Abstract
The different distance-based parameters are used to study the problems in various fields of computer science and chemistry such as pattern recognition, image processing, integer programming, navigation, drug discovery, and formation of different chemical compounds. In particular, distance among the nodes (vertices) of the networks plays a supreme role to study structural properties of networks such as connectivity, robustness, completeness, complexity, and clustering. Metric dimension is used to find the locations of machines with respect to minimum utilization of time, lesser number of the utilized nodes as places of the objects, and shortest distance among destinations. In this paper, lower bound of local fractional metric dimension for the connected networks is improved from unity and expressed in terms of ratio obtained by the cardinalities of the under-study network and the local resolving neighbourhood with maximum order for some edges of network. In the same context, the LFMDs of prism-related networks such as circular diagonal ladder, antiprism, triangular winged prism, and sun flower networks are computed with the help of obtained criteria. At the end, the bounded- and unboundedness of the obtained results is also shown numerically.
1. Introduction
For a connected network , Salter introduced the concept of resolving (locating) set with the cardinality of minimum resolving set which is called the location number of [1]. Harary and Melter introduced the concept of metric dimension for the connected networks [2]. The concept of metric independence number of a graph is introduced by Currie and Oellermann [3]. The metric dimension has been applied to solve the problems involving percolation in hierarchical lattice [4], coin weighting, and robot navigation [5]. It is also applied in subject of chemistry to find the structures of chemical compounds having similar characteristics in functional groups. These functional groups play a vital role in chemical and pharmaceutical industries to predict the various chemical properties of the molecular compounds that are used in the drug discovery [6]. Metric dimension of graph was formulated as integer programming problem by Charterand et al. [6]. Fehar et al. studied the metric dimension of Cayley digraphs [7]. For further studies of metric dimension of convex polytopes, Caylay and Toeplitz networks, see [8–11].
Currie and Oellermann defined the concept of fractional metric dimension (FMD) as an optimal solution of the linear relaxation of the integer programming problem (IPP) [3]. Later, Faher et al. presented the identical calculation of IPP with the help of FMD [7]. Arguman and Matthew introduced many different properties of FMD for connected networks with respect to their order [12]. FMDs of hierarchical product of graphs were computed by Feng and Wang [13]. Liu et al. [14] computed the FMD of generalized Jahangir graph. The concept of local fractional metric dimension (LFMD) is introduced by Aisyah et al. [15]. They also computed it for the connected networks which are obtained by the operation of the corona product. The results for the LFMDs of some cycle-related networks and rotationally symmetric and planar networks can be found in [16, 17]. Javaid et al. (2020) computed the sharp bounds of LFMD of connected networks and illustrated the obtained results with the help of wheel-related networks. They also compared the bounded- and unboundedness of the obtained results [18].
In this paper, lower bound of LFMD for connected networks is improved from unity and expressed in terms of ratio obtained by the cardinalities of the under-study network and the local resolving neighbourhood with maximum order for some edges of network. In the outcome of the obtained result, the LFMDs of prism-related networks as exact values and sharp bounds are computed. The rest of the article is organised as follows: Section 2 consists the preliminaries, Section 3 consists of main results of LFMD of connected networks, Section 4 deals with the local resolving neighbourhoods of prism-related networks, Section 5 presents LFMD of prism-related networks and Section 6 consists of conclusion and comparison among the main results.
2. Preliminaries
Let be a network with and as set of vertices and edges, respectively. A walk is defined as a sequence of alternating vertices and edges. A walk in which the vertices are all distinct is a path between vertices and and a closed path is called a cycle. For any two vertices and of , the distance is the length of shortest path in . A pair of vertices and in a network is a connected pair if there is a path between them and the network is a connected network. For a connected network and , a vertex distinguishes two vertices and if is known as symmetric vertex. Moreover, resolves the edge in if . For and , the -tuple metric form of in terms of is given by . The set becomes resolving set having elements of graph if each pair of vertices in bears a distinct metric form with respect to . The resolving set with least number of vertices is referred as metric basis for and cardinality of such resolving set is called metric dimension of defined by
For an edge , the local resolving neighbourhood is defined as , where . A function is called an upper local resolving function (ULRF) if and for each of , where . On the other hand, a function is called lower local resolving function (LLRF) if and for each of G, where . Then, LFMD is defined as , where is min{ which is the upper local minimal resolving function of } or max{ is the lower local maximal resolving function of }.
