
The properties of various fractal and multifractal measures and dimensions have been under extensive study in the real-line and higher-dimensional Euclidean spaces. In non-Euclidean spaces, it is often impossible to construct non-trivial self-similar or self-conformal sets, etc. We consider in the present paper the proper way to phrase the definitions for use in general metric spaces. We investigate the relative Hausdorff measures Hq,tμ and the relative packing measures Pq,tμ defined in a separable metric space. We give some product inequalities which are a consequence of a new version of density theorems for these measures. Moreover, we prove that Hq,tμ and Pq,tμ can be expressed as Henstock-Thomson variation measures. The question of the weak-Vitali property arises in this context.
Citation: Najmeddine Attia, Bilel Selmi. Different types of multifractal measures in separable metric spaces and their applications[J]. AIMS Mathematics, 2023, 8(6): 12889-12921. doi: 10.3934/math.2023650
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The properties of various fractal and multifractal measures and dimensions have been under extensive study in the real-line and higher-dimensional Euclidean spaces. In non-Euclidean spaces, it is often impossible to construct non-trivial self-similar or self-conformal sets, etc. We consider in the present paper the proper way to phrase the definitions for use in general metric spaces. We investigate the relative Hausdorff measures Hq,tμ and the relative packing measures Pq,tμ defined in a separable metric space. We give some product inequalities which are a consequence of a new version of density theorems for these measures. Moreover, we prove that Hq,tμ and Pq,tμ can be expressed as Henstock-Thomson variation measures. The question of the weak-Vitali property arises in this context.
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 z−1 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 x∈V(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 ab∈E(Γ) the local resolving neighbourhood set (RLN) Rx(ab) of ab is defined as Rx(ab)={c∈V(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 μ≤η and μ(a)≠η(a) for at least one a∈Γ(V) that is not a local resolving function of Γ. If |η|=∑a∈Rx(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 ab∈E(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, ∀e∈E(Γ) then
1≤dimlf(Γ)≤|V(Γ)|ω |
where ω=min{|Rx(e)|:e∈E(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)|:e∈E(Γ)}.
For z≥5 the modified prism network MPz,1,2 with vertex set vertex set V(MPz,1,2)={aj,a′j:1≤j≤z} and edge set E(MPz,1,2)={ajaj+2:1≤j≤z−2}∪{a′ja′j+1:1≤j≤z}∪{aja′j:1≤j≤z}∪{ajaj+1:1≤j≤z}, where |V(MPz,1,2)|=2z and |E(MPz,1,2)|=4z. Fore more details see Figure 1.
For z≥5 the modified prism network MQz,1,2 with vertex set vertex set V(MPz,1,2)={aj,a′j:1≤j≤z} and edge set E(MPz,1,2)={ajaj+2:1≤j≤z−2}∪{a′ja′j+1:1≤j≤z}∪{aja′j:1≤j≤z}∪{ajaj+1:1≤j≤z}∪{a′jbj:1≤j≤z}∪{ajaj+1:1≤j≤z}∪{bjbj+1:1≤j≤z}, where |V(MPz,1,2)|=3z and |E(MPz,1,2)|=6z. Fore more details see Figure 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 z≅1(mod4). Then
(a) |Rx(ajaj+1)|=z−1 and z⋃j=1Rx(ajaj+1)=V(MPz,1,2).
(b) |Rx(ajaj+1)|<|Rx(y)|, and |z⋃j=1Rx(ajaj+1)∩Rx(y)|>|Rx(ajaj+1)| where |Rx(y)| are the other possible resolving local neighbourhood sets.
Proof. Let aj inner, a′j be the outer vertices of modified generalized Prism network, for 1≤j≤z, where z+1≅(1modz), we have following possibilities
(a) Rx(ajaj+1) =V(MPz,1,2)−{aj+2,aj+4,aj+6.....,az+i−5,az+i−3,az+i−1}∪{a′j+2,a′j+4,a′j+6,.....,a′z+i−5,a′z+i−3,a′z+i−1}∪{az+2i+22}∪{a′z+2i+22} and |Rx(ajaj+1)|=z−1 and |z⋃j=1Rx(ajaj+1)|=3z=|V(MPz,1,2)|.
(b) Rx(aja′j)=V(MPz,1,2)−{a′j+2,a′j+3,a′z+j−3,az+j−4}, Rx(ajaj+2)=V(MPz,1,2)−{aj+1,a′j+1}, Rx(a′ja′j+1)=V(MPz,1,2)−{aj+2,aj+4,aj+6,....,az+j−3,a′i+4,a′i+6,a′i+8,a′i+10,...,a′z+i−5}.
The cardinalities among all these RLN sets are classified in Table 1.
RLN Set | Cardinality |
Rx(aja′j) | 2z−4>z−1 |
Rx(ajaj+2) | 2z−2>z−1 |
Rx(a′ja′j+1) | z+3>z−1 |
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 z≅1(mod4). Then
zz−1≤dimLF(MPz,1,2)≤2zz−1. |
Proof. Case 1. For z=5, we have the following RLN sets
Rx(a1a2)=Rx(a′1a′2)={a1,a2,a′1,a′2},
Rx(a2a3)=Rx(a′2a′3)={a2,a3,a′2,a′3},
Rx(a3a4)=Rx(a′3a′4)={a3,a4,a′3,a′4},
Rx(a4a5)=Rx(a′4a′5)={a4,a5,a′4,a′5},
Rx(a5a1)=Rx(a′5a′1)={a1,a5,a′1,a′5},
Rx(a1a3)={a1,a3,a′1,a′3},
Rx(a1a4)={a1,a4,a′1,a′4},
Rx(a2a4)={a2,a4,a′2,a′4},
Rx(a2a5)={a2,a5,a′2,a′5},
Rx(a3a5)={a3,a5,a′3,a′5},
Rx(a1a′1)=V(MP5,1,2)−{a′3,a′4},
Rx(a2a′2)=V(MP5,1,2)−{a′4,a′5},
Rx(a3a′3)=V(MP5,1,2)−{a′5,a′1},
Rx(a4a′4)=V(MP5,1,2)−{a′1,a′2},
Rx(a5a′5)=V(MP5,1,2)−{a′2,a′3}.
For 1≤j≤5 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 y∈V(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)≤10∑114=52. In the same context, for 1≤j≤z it is clear from the above RLN sets that |Rx(aja′j)|=8 and |Rx(aja′j)|≥|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 y∈V(MP5,1,2) hence dimLF(MP5,1,2)≥10∑118=54. Consequently,
54≤dimLF(MP5,1,2)≤52. |
Case 2. For 1≤j≤z 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)=1z−1 for each y∈V(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)≤2z∑j=11z−1=2zz−1. In the same way, for 1≤j≤z it is clear from Lemma 4.1 |Rx(ajaj+1)|=2z−2 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)=12z−4 for each y∈V(MPz,1,2) hence by Lemma Y dimLF(MPz,1,2)≥2z∑j=112z−2=zz−2. Consequently,
zz−2≤dimLF(MPz,1,2)≤2zz−1. |
Lemma 4.3. Let MPz,1,2 be a modified prism network, where z≅3(mod4). Then
(a) |Rx(ajaj+1)|=z+1 and z⋃j=1Rx(ajaj+1)=V(MPz,1,2).
