,a1,a6},

    Rx(a3a4)=V(MP7,1,2){a5,a7,a2,a7},

    Rx(a4a5)=V(MP7,1,2){a6,a1,a3,a1},

    Rx(a5a6)=V(MP7,1,2){a7,a2,a4,a2},

    Rx(a6a7)=V(MP7,1,2){a1,a3,a5,a3},

    Rx(a7a1)=V(MP7,1,2){a2,a4,a6,a4}.

    For 1j7 it is clear that |Rx(ajaj+1)|=8 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MP7,1,2). Therefore, an upper LRF η:V(MP7,1,2)[0,1] is defined as η(y)=18 for each yV(MP7,1,2). In order to show that η(y) is a minimal upper LRF, we define another LRF η(y):V(MP7,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a local resolving function of P7,1,2). Therefore, dimLF(MP7,1,2)14118=74. In the same context, for 1jz it is clear from the above RLN sets that |Rx(ajaj+2)|=12 and |Rx(ajaj+2)||Rx(e)|, where Rx(e) are the other RLN sets of MP7,1,2). Therefore, a lower LRF η:V(MP7,1,2)[0,1] is defined as η(y)=121 for each yV(MP7,1,2) hence dimLF(MP7,1,2)141112=76. Since MP7,1,2 is a non-bipartite network so its lower bound must be greater then 1. Consequently,

    76dimLF(MP7,1,2)74.

    Case 2. For 1jz from Lemma 4.3, it is clear that |Rx(ajaj+1)|=z+1 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MPz,1,2. Therefore, an upper LRF η:V(MPz,1,2)[0,1] is defined as η(y)=23n+6 for each yV(MPz,1,2). In order to show that η is a minimal LRF, we define another LRF η:V(MPz,1,2)[0,1] as |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MP7,1,2 hence by Lemma X dimLF2zj=11z+1=2zz+1. In the same way, for 1jz it is clear from Lemma 4.3 |Rx(ajaj+1)|=2z2 and |Rx(ajaj+2)||Rx(e)|, where Rx(e) are the other RLN of MPz,1,2. Therefore, a maximal lower LRF η:V(MPz,1,2)[0,1] is defined as η(y)=12z2 for each yV(MPz,1,2) hence by Lemma Y dimLF(MPz,1,2)2zj=112z2=zz1. Consequently,

    zz1dimLF(MPz,1,2)2zz+1.

    Lemma 4.5. Let MPz,1,2 be a modified generalized prism network, where z0(mod4). Then

    (a) |Rx(ajaj+1)|=z and zj=1Rx(ajaj+1)=V(MPz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)|, where |Rx(y)| are the other possible RLN sets.

    Proof. Let aj inner, aj be the outer vertices of modified generalized Prism network, for 1jz, where z+1(1modz), we have following possibilities

    (a) Rx(ajaj+1)=V(MPz,1,2){aj+2, aj+4,aj+6.....,az+2j2,az+2j+22, az+2j+62, az+2j+102, .....az+i5, az+i3,az+i1}{aj+2, aj+4,aj+6.....,az+2j2, az+2j+22,az+2j+62, az+2j+102, .....az+i5,az+i3, az+i1} and |Rx(ajaj+1)|=z and |zj=1Rx(ajaj+1)|=2z=|V(MPz,1,2)|.

    (b)Rx(ajaj)=V(MPz,1,2)-{aj+2,aj+3,az+j2, az+j3}, Rx(ajaj+2)=V(MPz,1,2)-{aj+1,aj+1,an+2j+22, an+2j+22}, Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4, aj+6,....,az+2j2,az+2j+22, az+2j+62,...., az+j6,az+j3,az+j1}.

    The RLN sets are classified in Table 3 and it is clear that cardinality of Rx(ajaj+1) is less then all other RLN sets of MPz,1,2.

    Table 3.  Cardinality of each LRN set.
    RLN Set Cardinality
    Rx(ajaj) 2z4>z
    Rx(ajaj+2) 2z2>z
    Rx(ajaj+1) z+3>z

     | Show Table
    DownLoad: CSV

    Theorem 4.6. Let MPz,1,2 be a modified prism network, where z0(mod4). Then

    zz2dimLF(MPz,1,2)2.

    Proof. Case 1. For z=8, we have the following RLN sets;

