{"title":"Distance-Based Fractional Dimension of Certain Wheel Networks","authors":"Hassan Zafar, Muhammad Javaid, Mamo Abebe Ashebo","doi":"10.1155/2024/8870335","DOIUrl":null,"url":null,"abstract":"Metric dimension is one of the distance-based parameters which are used to find the position of the robot in a network space by utilizing lesser number of notes and minimum consumption of time. It is also used to characterize the chemical compounds. The metric dimension has a wide range of applications in the field of computer science such as integer programming, radar tracking, pattern recognition, robot navigation, and image processing. A vertex <svg height=\"6.1673pt\" style=\"vertical-align:-0.2063904pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.39387 6.1673\" width=\"7.39387pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg> in a network <svg height=\"8.73791pt\" style=\"vertical-align:-0.04981995pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.68809 12.4829 8.73791\" width=\"12.4829pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg> resolves the adjacent pair of vertices <svg height=\"6.1934pt\" style=\"vertical-align:-0.2324901pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 13.0048 6.1934\" width=\"13.0048pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,6.994,0)\"></path></g></svg> if <svg height=\"6.1673pt\" style=\"vertical-align:-0.2063904pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.39387 6.1673\" width=\"7.39387pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-121\"></use></g></svg> attains an unequal distance from end points of <span><svg height=\"6.1934pt\" style=\"vertical-align:-0.2324901pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 13.0048 6.1934\" width=\"13.0048pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-118\"></use></g><g transform=\"matrix(.013,0,0,-0.013,6.994,0)\"><use xlink:href=\"#g185-40\"></use></g></svg>.</span> A local resolving neighbourhood set <svg height=\"12.4698pt\" style=\"vertical-align:-3.18147pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 35.7732 12.4698\" width=\"35.7732pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,8.086,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,13.708,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,18.206,0)\"><use xlink:href=\"#g113-118\"></use></g><g transform=\"matrix(.013,0,0,-0.013,25.201,0)\"><use xlink:href=\"#g185-40\"></use></g><g transform=\"matrix(.013,0,0,-0.013,31.064,0)\"></path></g></svg> is a set of vertices of <svg height=\"8.73791pt\" style=\"vertical-align:-0.04981995pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.68809 12.4829 8.73791\" width=\"12.4829pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g197-34\"></use></g></svg> which resolve <span><svg height=\"6.1934pt\" style=\"vertical-align:-0.2324901pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 13.0048 6.1934\" width=\"13.0048pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-118\"></use></g><g transform=\"matrix(.013,0,0,-0.013,6.994,0)\"><use xlink:href=\"#g185-40\"></use></g></svg>.</span> A mapping <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 13.981 11.5564\" width=\"13.981pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,11.017,0)\"></path></g></svg><span></span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"17.5631838 -9.28833 52.368 11.5564\" width=\"52.368pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,17.613,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,26.837,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,31.335,0)\"><use xlink:href=\"#g197-34\"></use></g><g transform=\"matrix(.013,0,0,-0.013,43.674,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,51.803,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,57.579,0)\"></path></g></svg><span></span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"73.6401838 -9.28833 13.689 11.5564\" width=\"13.689pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,73.69,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,78.175,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,84.415,0)\"></path></g></svg><span></span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"89.50818380000001 -9.28833 11.06 11.5564\" width=\"11.06pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,89.558,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,95.798,0)\"></path></g></svg></span> is called local resolving function of <svg height=\"8.73791pt\" style=\"vertical-align:-0.04981995pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.68809 12.4829 8.73791\" width=\"12.4829pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g197-34\"></use></g></svg> if <span><svg height=\"12.4698pt\" style=\"vertical-align:-3.18147pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 63.206 12.4698\" width=\"63.206pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-223\"></use></g><g transform=\"matrix(.013,0,0,-0.013,7.385,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,11.883,0)\"><use