(b) |Rx(ajaj+1)|<|Rx(y)|, and |z⋃j=1Rx(ajaj+1)∩Rx(y)|>|Rx(ajaj+1)| where |Rx(e)| are the other possible RLN sets.
Proof. Let aj inner, a′j be the outer vertices of modified prism network, for 1≤j≤z, where z+1≅1(modz), we have following possibilities,
(a)Rx(ajaj+1)=V(MPz,1,2)−{aj+2,aj+4,aj+6,.....,az+i−5,az+i−3,az}∪{a′j+2,a′j+4,a′j+6,.....,a′z+i−5,a′z+i−3,a′z} and |Rx(ajaj+1)|=z+1 and |z⋃j=1Rx(ajaj+1)|=3z=|V(MPz,1,2)|.
(b) Rx(aja′j)=V(MPz,1,2)−{a′j+2,a′j+3,a′z+j−3,az+j−4}, Rx(ajaj+2)=V(MPz,1,2)−{aj+1,a′j+1}, Rx(a′ja′j+1)=V(MPz,1,2)−{aj+2,aj+4,aj+6,....,az+j−3,a′i+4,a′i+6,a′i+8,a′i+10,...,a′z+i−5}.
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.
RLN Set | Cardinality |
Rx(aja′j) | 2z−4>z+1 |
Rx(ajaj+2) | 2z−2>z+1 |
Rx(a′ja′j+1) | z+3>z+1 |
Theorem 4.4. Let MPz,1,2 be a modified prism network, where z≅3(mod4). Then
zz−1≤dimLF(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,a′3,a′5,a′7},
Rx(a2a3)=V(MP7,1,2)−{a4,a6,a1,a′4,a′6,a′1},
Rx(a3a4)=V(MP7,1,2)−{a5,a7,a2,a′5,a′7,a′2},
Rx(a4a5)=V(MP7,1,2)−{a6,a1,a3,a′6,a′1,a′3},
Rx(a5a6)=V(MP7,1,2)−{a7,a2,a4,a′7,a′2,a′4},
Rx(a6a7)=V(MP7,1,2)−{a1,a3,a5,a′1,a′3,a′5},
Rx(a7a1)=V(MP7,1,2)−{a2,a4,a6,a′2,a′4,a′6},
Rx(a1a′1)=V(MP7,1,2)−{a′3,a′4,a′5,a′6},
Rx(a2a′2)=V(MP7,1,2)−{a′4,a′5,a′6,a′7},
Rx(a3a′3)=V(MP7,1,2)−{a′5,a′6,a′7,a′1},
Rx(a4a′4)=V(MP7,1,2)−{a′6,a′7,a′1,a′2},
Rx(a5a′5)=V(MP7,1,2)−{a′7,a′1,a′2,a′3},
Rx(a6a′6)=V(MP7,1,2)−{a′1,a′2,a′3,a′4},
Rx(a7a′7)=V(MP7,1,2)−{a′2,a′3,a′4,a′5,},
Rx(a1a3)=V(MP7,1,2)−{a2,a′2},
Rx(a2a4)=V(MP7,1,2)−{a3,a′3},
Rx(a3a5)=V(MP7,1,2)−{a4,a′4},
Rx(a4a6)=V(MP7,1,2)−{a5,a′5},
Rx(a5a7)=V(MP7,1,2)−{a6,a′6},
Rx(a6a1)=V(MP7,1,2)−{a7,a′7},
Rx(a7a2)=V(MP7,1,2)−{a1,a′1},
Rx(a′1a′2)=V(MP7,1,2)−{a3,a5,a7,a′5},
Rx(a′2a′3)=V(MP7,1,2)−{a4,a6,a1,a′6},
Rx(a′3a′4)=V(MP7,1,2)−{a5,a7,a2,a′7},
Rx(a′4a′5)=V(MP7,1,2)−{a6,a1,a3,a′1},
Rx(a′5a′6)=V(MP7,1,2)−{a7,a2,a4,a′2},
Rx(a′6a′7)=V(MP7,1,2)−{a1,a3,a5,a′3},
Rx(a′7a′1)=V(MP7,1,2)−{a2,a4,a6,a′4}.
For 1≤j≤7 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 y∈V(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)≤14∑118=74. In the same context, for 1≤j≤z 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 y∈V(MP7,1,2) hence dimLF(MP7,1,2)≥14∑1112=76. Since MP7,1,2 is a non-bipartite network so its lower bound must be greater then 1. Consequently,
76≤dimLF(MP7,1,2)≤74. |
Case 2. For 1≤j≤z 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 y∈V(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 dimLF≤2z∑j=11z+1=2zz+1. In the same way, for 1≤j≤z it is clear from Lemma 4.3 |Rx(ajaj+1)|=2z−2 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)=12z−2 for each y∈V(MPz,1,2) hence by Lemma Y dimLF(MPz,1,2)≥2z∑j=112z−2=zz−1. Consequently,
zz−1≤dimLF(MPz,1,2)≤2zz+1. |
Lemma 4.5. Let MPz,1,2 be a modified generalized prism network, where z≅0(mod4). Then
(a) |Rx(ajaj+1)|=z and z⋃j=1Rx(ajaj+1)=V(MPz,1,2).
(b) |Rx(ajaj+1)|<|Rx(y)|, and |z⋃j=1Rx(ajaj+1)∩Rx(y)|>|Rx(ajaj+1)|, where |Rx(y)| are the other possible RLN sets.
Proof. Let aj inner, a′j be the outer vertices of modified generalized Prism network, for 1≤j≤z, 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+i−5, az+i−3,az+i−1}∪{a′j+2, a′j+4,a′j+6.....,a′z+2j2, a′z+2j+22,a′z+2j+62, a′z+2j+102, .....a′z+i−5,a′z+i−3, a′z+i−1} and |Rx(ajaj+1)|=z and |z⋃j=1Rx(ajaj+1)|=2z=|V(MPz,1,2)|.
(b)Rx(aja′j)=V(MPz,1,2)-{a′j+2,a′j+3,a′z+j−2, az+j−3}, Rx(ajaj+2)=V(MPz,1,2)-{aj+1,a′j+1,an+2j+22, a′n+2j+22}, Rx(a′ja′j+1)=V(MPz,1,2)−{aj+2,aj+4, aj+6,....,az+2j2,az+2j+22, az+2j+62,...., az+j−6,az+j−3,az+j−1}.