    Rx(a1a2)=V(MP8,1,2){a3,a5,a6,a8,a3,a5,a6,a8},

    Rx(a2a3)=V(MP8,1,2){a4,a6,a1,a2,a4,a6,a7,a1},

    Rx(a3a4)=V(MP8,1,2){a5,a7,a8,a3,a5,a7,a8,a2},

    Rx(a4a5)=V(MP8,1,2){a6,a8,a1,a4,a6,a8,a1,a3},

    Rx(a5a6)=V(MP8,1,2){a7,a1,a2,a5,a7,a1,a2,a4},

    Rx(a6a7)=V(MP8,1,2){a8,a2,a3,a6,a8,a2,a3,a5},

    Rx(a7a8)=V(MP8,1,2){a1,a3,a4,a7,a1,a3,a4,a6},

    Rx(a1a8)=V(MP8,1,2){a2,a4,a5,a8,a2,a4,a5,a7},

    Rx(a1a1)=V(MP8,1,2){a3,a4,a6,a7},

    Rx(a2a2)=V(MP8,1,2){a4,a5,a7,a8},

    Rx(a3a3)=V(MP8,1,2){a5,a6,a8,a1},

    Rx(a4a4)=V(MP8,1,2){a6,a7,a1,a2},

    Rx(a5a5)=V(MP8,1,2){a7,a8,a2,a3},

    Rx(a6a6)=V(MP8,1,2){a8,a1,a3,a4},

    Rx(a7a7)=V(MP8,1,2){a1,a2,a4,a5},

    Rx(a8a8)=V(MP8,1,2){a2,a3,a5,a6},

    Rx(a1a2)=V(MP8,1,2){a3,a5,a6,a8},

    Rx(a2a3)=V(MP8,1,2){a4,a6,a7,a1},

    Rx(a3a4)=V(MP8,1,2){a5,a7,a8,a2},

    Rx(a4a5)=V(MP8,1,2){a6,a8,a1,a3},

    Rx(a5a6)=V(MP8,1,2){a7,a1,a2,a4},

    Rx(a6a7)=V(MP8,1,2){a8,a2,a3,a5},

    Rx(a7a8)=V(MP8,1,2){a1,a3,a4,a6},

    Rx(a8a1)=V(MP8,1,2){a2,a4,a5,a7},

    Rx(a1a3)=V(MP8,1,2){a2,a6,a2,a6},

    Rx(a2a4)=V(MP8,1,2){a3,a7,a3,a7},

    Rx(a3a5)=V(MP8,1,2){a4,a8,a4,a8},

    Rx(a4a6)=V(MP8,1,2){a5,a1,a5,a1},

    Rx(a5a7)=V(MP8,1,2){a6,a2,a6,a2},

    Rx(a6a8)=V(MP8,1,2){a3,a7,a3,a7},

    Rx(a7a1)=V(MP8,1,2){a8,a1,a8,a1},

    Rx(a8a2)=V(MP8,1,2){a1,a5,a1,a5}.

    For 1j8 it is clear that |Rx(ajaj+1)|=8 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the RLN sets of MP8,1,2. Then there exits an upper LRF η:V(MP8,1,2)[0,1] and it is defined as η(y)=18 for each yV(MP8,1,2). In order to show that η(y) is a minimal LRF, we define another LRF η(y):V(MP8,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MP8,1,2. Therefore, dimLF(MP8,1,2)16118=2. In the same context, for 1jz it is clear from RLN sets that |Rx(ajaj+2)|=12 and |Rx(ajaj+2)||Rx(e)|, where Rx(e) are the other RLN sets of MP8,1,2). Then there exist a lower LRF η:V(MP8,1,2)[0,1] and it is defined η(y)=121 for each yV(MP7,1,2) hence dimLF(MP8,1,2)161112=43. Since MP8,1,2 is a non-bipartite network so its lower bound must be greater then 1. Consequently,

    43dimLF(MP8,1,2)2.

    Case 2. For 1jz, it is clear from Lemma 4.5 it is that |Rx(ajaj+1)|=z and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MPz,1,2. Then there exits an upper LRF η:V(MPz,1,2)[0,1] an it is defined as η(y)=1z for each yV(MPz,1,2). In order to show that η is a minimal LRF, we define another LRF η:V(MPz,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MP8,1,2 hence by Lemma X dimLF(MPz,1,2)2zj=11z=2. In the same way, For 1jz it is clear from Lemma 4.5 |Rx(ajaj+1)|=2z4 and |Rx(ajaj+2)||Rx(e)|, where Rx(e) are the other RLN sets of MPz,1,2). Then there exits a maximal lower LRF η:V(MPz,1,2)[0,1] and it is defined as η(y)=1z1 for each yV(MPz,1,2) hence by Lemma Y dimLF(MPz,1,2)2zj=112z4=zz2. Consequently,

    zz2dimLF(MPz,1,2)2.

    Lemma 4.7. Let MPz,1,2 be a modified prism network, where z2(mod4). Then

    (a) |Rx(ajaj+1)|=z+2 and zj=1Rx(ajaj+1)=V(MPz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)| where |Rx(y)| are the other possible resolving local neighbourhood sets.

    Proof. Let aj inner, aj be the outer vertices of modified generalized Prism network, for 1jz, where z+1(1modz), we have following possibilities

    (a) Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4,aj+6....., az+j12,az+2j+22, az+2j+62,az+2j+102, .....az+i5, az+i3,az+i1}{aj+2,aj+4,aj+6.....,az+j12, az+2j+22,az+2j+62,az+2j+102, .....az+i5,az+i3,az+i1} and |Rx(ajaj+1)|=z and |zj=1Rx(ajaj+1)|=2z=|V(MPz,1,2)|.

    (b) Rx(ajaj)=V(MPz,1,2){aj+2,aj+3,az+j2,az+j3}, Rx(ajaj+2)=V(MPz,1,2){aj+1,aj+1, an+2j+22,an+2j+22}, Rx(ajaj+1)=V(MPz,1,2){aj+2, aj+4,aj+6, ....,az+j12,az+2j+22, az+2j+62,....,az+j6,az+j3,az+j1}.

    The RLN sets are classified in Table 4 and it is clear that |Rx(ajaj+1)| is less then all other RLN sets of MPz,1,2.

    Table 4.  Cardinality of each LRN set.
    RLN Set Cardinality
    Rx(ajaj) 2z4>z+2
    Rx(ajaj+2) 2z4>z+2
    Rx(ajaj+1) 2z4>z+2

     | Show Table
    DownLoad: CSV

    Theorem 4.8. Let MPz,1,2 be a modified prism network, where z2(mod4). Then

    zz2dimLF(MPz,1,2)2zz+2.

    Proof. Case 1. For z=6, we have the following RLN sets;

