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.
Highlights
For a connected network G, Salter introduced the concept of resolving set with the cardinality of minimum resolving set which is called the location number of G [1]
Harary and Melter introduced the concept of metric dimension for the connected networks [2]. e concept of metric independence number mi(G) of a graph G is introduced by Currie and Oellermann [3]. e metric dimension has been applied to solve the problems involving percolation in hierarchical lattice [4], coin weighting, and robot navigation [5]
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
Summary
For a connected network G, Salter introduced the concept of resolving (locating) set with the cardinality of minimum resolving set which is called the location number of G [1]. 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]. 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. 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. In the outcome of the obtained result, the LFMDs of prism-related networks as exact values and sharp bounds are computed. e 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
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