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.
RLN Set | Cardinality |
Rx(aja′j) | 2z−4>z |
Rx(ajaj+2) | 2z−2>z |
Rx(a′ja′j+1) | z+3>z |
Theorem 4.6. Let MPz,1,2 be a modified prism network, where z≅0(mod4). Then
zz−2≤dimLF(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,a′3,a′5,a′6,a′8},
Rx(a2a3)=V(MP8,1,2)−{a4,a6,a1,a2,a′4,a′6,a′7,a′1},
Rx(a3a4)=V(MP8,1,2)−{a5,a7,a8,a3,a′5,a′7,a′8,a′2},
Rx(a4a5)=V(MP8,1,2)−{a6,a8,a1,a4,a′6,a′8,a′1,a′3},
Rx(a5a6)=V(MP8,1,2)−{a7,a1,a2,a5,a′7,a′1,a′2,a′4},
Rx(a6a7)=V(MP8,1,2)−{a8,a2,a3,a6,a′8,a′2,a′3,a′5},
Rx(a7a8)=V(MP8,1,2)−{a1,a3,a4,a7,a′1,a′3,a′4,a′6},
Rx(a1a8)=V(MP8,1,2)−{a2,a4,a5,a8,a′2,a′4,a′5,a′7},
Rx(a1a′1)=V(MP8,1,2)−{a′3,a′4,a′6,a′7},
Rx(a2a′2)=V(MP8,1,2)−{a′4,a′5,a′7,a′8},
Rx(a3a′3)=V(MP8,1,2)−{a′5,a′6,a′8,a′1},
Rx(a4a′4)=V(MP8,1,2)−{a′6,a′7,a′1,a′2},
Rx(a5a′5)=V(MP8,1,2)−{a′7,a′8,a′2,a′3},
Rx(a6a′6)=V(MP8,1,2)−{a′8,a′1,a′3,a′4},
Rx(a7a′7)=V(MP8,1,2)−{a′1,a′2,a′4,a′5},
Rx(a8a′8)=V(MP8,1,2)−{a′2,a′3,a′5,a′6},
Rx(a′1a′2)=V(MP8,1,2)−{a3,a5,a6,a8},
Rx(a′2a′3)=V(MP8,1,2)−{a4,a6,a7,a1},
Rx(a′3a′4)=V(MP8,1,2)−{a5,a7,a8,a2},
Rx(a′4a′5)=V(MP8,1,2)−{a6,a8,a1,a3},
Rx(a′5a′6)=V(MP8,1,2)−{a7,a1,a2,a4},
Rx(a′6a′7)=V(MP8,1,2)−{a8,a2,a3,a5},
Rx(a′7a′8)=V(MP8,1,2)−{a1,a3,a4,a6},
Rx(a′8a′1)=V(MP8,1,2)−{a2,a4,a5,a7},
Rx(a1a3)=V(MP8,1,2)−{a2,a6,a′2,a′6},
Rx(a2a4)=V(MP8,1,2)−{a3,a7,a′3,a′7},
Rx(a3a5)=V(MP8,1,2)−{a4,a8,a′4,a′8},
Rx(a4a6)=V(MP8,1,2)−{a5,a1,a′5,a′1},
Rx(a5a7)=V(MP8,1,2)−{a6,a2,a′6,a′2},
Rx(a6a8)=V(MP8,1,2)−{a3,a7,a′3,a′7},
Rx(a7a1)=V(MP8,1,2)−{a8,a1,a′8,a′1},
Rx(a8a2)=V(MP8,1,2)−{a1,a5,a′1,a′5}.
For 1≤j≤8 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 y∈V(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)≤16∑118=2. In the same context, for 1≤j≤z 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 y∈V(MP7,1,2) hence dimLF(MP8,1,2)≥16∑1112=43. Since MP8,1,2 is a non-bipartite network so its lower bound must be greater then 1. Consequently,
43≤dimLF(MP8,1,2)≤2. |
Case 2. For 1≤j≤z, 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 y∈V(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)≤2z∑j=11z=2. In the same way, For 1≤j≤z it is clear from Lemma 4.5 |Rx(ajaj+1)|=2z−4 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)=1z−1 for each y∈V(MPz,1,2) hence by Lemma Y dimLF(MPz,1,2)≥2z∑j=112z−4=zz−2. Consequently,
zz−2≤dimLF(MPz,1,2)≤2. |
Lemma 4.7. Let MPz,1,2 be a modified prism network, where z≅2(mod4). Then
(a) |Rx(ajaj+1)|=z+2 and z⋃j=1Rx(ajaj+1)=V(MPz,1,2).
(b) |Rx(ajaj+1)|<|Rx(y)|, and |z⋃j=1Rx(ajaj+1)∩Rx(y)|>|Rx(ajaj+1)| where |Rx(y)| are the other possible resolving local neighbourhood sets.
Proof. Let aj inner, a′j be the outer vertices of modified generalized Prism network, for 1≤j≤z, where z+1≅(1modz), we have following possibilities
(a) Rx(ajaj+1)=V(MPz,1,2)−{aj+2,aj+4,aj+6....., az+j−12,az+2j+22, az+2j+62,az+2j+102, .....az+i−5, az+i−3,az+i−1}∪{a′j+2,a′j+4,a′j+6.....,a′z+j−12, a′z+2j+22,a′z+2j+62,a′z+2j+102, .....a′z+i−5,a′z+i−3,a′z+i−1} and |Rx(ajaj+1)|=z and |z⋃j=1Rx(ajaj+1)|=2z=|V(MPz,1,2)|.
(b) Rx(aja′j)=V(MPz,1,2)−{a′j+2,a′j+3,a′z+j−2,az+j−3}, Rx(ajaj+2)=V(MPz,1,2)−{aj+1,a′j+1, an+2j+22,a′n+2j+22}, Rx(a′ja′j+1)=V(MPz,1,2)−{aj+2, aj+4,aj+6, ....,az+j−12,az+2j+22, az+2j+62,....,az+j−6,az+j−3,az+j−1}.
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.