    Rx(a1a2)=V(MP6,1,2){a3,a6,a3,a5,a6},

    Rx(a2a3)=V(MP6,1,2){a4,a1,a4,a6,a1},

    Rx(a3a4)=V(MP6,1,2){a5,a2,a5,a1,a2},

    Rx(a4a5)=V(MP6,1,2){a6,a3,a6,a2,a3},

    Rx(a5a6)=V(MP6,1,2){a1,a4,a1,a3,a4},

    Rx(a6a1)=V(MP6,1,2){a2,a5,a2,a4,a5},

    Rx(a1a1)=V(MP6,1,2){a3,a4,a5},

    Rx(a2a2)=V(MP6,1,2){a4,a5,a6},

    Rx(a3a3)=V(MP6,1,2){a5,a6,a1},

    Rx(a4a4)=V(MP6,1,2){a6,a1,a2},

    Rx(a5a5)=V(MP6,1,2){a1,a2,a3},

    Rx(a6a6)=V(MP6,1,2){a2,a3,a4},

    Rx(a1a3)=V(MP6,1,2){a2,a5,a2,a5},

    Rx(a2a4)=V(MP6,1,2){a3,a6,a3,a6},

    Rx(a3a5)=V(MP6,1,2){a4,a1,a4,a1},

    Rx(a4a6)=V(MP6,1,2){a5,a2,a5,a2},

    Rx(a5a1)=V(MP6,1,2){a6,a3,a6,a3},

    Rx(a6a2)=V(MP6,1,2){a1,a4,a1,a4},

    Rx(a1a2)=V(MP6,1,2){a3,a6},

    Rx(a2a3)=V(MP6,1,2){a4,a1},

    Rx(a3a4)=V(MP6,1,2){a5,a2},

    Rx(a4a5)=V(MP6,1,2){a6,a3},

    Rx(a5a6)=V(MP6,1,2){a1,a4},

    Rx(a1a6)=V(MP6,1,2){a2,a5}.

    For 1j6 it is clear that |Rx(ajaj+1)|=7 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MP6,1,2). Then there exits an upper LRF η:V(MP6,1,2)[0,1] is defined as η(y)=17 for each yV(MP6,1,2). In order to show that η(y) is a minimal LRF, we define another LRF η(y):V(MP6,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MP6,1,2 hence dimLF(MP6,1,2)12118=32. In the same context, for 1jz it is clear that |Rx(ajaj+2)|=12 and |Rx(ajaj+2)||Rx(e)|, where Rx(e) are the other resolving local neighbour sets of MP6,1,2). Then there exits a lower LRF η:V(MP6,1,2)[0,1] and it is defined as η(y)=121 for each yV(MP6,1,2) hence dimLF(MP6,1,2)121110=65. Since MP6,1,2 is a non bipartite network so its lower bound must be greater then 1. Consequently,

    65<dimLF(MP6,1,2)32.

    Case 2. For 1jz from Lemma 4.7 that |Rx(ajaj+1)|=z and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MPz,1,2. Then there exits an upper LRF η:V(MPz,1,2)[0,1] and it is defined as η(y)=1z+2 for each yV(MPz,1,2). In order to show that η is a minimal upper LRF, we define another LRF η:V(MPz,1,2)[0,1] as |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MP6,1,2 hence by Lemma X dimLF(MPz,1,2)2zj=11z+2=2zz+2. In the same way, for 1jz it is clear from Lemma 4.7 |Rx(ajaj+1)|=2z4 and |Rx(ajaj+2)||Rx(e)|, where Rx(e) are the other RLN sets of MPz,1,2). Then there exits a lower LRF η:V(MPz,1,2)[0,1] and it is defined as η(y)=1z1 for each yV(MPz,1,2) hence by Lemma YdimLF(MPz,1,2)2zj=112z4=zz2. Consequently,

    zz2dimLF(MPz,1,2)2zz+2.

    In this section, we compute RLN sets and LFMD of modified prism network MQz,1,2 in the form of bounds.

    Lemma 5.1. Let MQz,1,2 be a modified prism network, where z2(mod4). Then

    (a) |Rx(ajaj+1)|=3z+62 and 3zj=1Rx(ajaj+1)=V(MQz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |3zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)| where |Rx(e)| are the other possible RLN sets.

    Proof. Let ai inner, ai middle and bi be the outer vertices of modified generalized Prism network, for 1jz, where z+11(modz), we have the following possibilities

    (a) Rx(ajaj+1)=V(MQz,1,2){aj+2,aj+4,aj+6....., az+j12,az+2j+22,az+2j+62,az+2j+102, .....az+i5,az+i3, az+i1}{aj+2,aj+4,aj+6....., az+j12,az+2j+22, az+2j+62, az+2j+102,.....az+i5, az+i3,az+i1}{bj+2,bj+4,bj+6....., bz+j12,bz+2j+22,bz+2j+62, bz+2j+102,.....bz+i5, bz+i3,bz+i1} and |Rx(ajaj+1)|=3z+62 and |3zj=1Rx(ajaj+1)|=3z=|V(MQz,1,2)|.

    (b)Rx(ajaj)=V(MQz,1,2){aj+2,ai+3,az+j3,az+j2, bj+2,bj+3,bz+j3,bz+j2}, Rx(ajaj+2)=V(MQz,1,2){aj+1,az+2j+22, aj+1,az+2j+22, bj+1,bz+2j+22}, Rx(ajaj+1)=V(MQz,1,2){az+j1}, Rx(bjbj+1)=V(MQz,1,2){aj+2}, Rx(ajbj)=V(MQz,1,2).

    The RLN sets classified in Table 5 and it is clear that |Rx(ajaj+1| is less then all other RLN sets of MQz,1,2.

    Table 5.  Cardinality of each LRN set.
    RLN Set Cardinality
    Rx(ajaj) 3z4>3z+62
    Rx(ajaj+2) 3z4>3z+62
    Rx(ajbj) 3z>3z+62
    Rx(ajaj+1) 3z1>3z+62
    Rx(bjbj+1) 3z1>3z+62

     | Show Table
    DownLoad: CSV

    Theorem 5.2. Let MQz,1,2 be a modified prism network, where z2(mod4). Then

    1<dimLF(MPz,1,2)2zz+2.