For , now we present some prism-related networks. The circular diagonal ladder is obtained from prism network of order and size by adding some double crossing edges and , as shown in Figure 1. The antiprism network of order and size is obtained by prism network by adding some crossing edges , see Figure 2. The sun flower network of order and size , we mean a network, is isomorphic to the network obtained from by deleting edges , see Figure 3 [19].
Theorem 1. (see [18]). Let be a connected network. Let be a local resolving neighbourhood for the edge of . If , thenwhere , , and .
Proposition 1. (see [18]). Let be a connected network. For each , if , then , where , and is a LRN set of .
3. Main Results
Main results of LFMD are as follows.
Proposition 2. Let be a connected network and be the local resolving neighbourhood set of the edge of . For , if , then for each local resolving neighbourhood of .
Theorem 2. Let be a connected network and be the local resolving neighbourhood set. Then,where and .
Proof. Define as for . By Proposition 2, for , we haveThis shows that is a lower local resolving function (LLRF). To show that is maximal, suppose on contrary, there exists another LLRF such that , where , for at least one . such that , we haveThus, . This shows that is not LLRF and consequently is maximal LLRF. Let be another maximal of . Then,Now, we consider three cases (i) for each , (ii) for each , and (iii) for some . Case 1: if , for each . For such that , we have . This shows is not . Thus, this case does not hold. Case 2: let . Then, Consequently, Case 3: assume that for some . Suppose that and . We note that ; otherwise, for , which implies that is not a LLRF. ConsiderSince , therefore,Consequently,Thus, from all the cases,Now, we present the following two corollaries as the direct consequences of the above result.
Corollary 1. Let be a connected network, be LRN of , , , and . If and , then
Proof. Since and , therefore, by Theorem A, . Also, by Theorem 1, . Consequently,
Corollary 2. Let be a connected network, be the LRN of , , , and . If and , then .
Proof. Since , then by Theorem 2, . Also, as , therefore, by Theorem 1, . Consequently, .
Remark 1. Corollary 2 strengthens the result proved in [18].
4. LRNs of the Prism-Related Networks
In this section, the local resolving neighbourhoods of prism-related networks are classified.
Lemma 1. Let CD with be a circular diagonal ladder network, where n and , where We have(a) and ,(b) and ,(c) and ,(d) and ,(e) and ,(f) and .
Proof. Assume that and are inner and outer vertices, respectively, for and . We have the following:(a)Consider with . Moreover, .(b) with and .(c) with and .(d) with and .(e) with and .(f) with and .
Lemma 2. Let with be a circular diagonal ladder network, where n and , where . We have(a) and ,(b) and ,(c) and ,(d) and ,(e) and ,(f) and .
Proof. Assume that and are inner and outer vertices, respectively, for and , . We have the following:(a)Consider with . Moreover, .(b)As with and .(c)As with and .(d)As with and .(e)As with and .(f)As with and .
Lemma 3. Let with be an antiprism network, where and and . We have(a) and and ,(b) and ,(c) and .
Proof. Assume that and are inner and outer vertices, respectively, for , where , , and . We have(a) and with . Moreover, ,(b)As with and ,(c)As with and .
Lemma 4. Let with be an antiprism network, where with . For , we have(a) and and ,(b) and ,(c) and .
Proof. Assume that and are inner and outer vertices, respectively, for , where , , , and . We have(a) and with . Moreover, .(b) = and .(c) = and .
Lemma 5. Let with be a sun flower network, where and . For , we have(a) and and ,(b) and .
Proof. Assume that and are inner and outer vertices, respectively, for , where , , , , and . Now, we have(a) and with . Moreover, .(b) and with and .
Lemma 6. Let with be a sun flower network, where and . For , we have(a) and and ,(b) and .
Proof. Assume that and are inner and outer vertices, respectively, for , where , , , , and . We have(a) and with . Moreover, .(b) and with and .