RLN Set | Cardinality |
Rx(aja′j) | 2z−4>z+2 |
Rx(ajaj+2) | 2z−4>z+2 |
Rx(a′ja′j+1) | 2z−4>z+2 |
Theorem 4.8. Let MPz,1,2 be a modified prism network, where z≅2(mod4). Then
zz−2≤dimLF(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,a′3,a′5,a′6},
Rx(a2a3)=V(MP6,1,2)−{a4,a1,a′4,a′6,a′1},
Rx(a3a4)=V(MP6,1,2)−{a5,a2,a′5,a′1,a′2},
Rx(a4a5)=V(MP6,1,2)−{a6,a3,a′6,a′2,a′3},
Rx(a5a6)=V(MP6,1,2)−{a1,a4,a′1,a′3,a′4},
Rx(a6a1)=V(MP6,1,2)−{a2,a5,a′2,a′4,a′5},
Rx(a1a′1)=V(MP6,1,2)−{a′3,a′4,a′5},
Rx(a2a′2)=V(MP6,1,2)−{a′4,a′5,a′6},
Rx(a3a′3)=V(MP6,1,2)−{a′5,a′6,a′1},
Rx(a4a′4)=V(MP6,1,2)−{a′6,a′1,a′2},
Rx(a5a′5)=V(MP6,1,2)−{a′1,a′2,a′3},
Rx(a6a′6)=V(MP6,1,2)−{a′2,a′3,a′4},
Rx(a1a3)=V(MP6,1,2)−{a2,a5,a′2,a′5},
Rx(a2a4)=V(MP6,1,2)−{a3,a6,a′3,a′6},
Rx(a3a5)=V(MP6,1,2)−{a4,a1,a′4,a′1},
Rx(a4a6)=V(MP6,1,2)−{a5,a2,a′5,a′2},
Rx(a5a1)=V(MP6,1,2)−{a6,a3,a′6,a′3},
Rx(a6a2)=V(MP6,1,2)−{a1,a4,a′1,a′4},
Rx(a′1a′2)=V(MP6,1,2)−{a3,a6},
Rx(a′2a′3)=V(MP6,1,2)−{a4,a1},
Rx(a′3a′4)=V(MP6,1,2)−{a5,a2},
Rx(a′4a′5)=V(MP6,1,2)−{a6,a3},
Rx(a′5a′6)=V(MP6,1,2)−{a1,a4},
Rx(a′1a′6)=V(MP6,1,2)−{a2,a5}.
For 1≤j≤6 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 y∈V(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)≤12∑118=32. In the same context, for 1≤j≤z 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 y∈V(MP6,1,2) hence dimLF(MP6,1,2)≥12∑1110=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 1≤j≤z 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 y∈V(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)≤2z∑j=11z+2=2zz+2. In the same way, for 1≤j≤z it is clear from Lemma 4.7 |Rx(ajaj+1)|=2z−4 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)=1z−1 for each y∈V(MPz,1,2) hence by Lemma YdimLF(MPz,1,2)≥2z∑j=112z−4=zz−2. Consequently,
zz−2≤dimLF(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 z≅2(mod4). Then
(a) |Rx(ajaj+1)|=3z+62 and 3z⋃j=1Rx(ajaj+1)=V(MQz,1,2).
(b) |Rx(ajaj+1)|<|Rx(y)|, and |3z⋃j=1Rx(ajaj+1)∩Rx(y)|>|Rx(ajaj+1)| where |Rx(e)| are the other possible RLN sets.
Proof. Let ai inner, a′i middle and bi be the outer vertices of modified generalized Prism network, for 1≤j≤z, where z+1≅1(modz), we have the following possibilities
(a) Rx(ajaj+1)=V(MQz,1,2)−{aj+2,aj+4,aj+6....., az+j−12,az+2j+22,az+2j+62,az+2j+102, .....az+i−5,az+i−3, az+i−1}∪{a′j+2,a′j+4,a′j+6....., a′z+j−12,a′z+2j+22, a′z+2j+62, a′z+2j+102,.....a′z+i−5, a′z+i−3,a′z+i−1}∪{bj+2,bj+4,bj+6....., bz+j−12,bz+2j+22,bz+2j+62, bz+2j+102,.....bz+i−5, bz+i−3,bz+i−1} and |Rx(ajaj+1)|=3z+62 and |3z⋃j=1Rx(ajaj+1)|=3z=|V(MQz,1,2)|.
(b)Rx(aja′j)=V(MQz,1,2)−{aj+2,ai+3,az+j−3,az+j−2, bj+2,bj+3,bz+j−3,bz+j−2}, Rx(ajaj+2)=V(MQz,1,2)−{aj+1,az+2j+22, a′j+1,a′z+2j+22, bj+1,bz+2j+22}, Rx(a′ja′j+1)=V(MQz,1,2)−{az+j−1}, Rx(bjbj+1)=V(MQz,1,2)−{aj+2}, Rx(a′jbj)=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.
RLN Set | Cardinality |
Rx(aja′j) | 3z−4>3z+62 |
Rx(ajaj+2) | 3z−4>3z+62 |
Rx(a′jbj) | 3z>3z+62 |
Rx(a′ja′j+1) | 3z−1>3z+62 |
Rx(bjbj+1) | 3z−1>3z+62 |
Theorem 5.2. Let MQz,1,2 be a modified prism network, where z≅2(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,a′3,a′6,b3,b6},
Rx(a2a3)=V(MQ6,1,2)−{a4,a1,a′4,a′1,b4,b1},
Rx(a3a4)=V(MQ6,1,2)−{a5,a6,a′5,a′2,b5,b2},
Rx(a4a5)=V(MQ6,1,2)−{a6,a1,a′6,a′3,b6,b3},
Rx(a5a6)=V(MQ6,1,2)−{a1,a2,a′1,a′2,b1,b4},
Rx(a6a1)=V(MQ6,1,2)−{a2,a3,a′2,a′3,b2,b5},
Rx(a1a′1)=V(MQ6,1,2)−{a′3,a′4,a′5,b3,b4,b5},
Rx(a2a′2)=V(MQ6,1,2)−{a′4,a′5,a′6,b4,b5,b6},
Rx(a3a′4)=V(MQ6,1,2)−{a′5,a′6,a′1,b5,b6,b1},
Rx(a4a′4)=V(MQ6,1,2)−{a′6,a′1,a′2,b6,b1,b2},
Rx(a5a′5)=V(MQ6,1,2)−{a′1,a′2,a′3,b1,b2,b3},
Rx(a6a′6)=V(MQ6,1,2)−{a′2,a′3,a′4,b2,b3,b4},
Rx(a1a3)=V(MQ6,1,2)−{a2,a5,a′2,a′5,b2,b5},
Rx(a2a4)=V(MQ6,1,2)−{a3,a6,a′3,a′6,b3,b6},
Rx(a3a5)=V(MQ6,1,2)−{a4,a1,a′4,a′1,b4,b1},
Rx(a4a6)=V(MQ6,1,2)−{a5,a2,a′5,a′2,b5,b2},
Rx(a5a1)=V(MQ6,1,2)−{a6,a3,a′6,a′3,b6,b3},
Rx(a6a2)=V(MQ6,1,2)−{a1,a4,a′6,a′4,b1,b4},
Rx(a′1a′2)=V(MQ6,1,2)−{a3,a6},
Rx(a′2a′3)=V(MQ6,1,2)−{a4,a5},
Rx(a′3a′4)=V(MQ6,1,2)−{a5,a6},
Rx(a′4a′5)=V(MQ6,1,2)−{a6,a1},
Rx(a′5a′6)=V(MQ6,1,2)−{a1,a2},
Rx(a′6a′1)=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(a′1b1)=V(MQ6,1,2),
Rx(a′2b2)=V(MQ6,1,2),
Rx(a′3b3)=V(MQ6,1,2),
Rx(a′4b4)=V(MQ6,1,2),
Rx(a′5b5)=V(MQ6,1,2),
Rx(a′6b6)=V(MQ6,1,2),
Rx(a′6b6)=V(MP6,1,2).