    Proof. Case 1. For z=6, we have the following RLN sets

    Rx(a1a2)=V(MQ6,1,2){a3,a6,a3,a6,b3,b6},

    Rx(a2a3)=V(MQ6,1,2){a4,a1,a4,a1,b4,b1},

    Rx(a3a4)=V(MQ6,1,2){a5,a6,a5,a2,b5,b2},

    Rx(a4a5)=V(MQ6,1,2){a6,a1,a6,a3,b6,b3},

    Rx(a5a6)=V(MQ6,1,2){a1,a2,a1,a2,b1,b4},

    Rx(a6a1)=V(MQ6,1,2){a2,a3,a2,a3,b2,b5},

    Rx(a1a1)=V(MQ6,1,2){a3,a4,a5,b3,b4,b5},

    Rx(a2a2)=V(MQ6,1,2){a4,a5,a6,b4,b5,b6},

    Rx(a3a4)=V(MQ6,1,2){a5,a6,a1,b5,b6,b1},

    Rx(a4a4)=V(MQ6,1,2){a6,a1,a2,b6,b1,b2},

    Rx(a5a5)=V(MQ6,1,2){a1,a2,a3,b1,b2,b3},

    Rx(a6a6)=V(MQ6,1,2){a2,a3,a4,b2,b3,b4},

    Rx(a1a3)=V(MQ6,1,2){a2,a5,a2,a5,b2,b5},

    Rx(a2a4)=V(MQ6,1,2){a3,a6,a3,a6,b3,b6},

    Rx(a3a5)=V(MQ6,1,2){a4,a1,a4,a1,b4,b1},

    Rx(a4a6)=V(MQ6,1,2){a5,a2,a5,a2,b5,b2},

    Rx(a5a1)=V(MQ6,1,2){a6,a3,a6,a3,b6,b3},

    Rx(a6a2)=V(MQ6,1,2){a1,a4,a6,a4,b1,b4},

    Rx(a1a2)=V(MQ6,1,2){a3,a6},

    Rx(a2a3)=V(MQ6,1,2){a4,a5},

    Rx(a3a4)=V(MQ6,1,2){a5,a6},

    Rx(a4a5)=V(MQ6,1,2){a6,a1},

    Rx(a5a6)=V(MQ6,1,2){a1,a2},

    Rx(a6a1)=V(MQ6,1,2){a2,a3},

    Rx(b1b2)=V(MQ6,1,2){a3,a6},

    Rx(b2b3)=V(MQ6,1,2){a4,a1},

    Rx(b3b4)=V(MQ6,1,2){a5,a2},

    Rx(b4b5)=V(MQ6,1,2){a6,a1},

    Rx(b5b6)=V(MQ6,1,2){a1,a2},

    Rx(b1b6)=V(MQ6,1,2){a2,a1},

    Rx(a1b1)=V(MQ6,1,2),

    Rx(a2b2)=V(MQ6,1,2),

    Rx(a3b3)=V(MQ6,1,2),

    Rx(a4b4)=V(MQ6,1,2),

    Rx(a5b5)=V(MQ6,1,2),

    Rx(a6b6)=V(MQ6,1,2),

    Rx(a6b6)=V(MP6,1,2).

    For 1j6 it is clear that |Rx(ajaj+1)|=12 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQ6,1,2. Then there exits an upper LRF η:V(MP6,1,2)[0,1] and is defined as η(y)=112 for each yV(MQ6,1,2). In order to show that η(y) is a minimal LRF, we define another LRF η(y):V(MP6,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not LRF. Therefore, dimLF(MQ6,1,2)181112=32. For 1j6 it is clear from the above RLN sets that |Rx(bjbj+1)|=18 and |Rx(bjbj+1)||Rx(e)|, where Rx(e) are other RLN sets of MQ6,1,2). Then there exits a lower LRF η:V(MQ6,1,2)[0,1] and it is defined as η(y)=118 for each yV(MQ6,1,2) hence dimLF(MQ6,1,2)181118=1. Since MQ6,1,2 is a non-bipartite network so its lower bound must be greater then 1. Consequently,

    1<dimLF(MQ6,1,2)32.

    Case 2. For 1jz from Lemma 5.1 it is clear from the above resolving local neighbourhood sets that |Rx(ajaj+1)|=23z+6 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQz,1,2). Then there exits an upper LRF η:V(MQz,1,2)[0,1] and it is defined as η(y)=23n+6 for each yV(MQz,1,2). In order to show that η is a minimal LRF, we define another LRF η:V(MQz,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MQ6,1,2. Therefore by Lemma X dimLF(MQz,1,2)3zj=123z+6=2zz+2.

    For 1jz it is clear from Lemma 5.1 |Rx(ajbj)|=3z and |Rx(bjbj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQz,1,2. Then there exits a maximal lower LRF η:V(MPz,1,2)[0,1] and it is defined as η(y)=13z for each yV(MQz,1,2). Hence by Lemma Y dimLF(MQz,1,2)3zj=113z=1. Since MQz,1,2 is a non-bipartite network so its lower of LFMD bound must be greater then 1. Consequently,

    1<dimLF(MQz,1,2)2zz+2.

    Lemma 5.3. Let MQz,1,2 be a modified prism network, where z0(mod4). Then

    (a) |Rx(ajaj+1)|=3z2 and 3zj=1Rx(ajaj+1)=V(MQz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |3zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)|, where |Rx(y)| are the other possible RLN sets.

    Proof. Let ai inner, ai middle and bi be the outer vertices of modified generalized Prism network, for 1jz, where z+11(modz), we have following possibilities

    (a) Rx(ajaj+1) = V(MQz,1,2){aj+2,aj+4,aj+6.....,az+j12, az+2j+22, az+2j+62,az+2j+102,.....az+i5, az+i3,az+i1}{aj+2,aj+4,aj+6.....,az+j12, az+2j+22, az+2j+62,az+2j+102, .....az+i5,az+i3,az+i1}{bj+2,bj+4,bj+6....., bz+j12, bz+2j+22, bz+2j+62,bz+2j+102, .....bz+i5,bz+i3, bz+i1} and |Rx(ajaj+1)|=3z2 and |3zj=1Rx(ajaj+1)| = 3z=|V(MQz,1,2)|.

    (b) Rx(ajaj)=V(MQz,1,2){aj+2,ai+3,az+j3,az+j2,bj+2,bj+3,bz+j3,bz+j2}, Rx(ajaj+2)=V(MQz,1,2)-{aj+1,az+2j+22, aj+1,az+2j+22,bj+1, bz+2j+22}, Rx(ajaj+1)=V(MQz,1,2){az+j1}, Rx(bjbj+1)=V(MQz,1,2){aj+2}, Rx(ajbj)=V(MQz,1,2).