xlink:href=\"#g113-83\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,19.969,3.132)\"><use xlink:href=\"#g50-77\"></use></g><g transform=\"matrix(.013,0,0,-0.013,25.592,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,30.09,0)\"><use xlink:href=\"#g113-118\"></use></g><g transform=\"matrix(.013,0,0,-0.013,37.084,0)\"><use xlink:href=\"#g185-40\"></use></g><g transform=\"matrix(.013,0,0,-0.013,42.947,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,47.445,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,55.575,0)\"></path></g></svg><span></span><svg height=\"12.4698pt\" style=\"vertical-align:-3.18147pt\" version=\"1.1\" viewbox=\"66.7881838 -9.28833 6.563 12.4698\" width=\"6.563pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,66.838,0)\"><use xlink:href=\"#g113-50\"></use></g></svg></span> for any adjacent pair of vertices of <svg height=\"6.1934pt\" style=\"vertical-align:-0.2324901pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 13.0048 6.1934\" width=\"13.0048pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-118\"></use></g><g transform=\"matrix(.013,0,0,-0.013,6.994,0)\"><use xlink:href=\"#g185-40\"></use></g></svg> of <svg height=\"8.73791pt\" style=\"vertical-align:-0.04981995pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.68809 12.4829 8.73791\" width=\"12.4829pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g197-34\"></use></g></svg> and the minimal value of <svg height=\"12.4698pt\" style=\"vertical-align:-3.18147pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 52.2146 12.4698\" width=\"52.2146pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-223\"></use></g><g transform=\"matrix(.013,0,0,-0.013,7.385,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,11.883,0)\"><use xlink:href=\"#g113-83\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,19.969,3.132)\"><use xlink:href=\"#g50-77\"></use></g><g transform=\"matrix(.013,0,0,-0.013,25.592,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,30.09,0)\"><use xlink:href=\"#g113-118\"></use></g><g transform=\"matrix(.013,0,0,-0.013,37.084,0)\"><use xlink:href=\"#g185-40\"></use></g><g transform=\"matrix(.013,0,0,-0.013,42.947,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,47.445,0)\"><use xlink:href=\"#g113-42\"></use></g></svg> for all local resolving functions <svg height=\"6.1673pt\" style=\"vertical-align:-0.2063904pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.51131 6.1673\" width=\"7.51131pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-223\"></use></g></svg> of <svg height=\"8.73791pt\" style=\"vertical-align:-0.04981995pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.68809 12.4829 8.73791\" width=\"12.4829pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g197-34\"></use></g></svg> is called local fractional metric dimension of <span><svg height=\"8.73791pt\" style=\"vertical-align:-0.04981995pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.68809 12.4829 8.73791\" width=\"12.4829pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g197-34\"></use></g></svg>.</span> In this paper, we have studied the local fractional metric dimension of wheel-related networks such as web-wheel network, subdivision of wheel network, line network of subdivision of wheel network, and double-wheel network and also examined their boundedness.","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"171 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/8870335","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Metric dimension is one of the distance-based parameters which are used to find the position of the robot in a network space by utilizing lesser number of notes and minimum consumption of time. It is also used to characterize the chemical compounds. The metric dimension has a wide range of applications in the field of computer science such as integer programming, radar tracking, pattern recognition, robot navigation, and image processing. A vertex in a network resolves the adjacent pair of vertices if attains an unequal distance from end points of . A local resolving neighbourhood set is a set of vertices of which resolve . A mapping is called local resolving function of if for any adjacent pair of vertices of of and the minimal value of for all local resolving functions of is called local fractional metric dimension of . In this paper, we have studied the local fractional metric dimension of wheel-related networks such as web-wheel network, subdivision of wheel network, line network of subdivision of wheel network, and double-wheel network and also examined their boundedness.
期刊介绍：
Journal of Mathematics is a broad scope journal that publishes original research articles as well as review articles on all aspects of both pure and applied mathematics. As well as original research, Journal of Mathematics also publishes focused review articles that assess the state of the art, and identify upcoming challenges and promising solutions for the community.
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