For 1≤j≤6 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 y∈V(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)≤18∑1112=32. For 1≤j≤6 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 y∈V(MQ6,1,2) hence dimLF(MQ6,1,2)≥18∑1118=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 1≤j≤z 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 y∈V(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)≤3z∑j=123z+6=2zz+2.
For 1≤j≤z it is clear from Lemma 5.1 |Rx(a′jb′j)|=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 y∈V(MQz,1,2). Hence by Lemma Y dimLF(MQz,1,2)≥3z∑j=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 z≅0(mod4). Then
(a) |Rx(ajaj+1)|=3z2 and 3z⋃j=1Rx(ajaj+1)=V(MQz,1,2).
(b) |Rx(ajaj+1)|<|Rx(y)|, and |3z⋃j=1Rx(ajaj+1)∩Rx(y)|>|Rx(ajaj+1)|, where |Rx(y)| are the other possible RLN sets.
Proof. Let ai inner, a′i middle and bi be the outer vertices of modified generalized Prism network, for 1≤j≤z, where z+1≅1(modz), we have following possibilities
(a) Rx(ajaj+1) = V(MQz,1,2)−{aj+2,aj+4,aj+6.....,az+j−12, az+2j+22, az+2j+62,az+2j+102,.....az+i−5, az+i−3,az+i−1}∪{a′j+2,a′j+4,a′j+6.....,a′z+j−12, a′z+2j+22, a′z+2j+62,a′z+2j+102, .....a′z+i−5,a′z+i−3,a′z+i−1}{bj+2,bj+4,bj+6....., bz+j−12, bz+2j+22, bz+2j+62,bz+2j+102, .....bz+i−5,bz+i−3, bz+i−1} and |Rx(ajaj+1)|=3z2 and |3z⋃j=1Rx(ajaj+1)| = 3z=|V(MQz,1,2)|.
(b) Rx(aja′j)=V(MQz,1,2)−{aj+2,ai+3,az+j−3,az+j−2,bj+2,bj+3,bz+j−3,bz+j−2}, Rx(ajaj+2)=V(MQz,1,2)-{aj+1,az+2j+22, a′j+1,a′z+2j+22,bj+1, bz+2j+22}, Rx(a′ja′j+1)=V(MQz,1,2)−{az+j−1}, Rx(bjbj+1)=V(MQz,1,2)−{aj+2}, Rx(a′jbj)=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.
RLN Set | Cardinality |
Rx(aja′j) | 3z−4>3z2 |
Rx(ajaj+2) | 3z−4>3z2 |
Rx(a′jbj) | 3z>3z2 |
Rx(a′ja′j+1) | 3z−1>3z2 |
Rx(bjbj+1) | 3z−1>3z2 |
Theorem 5.4. Let MQz,1,2 be a modified prism network, where z≅0(mod4). Then
1≤dimLF(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,a′3,a′5,a′6,a′8,b3,b5,b7,b8},
Rx(a2a3)=V(MQ8,1,2)−{a4,a6,a1,a2,a′4,a′6,a′7,a′1,b4,b6,b8,b1},
Rx(a3a4)=V(MQ8,1,2)−{a5,a7,a8,a3,a′5,a′7,a′8,a′2,b5,b7,b1,b2},
Rx(a4a5)=V(MQ8,1,2)−{a6,a8,a1,a4,a′6,a′8,a′1,a′3,b6,b8,b2,b3},
Rx(a5a6)=V(MQ8,1,2)−{a7,a1,a2,a5,a′7,a′1,a′2,a′4,b7,b1,b3,b4},
Rx(a6a7)=V(MQ8,1,2)−{a8,a2,a3,a6,a′8,a′2,a′3,a′5,b8,b2,b4,b5},
Rx(a7a8)=V(MQ8,1,2)−{a1,a3,a4,a7,a′1,a′3,a′4,a′6,b1,b3,b5,b6},
Rx(a1a8)=V(MQ8,1,2)−{a2,a4,a5,a8,a′2,a′4,a′5,a′7,b2,b4,b6,b7},
Rx(a1a′1)=V(MQ8,1,2)−{a′3,a′4,a′6,a′7,b3,b4,b6,b7},
Rx(a2a′2)=V(MQ8,1,2)−{a′4,a′5,a′7,a′8,b4,b5,b7,b8},
Rx(a3a′3)=V(MQ8,1,2)−{a′5,a′6,a′8,a′1,b5,b6,b8,b1},
Rx(a4a′4)=V(MQ8,1,2)−{a′6,a′7,a′1,a′2,b6,b7,b1,b2},
Rx(a5a′5)=V(MQ8,1,2)−{a′7,a′8,a′2,a′3,b7,b8,b2,b3},
Rx(a6a′6)=V(MQ8,1,2)−{a′8,a′1,a′3,a′4,b8,b1,b3,b4},
Rx(a7a′7)=V(MQ8,1,2)−{a′1,a′2,a′4,a′5,b1,b2,b4,b5},
Rx(a8a′8)=V(MQ8,1,2)−{a′2,a′3,a′5,a′6,b2,b3,b5,b6},
Rx(a′1a′2)=V(MQ8,1,2)−{a3,a5,a6,a8,},
Rx(a′2a′3)=V(MQ8,1,2)−{a4,a6,a7,a1},
Rx(a′3a′4)=V(MQ8,1,2)−{a5,a7,a8,a2},
Rx(a′4a′5)=V(MQ8,1,2)−{a6,a8,a1,a3},
Rx(a′5a′6)=V(MQ8,1,2)−{a7,a1,a2,a4},
Rx(a′6a′7)=V(MQ8,1,2)−{a8,a2,a3,a5},
Rx(a′7a′8)=V(MQ8,1,2)−{a1,a3,a4,a6},
Rx(a′8a′1)=V(MQ8,1,2)−{a2,a4,a5,a7},
Rx(a1a3)=V(MQ8,1,2)−{a2,a6,a′2,a′6,b2,b6},
Rx(a2a4)=V(MQ8,1,2)−{a3,a7,a′3,a′7,b3,b7},
Rx(a3a5)=V(MQ8,1,2)−{a4,a8,a′4,a′8,b4,b8},
Rx(a4a6)=V(MQ8,1,2)−{a5,a1,a′5,a′1,b5,b1},
Rx(a5a7)=V(MQ8,1,2)−{a6,a2,a′6,a′2,b6,b2},
Rx(a6a8)=V(MQ8,1,2)−{a3,a7,a′3,a′7,b7,b3},
Rx(a7a1)=V(MQ8,1,2)−{a8,a1,a′8,a′1,b8,b4},
Rx(a8a2)=V(MQ8,1,2)−{a1,a5,a′1,a′5,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(a′1b1)=V(MQ8,1,2),
Rx(a′2b2)=V(MQ8,1,2),
Rx(a′3b3)=V(MQ8,1,2),
Rx(a′4b4)=V(MQ8,1,2),
Rx(a′5b5)=V(MQ8,1,2),
Rx(a′6b6)=V(MQ8,1,2),
Rx(a′7b7)=V(MQ8,1,2),
Rx(a′8b8)=V(MQ8,1,2).