    The RLN sets are classified in Table 6 and it is clear that |Rx(ajaj+1| is less then all other RLN sets of MQz,1,2.

    Table 6.  Cardinality of each LRN set.
    RLN Set Cardinality
    Rx(ajaj) 3z4>3z2
    Rx(ajaj+2) 3z4>3z2
    Rx(ajbj) 3z>3z2
    Rx(ajaj+1) 3z1>3z2
    Rx(bjbj+1) 3z1>3z2

     | Show Table
    DownLoad: CSV

    Theorem 5.4. Let MQz,1,2 be a modified prism network, where z0(mod4). Then

    1dimLF(MQz,1,2)2.

    Proof. Case 1. For z=8, we have the following RLN sets;

    Rx(a1a2)=V(MQ8,1,2){a3,a5,a6,a8,a3,a5,a6,a8,b3,b5,b7,b8},

    Rx(a2a3)=V(MQ8,1,2){a4,a6,a1,a2,a4,a6,a7,a1,b4,b6,b8,b1},

    Rx(a3a4)=V(MQ8,1,2){a5,a7,a8,a3,a5,a7,a8,a2,b5,b7,b1,b2},

    Rx(a4a5)=V(MQ8,1,2){a6,a8,a1,a4,a6,a8,a1,a3,b6,b8,b2,b3},

    Rx(a5a6)=V(MQ8,1,2){a7,a1,a2,a5,a7,a1,a2,a4,b7,b1,b3,b4},

    Rx(a6a7)=V(MQ8,1,2){a8,a2,a3,a6,a8,a2,a3,a5,b8,b2,b4,b5},

    Rx(a7a8)=V(MQ8,1,2){a1,a3,a4,a7,a1,a3,a4,a6,b1,b3,b5,b6},

    Rx(a1a8)=V(MQ8,1,2){a2,a4,a5,a8,a2,a4,a5,a7,b2,b4,b6,b7},

    Rx(a1a1)=V(MQ8,1,2){a3,a4,a6,a7,b3,b4,b6,b7},

    Rx(a2a2)=V(MQ8,1,2){a4,a5,a7,a8,b4,b5,b7,b8},

    Rx(a3a3)=V(MQ8,1,2){a5,a6,a8,a1,b5,b6,b8,b1},

    Rx(a4a4)=V(MQ8,1,2){a6,a7,a1,a2,b6,b7,b1,b2},

    Rx(a5a5)=V(MQ8,1,2){a7,a8,a2,a3,b7,b8,b2,b3},

    Rx(a6a6)=V(MQ8,1,2){a8,a1,a3,a4,b8,b1,b3,b4},

    Rx(a7a7)=V(MQ8,1,2){a1,a2,a4,a5,b1,b2,b4,b5},

    Rx(a8a8)=V(MQ8,1,2){a2,a3,a5,a6,b2,b3,b5,b6},

    Rx(a1a2)=V(MQ8,1,2){a3,a5,a6,a8,},

    Rx(a2a3)=V(MQ8,1,2){a4,a6,a7,a1},

    Rx(a3a4)=V(MQ8,1,2){a5,a7,a8,a2},

    Rx(a4a5)=V(MQ8,1,2){a6,a8,a1,a3},

    Rx(a5a6)=V(MQ8,1,2){a7,a1,a2,a4},

    Rx(a6a7)=V(MQ8,1,2){a8,a2,a3,a5},

    Rx(a7a8)=V(MQ8,1,2){a1,a3,a4,a6},

    Rx(a8a1)=V(MQ8,1,2){a2,a4,a5,a7},

    Rx(a1a3)=V(MQ8,1,2){a2,a6,a2,a6,b2,b6},

    Rx(a2a4)=V(MQ8,1,2){a3,a7,a3,a7,b3,b7},

    Rx(a3a5)=V(MQ8,1,2){a4,a8,a4,a8,b4,b8},

    Rx(a4a6)=V(MQ8,1,2){a5,a1,a5,a1,b5,b1},

    Rx(a5a7)=V(MQ8,1,2){a6,a2,a6,a2,b6,b2},

    Rx(a6a8)=V(MQ8,1,2){a3,a7,a3,a7,b7,b3},

    Rx(a7a1)=V(MQ8,1,2){a8,a1,a8,a1,b8,b4},

    Rx(a8a2)=V(MQ8,1,2){a1,a5,a1,a5,b1,b5},

    Rx(b1b2)=V(MQ8,1,2){a3,a5,a6,a8},

    Rx(b2b3)=V(MQ8,1,2){a4,a6,a7,a1},

    Rx(b3b4)=V(MQ8,1,2){a5,a7,a8,a2},

    Rx(b4b5)=V(MQ8,1,2){a6,a8,a1,a3},

    Rx(b5b6)=V(MQ8,1,2){a7,a1,a2,a4},

    Rx(b6b7)=V(MQ8,1,2){a8,a2,a3,a5},

    Rx(b7b8)=V(MQ8,1,2){a1,a3,a4,a6},

    Rx(b8b1)=V(MQ8,1,2){a2,a4,a5,a7},

    Rx(a1b1)=V(MQ8,1,2),

    Rx(a2b2)=V(MQ8,1,2),

    Rx(a3b3)=V(MQ8,1,2),

    Rx(a4b4)=V(MQ8,1,2),

    Rx(a5b5)=V(MQ8,1,2),

    Rx(a6b6)=V(MQ8,1,2),

    Rx(a7b7)=V(MQ8,1,2),

    Rx(a8b8)=V(MQ8,1,2).