5. LFMD of Prism-Related Networks
The LFMD of prism-related networks is computed as follows.
Theorem 3. Let with be a circular diagonal ladder network and . Then, .
Proof. Case 1: For LRNs are as follows: , , , , , , , , , , , , , , , , , , , , , , , .As, for , is 2 such that , where . Moreover, and . Define as is a LRF with minimum cardinality for each . Consequently, by Preposition 1, . Case 2: For , LRNs are as follows: , , , , , , , , , , , , , , , , , , , , , , , , , , , , , .As, for , such that , where . Moreover, ; this implies and . Define such that is the LRF with minimum cardinality for each . Consequently, by Proposition 1, . Case 3: For and by Lemma 1, and , where R are all other LRNs. Define such that is the LRF with minimum cardinality and , as . . Consequently, by Proposition 1, .
Theorem 4. Let with be an antiprism network, where and . Then, ,
Proof. Case 1: For , LRNs are as follows: , , , , , , , , .As, for , . Moreover, such that and . Define as is the LRF with minimum cardinality. Consequently, by Corollary 1, . Case 2: For , LRNs are as follows: , , , , , , , , , , , , , , , , , , , .As, for , such that and . ; this implies and . Define as is the LRF with minimum cardinality for each . Consequently, by Theorem 1, .
As, for , , , and . Moreover, . Define as is the LRF with minimum cardinality for each . Therefore, by Theorem 2, . Consequently, Case 3: For and by Lemma 4, and , where R are all other LRNs. There exists an ULRF , which is defined as , as . Consequently, by Theorem 1, . Case 4: For , moreover, . There exists a LLRF such that is defined as for each . Therefore, by Theorem 2, we have . Consequently,
Theorem 5. Let with be an antiprism network, where and . Then,
Proof. Case 1: For , LRNs are as follows: , , , , , , , , , , , , , , , .As, for , such that , where . Moreover, and . There exits an ULRF defined as which is the LRF with minimum cardinality . Consequently, by Theorem 1, .
As, for , such that , where. Moreover, . There exists a LLRF s which is defined as for each . Therefore, by Theorem 1, we have . Consequently, Case 2: For , , by Lemma 3, and , where R are all other LRNs. There exists an ULRF which is defined as , as V(G) . Consequently, by Theorem 1, . Case 3: As, , which is greater or equal to the cardinalities of all other LRNs, moreover, . There exists a LLRF which is defined as . Therefore, by Theorem 2, . Consequently,
Theorem 6. Let with be a sun flower network and . Then,
Proof. Case 1: For , LRNs are as follows: , , , , , , , , .As, for , and . There exists an ULRF which is defined by , . Consequently, by Corollary 1, . Case 2: For , LRNs are as follows: , , , , , , , , , , , .As, for , , such that , where . Moreover, . There exists defined as which is the LRF , as . Consequently, by Theorem 1, .
As, for , such that , where . There exists defined as , which is the LLRF with maximum cardinality, where , as . By Theorem 1, . Consequently, Case 3: As , , , and , where R are all other LRNs. There exists an ULRF defined as , as V(G) . Consequently, by Theorem 1, . Case 4: As which is greater than the cardinalities of all other LRNs of , there exists a LLRF defined by , , as V(G) . By Theorem 2, . Consequently,
6. Conclusion
In this article, we studied the various aspects of the LFMDs for the different connected networks including the existence of the exact values, lower and upper bonds and bounded- and unbounded behaviours. Mainly, the lower bond of LFMD for the arbitrary connected networks is improved from unity. As the applications of the main result, LFMDs of the prism-related networks are as follows:(i)The exact value LFMD of is and of is , where .(ii)The LFMDs of antiprism and sun flower networks are computed in bounds, see Table 1.
Moreover, we note that, for , the computed bounds are convergent which show their boundedness. Now, we close our discussion by proposing the following open problem: open problem. Characterize the connected networks whose LFMDs is , where is some integral value.
Data Availability
The data used to support the findings of this study are included within this article. However, the reader may contact the corresponding author for more details on the data.
Conflicts of Interest
The authors have no conflicts of interest.