For 1≤j≤8 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 y∈V(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)≤24∑1112=2. In the same context, for 1≤j≤z it is clear from the above RLN sets that |Rx(ajaj+2)|=12 and |Rx(a′ja′j+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 y∈V(MQ8,1,2) hence dimLF(MQ8,1,2)≥24∑1124=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 1≤j≤z 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 y∈V(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)≤3z∑j=123z=2. In the same way, for 1≤j≤z it is clear from Lemma 5.3 that |Rx(a′jbj)|=3z and |Rx(a′jbj)|≥|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 y∈V(MQz,1,2) hence by Lemma YdimLF(MQz,1,2)≥3z∑j=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 z≅1(mod4). Then
(a) |Rx(ajaj+1)|=3z−32 and z⋃j=1Rx(ajaj+1)=V(MQz,1,2).
(b) |Rx(ajaj+1)|<|Rx(y)|, and |z⋃j=1Rx(ajaj+1)∩Rx(y)|>|Rx(ajaj+1)| where |Rx(y)| are the other possible RLN sets.
Proof. Let aj inner, a′j middle and bj are be the outer vertices of modified prism network, for 1≤j≤z, where z+1≅(1modz), we have following possibilities
(a) Rx(ajaj+1) = V(MPz,1,2)−{aj+2,aj+4,aj+6.....,az+i−5, az+i−3,az+i−1}∪{a′j+2,a′j+4, a′j+6,.....,a′z+i−5, a′z+i−3,a′z+i−1}∪∪{bj+2,bj+4, bj+6,.....,bz+i−5,bz+i−3, bz+i−1}∪{az+2i+22}∪{a′z+2i+22} and |Rx(ajaj+1)|=z−1 and |3z⋃j=1Rx(ajaj+1)|=3z=|V(MPz,1,2)|.
(b) Rx(aja′j)=V(MPz,1,2)-{a′j+2,a′j+3,a′z+j−3,az+j−4,bj+2,bj+3,bz+j−3,bz+j−4}, Rx(ajaj+2)=V(MPz,1,2)-{aj+1,a′j+1,bj+1,az+2j+12,a′z+2j+12, bz+2j+12}, Rx(bjbj+1)=Rx(a′ja′j+1)=V(MPz,1,2)−{aj+2,aj+4, aj+6,....,az+j−3,az+j−1}∪{az+2j+12}∪{a′z+2j+12,bz+2j+12}. Rx(a′jbj)=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.
RLN Set | Cardinality |
Rx(aja′j) | 3z−8>3z−32 |
Rx(ajaj+2) | 3z−6>3z−32 |
Rx(a′ja′j+1) | 5x−25>3z−32 |
Rx(a′jbj) | 3z>3z−32 |
Rx(bjaj+1) | 5z−25>3z−32 |
Theorem 5.6. Let MPz,1,2 be a modified prism network, where z≅1(mod4). Then
1<dimLF(MPz,1,2)≤2zz−1. |
Proof. Case 1. For z=5, we have the following RLN sets
Rx(a1a2)={a1,a2,a′1,a′2,b1,b2},
Rx(a2a3)={a2,a3,a′2,a′3,b2,b3},
Rx(a3a4)={a3,a4,a′3,a′4,b3,b4},
Rx(a4a5)={a4,a5,a′4,a′5,b4,b5},
Rx(a5a1)={a1,a5,a′1,a′5,b5,b1},
Rx(a1a3)={a1,a3,a′1,a′3,b1,b3},
Rx(a1a4)={a1,a4,a′1,a′4,b2,b4},
Rx(a2a4)={a2,a4,a′2,a′4,b3,b5},
Rx(a2a5)={a2,a5,a′2,a′5,b4,b1},
Rx(a3a5)={a3,a5,a′3,a′5,b5,b2},
Rx(a1a′1)=V(MP5,1,2)−{a′3,a′4,b3,b4},
Rx(a2a′2)=V(MP5,1,2)−{a′4,a′5,b4,b5},
Rx(a3a′3)=V(MP5,1,2)−{a′5,a′1,b5,b1},
Rx(a4a′4)=V(MP5,1,2)−{a′1,a′2,b1,b2},
Rx(a5a′5)=V(MP5,1,2)−{a′2,a′3,b2,b3},
Rx(a′1a′2)=V(MP5,1,2)−{a3,a4,a5,a′4,b4},
Rx(a′2a′3)=V(MP5,1,2)−{a4,a5,a1,a′5,b5},
Rx(a′3a′4)=V(MP5,1,2)−{a5,a1,a2,a′1,b1},
Rx(a′4a′5)=V(MP5,1,2)−{a1,a2,a3,a′2,b5},
Rx(a′5a′1)=V(MP5,1,2)−{a2,a3,a4,a′3,b1},
Rx(b1b2)=V(MP5,1,2)−{a3,a4,a5,a′4,b4},
Rx(b2b3)=V(MP5,1,2)−{a4,a5,a1,a′5,b5},
Rx(b3b4)=V(MP5,1,2)−{a5,a1,a2,a′1,b1},
Rx(b4b5)=V(MP5,1,2)−{a1,a2,a3,a′2,b2},
Rx(b5b1)=V(MP5,1,2)−{a2,a3,a4,a′3,b3},
Rx(a′1b1)=V(MP5,1,2),
Rx(a′2b2)=V(MP5,1,2),
Rx(a′3b3)=V(MP5,1,2),
Rx(a′4b4)=V(MP5,1,2),
Rx(a′5b5)=V(MP5,1,2).