    For 1j8 it is clear that |Rx(ajaj+1)|=12 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQ8,1,2). Then there exits an upper LRF η:V(MQ8,1,2)[0,1] and it is defined as η(y)=18 for each yV(MQ8,1,2). In order to show that η(y) is a minimal LRF, we define another resolving function η(y):V(MQ8,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MQ8,1,2 hence by Lemma XdimLF(MQ8,1,2)241112=2. In the same context, for 1jz it is clear from the above RLN sets that |Rx(ajaj+2)|=12 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQ8,1,2. Then there exits a lower LRF η:V(MQ8,1,2)[0,1] such that η(y)=124 for each yV(MQ8,1,2) hence dimLF(MQ8,1,2)241124=1. Since MQ8,1,2 is non bipartite network so its lower bound of LFMD must be greater then 1. Consequently,

    1<dimLF(MQ8,1,2)2.

    Case 2. For 1jz it is clear that |Rx(ajaj+1)|=z and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQz,1,2. Then there exits an upper LRF η:V(MQz,1,2)[0,1] is defined as η(y)=1z for each yV(MQz,1,2). In order to show that η is a minimal LRF, we define another LRF η:V(MQz,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MQ8,1,2 hence by Lemma X dimLF(MPz,1,2)3zj=123z=2. In the same way, for 1jz it is clear from Lemma 5.3 that |Rx(ajbj)|=3z and |Rx(ajbj)||Rx(e)|, where Rx(e) are the other resolving local neighbour sets of MQz,1,2). Then there exits a maximal lower LRF η:V(MQz,1,2)[0,1] and it is defined as η(y)=13z for each yV(MQz,1,2) hence by Lemma YdimLF(MQz,1,2)3zj=113z=1. Since MQz,1,2 is a non-bipartite network so its lower bound of LFMD must be greater then 1. Consequently,

    1<dimLF(MQz,1,2)2.

    Lemma 5.5. Let MQz,1,2 be a modified prism network, where z1(mod4). Then

    (a) |Rx(ajaj+1)|=3z32 and zj=1Rx(ajaj+1)=V(MQz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)| where |Rx(y)| are the other possible RLN sets.

    Proof. Let aj inner, aj middle and bj are be the outer vertices of modified prism network, for 1jz, where z+1(1modz), we have following possibilities

    (a) Rx(ajaj+1) = V(MPz,1,2){aj+2,aj+4,aj+6.....,az+i5, az+i3,az+i1}{aj+2,aj+4, aj+6,.....,az+i5, az+i3,az+i1}{bj+2,bj+4, bj+6,.....,bz+i5,bz+i3, bz+i1}{az+2i+22}{az+2i+22} and |Rx(ajaj+1)|=z1 and |3zj=1Rx(ajaj+1)|=3z=|V(MPz,1,2)|.

    (b) Rx(ajaj)=V(MPz,1,2)-{aj+2,aj+3,az+j3,az+j4,bj+2,bj+3,bz+j3,bz+j4}, Rx(ajaj+2)=V(MPz,1,2)-{aj+1,aj+1,bj+1,az+2j+12,az+2j+12, bz+2j+12}, Rx(bjbj+1)=Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4, aj+6,....,az+j3,az+j1}{az+2j+12}{az+2j+12,bz+2j+12}. Rx(ajbj)=V(MPz,1,2).

    The RLN sets are classified in Table 7 and it is clear |Rx(ajaj+1)| is less then all other RLN sets of MQz,1,2.

    Table 7.  Cardinality of each LRN set.
    RLN Set Cardinality
    Rx(ajaj) 3z8>3z32
    Rx(ajaj+2) 3z6>3z32
    Rx(ajaj+1) 5x25>3z32
    Rx(ajbj) 3z>3z32
    Rx(bjaj+1) 5z25>3z32

     | Show Table
    DownLoad: CSV

    Theorem 5.6. Let MPz,1,2 be a modified prism network, where z1(mod4). Then

    1<dimLF(MPz,1,2)2zz1.

    Proof. Case 1. For z=5, we have the following RLN sets

    Rx(a1a2)={a1,a2,a1,a2,b1,b2},

    Rx(a2a3)={a2,a3,a2,a3,b2,b3},

    Rx(a3a4)={a3,a4,a3,a4,b3,b4},

    Rx(a4a5)={a4,a5,a4,a5,b4,b5},

    Rx(a5a1)={a1,a5,a1,a5,b5,b1},

    Rx(a1a3)={a1,a3,a1,a3,b1,b3},

    Rx(a1a4)={a1,a4,a1,a4,b2,b4},

    Rx(a2a4)={a2,a4,a2,a4,b3,b5},

    Rx(a2a5)={a2,a5,a2,a5,b4,b1},

    Rx(a3a5)={a3,a5,a3,a5,b5,b2},

    Rx(a1a1)=V(MP5,1,2){a3,a4,b3,b4},

    Rx(a2a2)=V(MP5,1,2){a4,a5,b4,b5},

    Rx(a3a3)=V(MP5,1,2){a5,a1,b5,b1},

    Rx(a4a4)=V(MP5,1,2){a1,a2,b1,b2},

    Rx(a5a5)=V(MP5,1,2){a2,a3,b2,b3},

    Rx(a1a2)=V(MP5,1,2){a3,a4,a5,a4,b4},

    Rx(a2a3)=V(MP5,1,2){a4,a5,a1,a5,b5},

    Rx(a3a4)=V(MP5,1,2){a5,a1,a2,a1,b1},

    Rx(a4a5)=V(MP5,1,2){a1,a2,a3,a2,b5},

    Rx(a5a1)=V(MP5,1,2){a2,a3,a4,a3,b1},

    Rx(b1b2)=V(MP5,1,2){a3,a4,a5,a4,b4},

    Rx(b2b3)=V(MP5,1,2){a4,a5,a1,a5,b5},

    Rx(b3b4)=V(MP5,1,2){a5,a1,a2,a1,b1},

    Rx(b4b5)=V(MP5,1,2){a1,a2,a3,a2,b2},

    Rx(b5b1)=V(MP5,1,2){a2,a3,a4,a3,b3},

    Rx(a1b1)=V(MP5,1,2),

    Rx(a2b2)=V(MP5,1,2),

    Rx(a3b3)=V(MP5,1,2),

    Rx(a4b4)=V(MP5,1,2),

    Rx(a5b5)=V(MP5,1,2).

    For 1j5 it is clear that |Rx(ajaj+1)|=8 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQ5,1,2. Then there exists an upper LRF η:V(MQ5,1,2)[0,1] and it is defined as η(y)=16 for each yV(MQ5,1,2. In order to show that η(y) is a minimal resolving local function, we define another resolving function η(y):V(MQ5,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MP5,1,2 hence dimLF(MQ5,1,2)15116=52. In the same context, for 1jz it is clear that |Rx(ajaj)|=8 and |Rx(ajbj)||Rx(e)|, where Rx(e) are the other RLN sets of MQ5,1,2. Then there exits a maximal lower LRF η:V(MQ5,1,2)[0,1] and it is defined as η(y)=115 for each yV(MQ5,1,2) hence dimLF(MQ5,1,2)151115=1. Since MQ5,1,2) is a non bipartite network so its lower bound must be greater then 1. Consequently,

    1<dimLF(MQ5,1,2)52.

    Case 2. For 1jz from Lemma 5.5 it is clear that |Rx(ajaj+1)|=z+1 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQz,1,2). Then there exists an upper LRF η:V(MQz,1,2)[0,1] and it is defined as η(y)=1z1 for each yV(MQz,1,2). In order to show that η is a minimal LRF, we define another LRF η:V(MPz,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MQ5,1,2. Therefore, by Lemma X dimLF(MQz,1,2)3zj=123z3=2zz1. In the same context, for 1jz it is clear from Lemma 5.5 that |Rx(ajbj)|=3z and |Rx(ajbj)||Rx(e)|, where Rx(e) are the other RLN sets of MQz,1,2. Then there exists an upper LRF η:V(MPz,1,2)[0,1] and it is defined as η(y)=13z for each yV(MQ) hence by Lemma Y dimLF(MQz,1,2)3zj=113z=1. Since MQz,1,2 is a non-bipartite network so its lower bound of LFMD must be greater then 1. Consequently,

    1<dimLF(MQz,1,2)2zz1.

    Lemma 5.7. Let MQz,1,2 be a modified prism network, where z3(mod4). Then

    (a) |Rx(ajaj+1)|=3z+32 and 3zj=1Rx(ajaj+1)=V(MQz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |3zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)| where |Rx(y)| are the other possible resolving local neighbourhood sets.

    Proof. Let ai inner, ai middle and bi be the outer vertices of modified generalized prism network, for 1jz, where z+1(1modz), we have following possibilities

    (a) Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4,aj+6,....,az+i1}{aj+2,aj+4,aj+6,....,az+i1}{bj+2,bj+4,bj+6,....,bz+i1} and |Rx(ajaj+1)|=3z+32 and |3zj=1Rx(ajaj+1)|=3z=|V(MPz,1,2)|.

    (b)Rx(ajaj) =V(MPz,1,2){aj+2,aj+3,az+j3,az+j2,bj+2,bj+3,bz+j3,bz+j2}, Rx(ajaj+2)=V(MPz,1,2){aj+1,aj+1,bj+1}, Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4,aj+6,...,az+j1,az+2j+12,bz+2j+12}, Rx(bjbj+1)=V(MPz,1,2){aj+2,aj+4,aj+6,...,az+i1,az+2j+12,bz+2j+12}, Rx(ajbj)=V(MPz,1,2).

    The RLN sets are classified in Table 8 and it is clear that |Rx(ajaj+1)| is less then all other RLN sets MQz,1,2.

    Table 8.  Cardinality of each LRN set.
    RLN Set Cardinality
    Rx(ajaj) 3z4>3z+32
    Rx(ajaj+2) 3z4>3z+32
    Rx(ajbj) 3z>3z+32
    Rx(ajaj+1) 3z1>3z+32
    Rx(bjbj+1) 3z1>3z+32

     | Show Table
    DownLoad: CSV

    Theorem 5.8. Let MQz,1,2 be a generalized modified prism network, where z3(mod4). Then

    1<dimLF(MQz,1,2)2zz+2.