For 1≤j≤5 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 y∈V(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)≤15∑116=52. In the same context, for 1≤j≤z it is clear that |Rx(aja′j)|=8 and |Rx(a′jbj)|≥|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 y∈V(MQ5,1,2) hence dimLF(MQ5,1,2)≥15∑1115=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 1≤j≤z 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)=1z−1 for each y∈V(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)≤3z∑j=123z−3=2zz−1. In the same context, for 1≤j≤z it is clear from Lemma 5.5 that |Rx(a′jbj)|=3z and |Rx(a′jbj)|≥|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 y∈V(MQ) hence by Lemma Y dimLF(MQz,1,2)≥3z∑j=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. |
Lemma 5.7. Let MQz,1,2 be a modified prism network, where z≅3(mod4). Then
(a) |Rx(ajaj+1)|=3z+32 and 3z⋃j=1Rx(ajaj+1)=V(MQz,1,2).
(b) |Rx(ajaj+1)|<|Rx(y)|, and |3z⋃j=1Rx(ajaj+1)∩Rx(y)|>|Rx(ajaj+1)| where |Rx(y)| are the other possible resolving local neighbourhood sets.
Proof. Let ai inner, a′i middle and bi be the outer vertices of modified generalized prism network, for 1≤j≤z, where z+1≅(1modz), we have following possibilities
(a) Rx(ajaj+1)=V(MPz,1,2)−{aj+2,aj+4,aj+6,....,az+i−1}∪{a′j+2,a′j+4,a′j+6,....,a′z+i−1}∪{bj+2,bj+4,bj+6,....,bz+i−1} and |Rx(ajaj+1)|=3z+32 and |3z⋃j=1Rx(ajaj+1)|=3z=|V(MPz,1,2)|.
(b)Rx(aja′j) =V(MPz,1,2)−{a′j+2,a′j+3,a′z+j−3,az+j−2,bj+2,bj+3,bz+j−3,bz+j−2}, Rx(ajaj+2)=V(MPz,1,2)−{aj+1,a′j+1,bj+1}, Rx(a′ja′j+1)=V(MPz,1,2)−{aj+2,aj+4,aj+6,...,az+j−1,a′z+2j+12,bz+2j+12}, Rx(bjbj+1)=V(MPz,1,2)−{aj+2,aj+4,aj+6,...,az+i−1,a′z+2j+12,bz+2j+12}, Rx(a′jbj)=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.
RLN Set | Cardinality |
Rx(aja′j) | 3z−4>3z+32 |
Rx(ajaj+2) | 3z−4>3z+32 |
Rx(a′jbj) | 3z>3z+32 |
Rx(a′ja′j+1) | 3z−1>3z+32 |
Rx(bjbj+1) | 3z−1>3z+32 |
Theorem 5.8. Let MQz,1,2 be a generalized modified prism network, where z≅3(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,a′3,a′5,a′7,b3,b5,b7},
Rx(a2a3)=V(MQ7,1,2)−{a4,a6,a1,a′4,a′6,a′1,b4,b6,b1},
Rx(a3a4)=V(MQ7,1,2)−{a5,a7,a2,a′5,a′7,a′2,b5,b7,b2},
Rx(a4a5)=V(MQ7,1,2)−{a6,a1,a3,a′6,a′1,a′3,b6,b1,b3},
Rx(a5a6)=V(MQ7,1,2)−{a7,a2,a4,a′7,a′2,a′4,b7,b2,b4},
Rx(a6a7)=V(MQ7,1,2)−{a1,a3,a5,a′1,a′3,a′5,b1,b3,b5},
Rx(a7a1)=V(MQ7,1,2)−{a2,a4,a6,a′2,a′4,a′6,b2,b4,b6},
Rx(a1a′1)=V(MQ7,1,2)−{a′3,a′4,a′5,a′6,b3,b4,b5,b6},
Rx(a2a′2)=V(MQ7,1,2)−{a′4,a′5,a′6,a′7,b4,b5,b6,b7},
Rx(a3a′3)=V(MQ7,1,2)−{a′5,a′6,a′7,a′1,b5,b6,b7,b1},
Rx(a4a′4)=V(MQ7,1,2)−{a′6,a′7,a′1,a′2,b5,b7,b1,b2},
Rx(a5a′5)=V(MQ7,1,2)−{a′7,a′1,a′2,a′3b6,b1,b2,b3},
Rx(a6a′6)=V(MQ7,1,2)−{a′1,a′2,a′3,a′4,b7,b2,b3,b4},
Rx(a7a′7)=V(MQ7,1,2)−{a′2,a′3,a′4,a′5,b1,b3,b4,b5},
Rx(a1a3)=V(MQ7,1,2)−{a2,a′2,b2},
Rx(a2a4)=V(MQ7,1,2)−{a3,a′3,b3},
Rx(a3a5)=V(MQ7,1,2)−{a4,a′4,b4},
Rx(a4a6)=V(MQ7,1,2)−{a5,a′5,b5},
Rx(a5a7)=V(MQ7,1,2)−{a6,a′6,b6},
Rx(a6a1)=V(MQ7,1,2)−{a7,a′7,b7},
Rx(a7a2)=V(MQ7,1,2)−{a1,a′1,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(a′1a′2)=V(MQ7,1,2)−{a3,a5,a7,a′5,b5},
Rx(a′2a′3)=V(MQ7,1,2)−{a4,a6,a1,a′6,b6},
Rx(a′3a′4)=V(MQ7,1,2)−{a5,a7,a2,a′7,b7},
Rx(a′4a′5)=V(MQ7,1,2)−{a6,a1,a3,a′1,b1},
Rx(a′5a′6)=V(MQ7,1,2)−{a7,a2,a4,a′2,b2},
Rx(a′6a′7)=V(MQ7,1,2)−{a1,a3,a5,a′3,b3},
Rx(a′7a′1)=V(MQ7,1,2)−{a2,a4,a6,a′4,b4},
Rx(a′1b1)=V(MQ7,1,2),
Rx(a′2b2)=V(MQ7,1,2),
Rx(a′3b3)=V(MQ7,1,2),
Rx(a′4b4)=V(MQ7,1,2),
Rx(a′5b5)=V(MQ7,1,2),
Rx(a′6b6)=V(MQ7,1,2),
Rx(a′7b7)=V(MQ7,1,2).