    Proof. Case 1. For z=7, we have the following RLN sets

    Rx(a1a2)=V(MQ7,1,2){a3,a5,a7,a3,a5,a7,b3,b5,b7},

    Rx(a2a3)=V(MQ7,1,2){a4,a6,a1,a4,a6,a1,b4,b6,b1},

    Rx(a3a4)=V(MQ7,1,2){a5,a7,a2,a5,a7,a2,b5,b7,b2},

    Rx(a4a5)=V(MQ7,1,2){a6,a1,a3,a6,a1,a3,b6,b1,b3},

    Rx(a5a6)=V(MQ7,1,2){a7,a2,a4,a7,a2,a4,b7,b2,b4},

    Rx(a6a7)=V(MQ7,1,2){a1,a3,a5,a1,a3,a5,b1,b3,b5},

    Rx(a7a1)=V(MQ7,1,2){a2,a4,a6,a2,a4,a6,b2,b4,b6},

    Rx(a1a1)=V(MQ7,1,2){a3,a4,a5,a6,b3,b4,b5,b6},

    Rx(a2a2)=V(MQ7,1,2){a4,a5,a6,a7,b4,b5,b6,b7},

    Rx(a3a3)=V(MQ7,1,2){a5,a6,a7,a1,b5,b6,b7,b1},

    Rx(a4a4)=V(MQ7,1,2){a6,a7,a1,a2,b5,b7,b1,b2},

    Rx(a5a5)=V(MQ7,1,2){a7,a1,a2,a3b6,b1,b2,b3},

    Rx(a6a6)=V(MQ7,1,2){a1,a2,a3,a4,b7,b2,b3,b4},

    Rx(a7a7)=V(MQ7,1,2){a2,a3,a4,a5,b1,b3,b4,b5},

    Rx(a1a3)=V(MQ7,1,2){a2,a2,b2},

    Rx(a2a4)=V(MQ7,1,2){a3,a3,b3},

    Rx(a3a5)=V(MQ7,1,2){a4,a4,b4},

    Rx(a4a6)=V(MQ7,1,2){a5,a5,b5},

    Rx(a5a7)=V(MQ7,1,2){a6,a6,b6},

    Rx(a6a1)=V(MQ7,1,2){a7,a7,b7},

    Rx(a7a2)=V(MQ7,1,2){a1,a1,b1},

    Rx(b1b2)=V(MQ7,1,2){a3,a5,a7,b5},

    Rx(b2b3)=V(MQ7,1,2){a4,a6,a1,b6},

    Rx(b3b4)=V(MQ7,1,2){a5,a7,a2,b7},

    Rx(b4b5)=V(MQ7,1,2){a6,a1,a3,b1},

    Rx(b5b6)=V(MQ7,1,2){a7,a2,a4,b2},

    Rx(b6b7)=V(MQ7,1,2){a1,a3,a5,b3},

    Rx(b7b1)=V(MQ7,1,2){a2,a4,a6,b4},

    Rx(a1a2)=V(MQ7,1,2){a3,a5,a7,a5,b5},

    Rx(a2a3)=V(MQ7,1,2){a4,a6,a1,a6,b6},

    Rx(a3a4)=V(MQ7,1,2){a5,a7,a2,a7,b7},

    Rx(a4a5)=V(MQ7,1,2){a6,a1,a3,a1,b1},

    Rx(a5a6)=V(MQ7,1,2){a7,a2,a4,a2,b2},

    Rx(a6a7)=V(MQ7,1,2){a1,a3,a5,a3,b3},

    Rx(a7a1)=V(MQ7,1,2){a2,a4,a6,a4,b4},

    Rx(a1b1)=V(MQ7,1,2),

    Rx(a2b2)=V(MQ7,1,2),

    Rx(a3b3)=V(MQ7,1,2),

    Rx(a4b4)=V(MQ7,1,2),

    Rx(a5b5)=V(MQ7,1,2),

    Rx(a6b6)=V(MQ7,1,2),

    Rx(a7b7)=V(MQ7,1,2).

    For 1j7 |Rx(ajaj+1)|=13 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQ7,1,2). Then there exists an upper LRF η:V(MQ7,1,2)[0,1] and it is defined as η(y)=113 for each yV(MQ7,1,2). In order to show that η(y) is a minimal LRF, we define another LRF η(y):V(MQ7,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MQ7,1,2 hence dimLF211112=712. In the same context, for 1j7 it is clear that |Rx(ajaj+1)|=21 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQ7,1,2). Then there exists a maximal LLRF η:V(MQ7,1,2)[0,1] and it is defined as η(y)=121 for each yV(MQ7,1,2) hence dimLF=211121=1. Since MQ7,1,2) is non-bipartite network so its lower bound must be greater then 1. Consequently,

    1<dimLF(MQ7,1,2)712.

    Case 2. For 1jz from Lemma 5.7 it is clear that |Rx(ajaj)|=23z+6 and |Rx(ajaj)||Rx(e)|, where Rx(e) are the other RLN sets of MQz,1,2). Then there exits an upper LRF η:V(MQz,1,2)[0,1] and it is defined as η(y)=23n+6 for each yV(MQz,1,2). In order to show that η is a minimal LRF of MQz,1,2), we define another LRF η:V(MQz,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MQz,1,2) hence by Lemma X dimLF3zj=123z+3=2zz+1. In the same context for 1jz it is clear from Lemma 5.7 that |Rx(ajbj)|=3z and |Rx(bjbj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQz,1,2. Then there exists a maximal lower LRF η:V(MQz,1,2)[0,1] and it is defined as η(y)=13z for each yV(MQz,1,2). Therefore, by Lemma Y dimLF3zj=113z=1. Since MQz,1,2 is a non-bipartite network so its lower bound of LFMD must be greater then 1. Consequently,

    1<dimLF(MQz,1,2)2zz+1.

    In this paper, we have computed the local fractional metric dimension of generalized modified prism networks (MPz,1,2,MQz,1,2) in the form of lower and upper bounds. The lower bounds of all the modified prism networks MQz,1,2 is strictly greater than 1 in all cases. Moreover, all of these modified prism networks remain bounded when z as shown in Table 9.

    Table 9.  Limiting values of LFMDs of modified prism networks.
    z LFMDs Limiting LFMDs as z Comment
    1(mod4) zz1dimLF(MPz,1,2)2zz1 1<dimLF(MPz,1,2)2 Bounded
    3(mod4) zz1dimLF(MPz,1,2)2zz+1 1<dimLF(MPz,1,2)2 Bounded
    0(mod4) zz2dimLF(MPz,1,2)2 1<dimLF(MPz,1,2)2 Bounded
    2(mod4) zz2dimLF(MPz,1,2)2zz+2 1<dimLF(MPz,1,2)2 Bounded
    2(mod4) 1<dimLF(MQz,1,2)2zz+2 1<dimLF(MQz,1,2)2 Bounded
    0(mod4) 1<dimLF(MQz,1,2)2 1<dimLF(MQz,1,2)2 Bounded
    1(mod4) 1<dimLF(MQz,1,2)2zz1 1<dimLF(MQz,1,2)2 Bounded
    3(mod4) 1<dimLF(MQz,1,2)2zz+2 1<dimLF(MQz,1,2)2 Bounded

     | Show Table
    DownLoad: CSV

    The authors appreciate the valuable comments and remarks of anonymous referees which helped to greatly improve the quality of the paper.

    The second author (Hassan Zafar) and the third author (Muhammad Javaid) are supported by the Higher Education Commission of Pakistan through the National Research Program for Universities (NRPU) Grant NO. 20-16188/NRPU/R & D/HEC/2021 2021.

    The authors declare that they have no conflicts of interest.



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