For 1≤j≤7 |Rx(aja′j+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 y∈V(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 dimLF≤21∑1112=712. In the same context, for 1≤j≤7 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 y∈V(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 1≤j≤z from Lemma 5.7 it is clear that |Rx(aja′j)|=23z+6 and |Rx(aja′j)|≤|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 y∈V(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 dimLF≤3z∑j=123z+3=2zz+1. In the same context for 1≤j≤z it is clear from Lemma 5.7 that |Rx(a′jbj)|=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 y∈V(MQz,1,2). Therefore, by Lemma Y dimLF3z∑j=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.
z≅ | LFMDs | Limiting LFMDs as z→∞ | Comment |
1(mod4) | zz−1≤dimLF(MPz,1,2)≤2zz−1 | 1<dimLF(MPz,1,2)≤2 | Bounded |
3(mod4) | zz−1≤dimLF(MPz,1,2)≤2zz+1 | 1<dimLF(MPz,1,2)≤2 | Bounded |
0(mod4) | zz−2≤dimLF(MPz,1,2)≤2 | 1<dimLF(MPz,1,2)≤2 | Bounded |
2(mod4) | zz−2≤dimLF(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)≤2zz−1 | 1<dimLF(MQz,1,2)≤2 | Bounded |
3(mod4) | 1<dimLF(MQz,1,2)≤2zz+2 | 1<dimLF(MQz,1,2)≤2 | Bounded |
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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RLN Set | Cardinality |
Rx(aja′j) | 2z−4>z−1 |
Rx(ajaj+2) | 2z−2>z−1 |
Rx(a′ja′j+1) | z+3>z−1 |
RLN Set | Cardinality |
Rx(aja′j) | 2z−4>z+1 |
Rx(ajaj+2) | 2z−2>z+1 |
Rx(a′ja′j+1) | z+3>z+1 |
RLN Set | Cardinality |
Rx(aja′j) | 2z−4>z |
Rx(ajaj+2) | 2z−2>z |
Rx(a′ja′j+1) | z+3>z |
RLN Set | Cardinality |
Rx(aja′j) | 2z−4>z+2 |
Rx(ajaj+2) | 2z−4>z+2 |
Rx(a′ja′j+1) | 2z−4>z+2 |
RLN Set | Cardinality |
Rx(aja′j) | 3z−4>3z+62 |
Rx(ajaj+2) | 3z−4>3z+62 |
Rx(a′jbj) | 3z>3z+62 |
Rx(a′ja′j+1) | 3z−1>3z+62 |
Rx(bjbj+1) | 3z−1>3z+62 |
RLN Set | Cardinality |
Rx(aja′j) | 3z−4>3z2 |
Rx(ajaj+2) | 3z−4>3z2 |
Rx(a′jbj) | 3z>3z2 |
Rx(a′ja′j+1) | 3z−1>3z2 |
Rx(bjbj+1) | 3z−1>3z2 |
RLN Set | Cardinality |
Rx(aja′j) | 3z−8>3z−32 |
Rx(ajaj+2) | 3z−6>3z−32 |
Rx(a′ja′j+1) | 5x−25>3z−32 |
Rx(a′jbj) | 3z>3z−32 |
Rx(bjaj+1) | 5z−25>3z−32 |
RLN Set | Cardinality |
Rx(aja′j) | 3z−4>3z+32 |
Rx(ajaj+2) | 3z−4>3z+32 |
Rx(a′jbj) | 3z>3z+32 |
Rx(a′ja′j+1) | 3z−1>3z+32 |
Rx(bjbj+1) | 3z−1>3z+32 |
z≅ | LFMDs | Limiting LFMDs as z→∞ | Comment |
1(mod4) | zz−1≤dimLF(MPz,1,2)≤2zz−1 | 1<dimLF(MPz,1,2)≤2 | Bounded |
3(mod4) | zz−1≤dimLF(MPz,1,2)≤2zz+1 | 1<dimLF(MPz,1,2)≤2 | Bounded |
0(mod4) | zz−2≤dimLF(MPz,1,2)≤2 | 1<dimLF(MPz,1,2)≤2 | Bounded |
2(mod4) | zz−2≤dimLF(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)≤2zz−1 | 1<dimLF(MQz,1,2)≤2 | Bounded |
3(mod4) | 1<dimLF(MQz,1,2)≤2zz+2 | 1<dimLF(MQz,1,2)≤2 | Bounded |
RLN Set | Cardinality |
Rx(aja′j) | 2z−4>z−1 |
Rx(ajaj+2) | 2z−2>z−1 |
Rx(a′ja′j+1) | z+3>z−1 |
RLN Set | Cardinality |
Rx(aja′j) | 2z−4>z+1 |
Rx(ajaj+2) | 2z−2>z+1 |
Rx(a′ja′j+1) | z+3>z+1 |
RLN Set | Cardinality |
Rx(aja′j) | 2z−4>z |
Rx(ajaj+2) | 2z−2>z |
Rx(a′ja′j+1) | z+3>z |
RLN Set | Cardinality |
Rx(aja′j) | 2z−4>z+2 |
Rx(ajaj+2) | 2z−4>z+2 |
Rx(a′ja′j+1) | 2z−4>z+2 |
RLN Set | Cardinality |
Rx(aja′j) | 3z−4>3z+62 |
Rx(ajaj+2) | 3z−4>3z+62 |
Rx(a′jbj) | 3z>3z+62 |
Rx(a′ja′j+1) | 3z−1>3z+62 |
Rx(bjbj+1) | 3z−1>3z+62 |
RLN Set | Cardinality |
Rx(aja′j) | 3z−4>3z2 |
Rx(ajaj+2) | 3z−4>3z2 |
Rx(a′jbj) | 3z>3z2 |
Rx(a′ja′j+1) | 3z−1>3z2 |
Rx(bjbj+1) | 3z−1>3z2 |
RLN Set | Cardinality |
Rx(aja′j) | 3z−8>3z−32 |
Rx(ajaj+2) | 3z−6>3z−32 |
Rx(a′ja′j+1) | 5x−25>3z−32 |
Rx(a′jbj) | 3z>3z−32 |
Rx(bjaj+1) | 5z−25>3z−32 |
RLN Set | Cardinality |
Rx(aja′j) | 3z−4>3z+32 |
Rx(ajaj+2) | 3z−4>3z+32 |
Rx(a′jbj) | 3z>3z+32 |
Rx(a′ja′j+1) | 3z−1>3z+32 |
Rx(bjbj+1) | 3z−1>3z+32 |
z≅ | LFMDs | Limiting LFMDs as z→∞ | Comment |
1(mod4) | zz−1≤dimLF(MPz,1,2)≤2zz−1 | 1<dimLF(MPz,1,2)≤2 | Bounded |
3(mod4) | zz−1≤dimLF(MPz,1,2)≤2zz+1 | 1<dimLF(MPz,1,2)≤2 | Bounded |
0(mod4) | zz−2≤dimLF(MPz,1,2)≤2 | 1<dimLF(MPz,1,2)≤2 | Bounded |
2(mod4) | zz−2≤dimLF(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)≤2zz−1 | 1<dimLF(MQz,1,2)≤2 | Bounded |
3(mod4) | 1<dimLF(MQz,1,2)≤2zz+2 | 1<dimLF(MQz,1,2)≤2 | Bounded |