{"title":"Some Inequalities between General Randić-Type Graph Invariants","authors":"Imran Nadeem, Saba Siddique, Yilun Shang","doi":"10.1155/2024/8204742","DOIUrl":null,"url":null,"abstract":"The Randić-type graph invariants are extensively investigated vertex-degree-based topological indices and have gained much prominence in recent years. The general Randić and zeroth-order general Randić indices are Randić-type graph invariants and are defined for a graph <svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.02496 8.8423\" width=\"9.02496pt\" 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> with vertex set <svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.35121 8.8423\" width=\"9.35121pt\" 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> as <span><svg height=\"17.1973pt\" style=\"vertical-align:-7.24091pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.95639 43.051 17.1973\" width=\"43.051pt\" 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,-0.013,13.9,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,18.398,0)\"><use xlink:href=\"#g113-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,27.29,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,35.42,0)\"></path></g></svg><span></span><svg height=\"17.1973pt\" style=\"vertical-align:-7.24091pt\" version=\"1.1\" viewbox=\"46.6331838 -9.95639 67.033 17.1973\" width=\"67.033pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,46.683,.007)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,56.47,3.466)\"></path></g><g transform=\"matrix(.0065,0,0,-0.0065,60.728,5.567)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,63.099,3.466)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,68.659,3.466)\"><use xlink:href=\"#g50-242\"></use></g><g transform=\"matrix(.0065,0,0,-0.0065,72.918,5.567)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,76.812,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,81.31,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,88.46,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,91.515,0)\"><use xlink:href=\"#g113-101\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,98.665,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,103.091,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,107.589,-5.741)\"><use xlink:href=\"#g50-223\"></use></g></svg></span> and <span><svg height=\"15.6315pt\" style=\"vertical-align:-5.67511pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.95639 44.286 15.6315\" width=\"44.286pt\" 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,-0.0091,9.321,3.132)\"><use xlink:href=\"#g50-223\"></use></g><g transform=\"matrix(.013,0,0,-0.013,15.135,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,19.633,0)\"><use xlink:href=\"#g113-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,28.525,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,36.655,0)\"><use xlink:href=\"#g117-34\"></use></g></svg><span></span><span><svg height=\"15.6315pt\" style=\"vertical-align:-5.67511pt\" version=\"1.1\" viewbox=\"47.8681838 -9.95639 41.872 15.6315\" width=\"41.872pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,47.918,.007)\"><use xlink:href=\"#g119-65\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,57.705,3.466)\"></path></g><g transform=\"matrix(.0065,0,0,-0.0065,61.772,5.567)\"><use xlink:href=\"#g176-106\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,64.143,3.466)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,69.412,3.466)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,76.503,0)\"><use xlink:href=\"#g113-101\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,83.718,-5.741)\"><use xlink:href=\"#g50-223\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,83.653,3.784)\"><use xlink:href=\"#g50-106\"></use></g></svg>,</span></span> respectively, where <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)\"></path></g></svg> is an arbitrary real number, <svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 10.3321 12.5794\" width=\"10.3321pt\" 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=\"#g113-101\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,7.15,3.132)\"><use xlink:href=\"#g50-106\"></use></g></svg> denotes the degree of a vertex <span><svg height=\"9.25202pt\" style=\"vertical-align:-3.29111pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 9.09247 9.25202\" width=\"9.09247pt\" 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,-0.0091,5.915,3.132)\"><use xlink:href=\"#g50-106\"></use></g></svg>,</span> and <span><svg height=\"11.4899pt\" style=\"vertical-align:-5.52899pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 20.233 11.4899\" width=\"20.233pt\" 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=\"#g113-242\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,5.915,3.132)\"><use xlink:href=\"#g50-106\"></use></g><g transform=\"matrix(.013,0,0,-0.013,12.602,0)\"></path></g></svg><span></span><svg height=\"11.4899pt\" style=\"vertical-align:-5.52899pt\" version=\"1.1\" viewbox=\"23.8151838 -5.96091 10.514 11.4899\" width=\"10.514pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,23.865,0)\"><use xlink:href=\"#g113-242\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,29.78,3.132)\"><use xlink:href=\"#g50-107\"></use></g></svg></span> represents the adjacency of vertices <svg height=\"9.25202pt\" style=\"vertical-align:-3.29111pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 9.09247 9.25202\" width=\"9.09247pt\" 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=\"#g113-242\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,5.915,3.132)\"><use xlink:href=\"#g50-106\"></use></g></svg> and <svg height=\"11.4899pt\" style=\"vertical-align:-5.52899pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 10.4626 11.4899\" width=\"10.4626pt\" 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=\"#g113-242\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,5.915,3.132)\"><use xlink:href=\"#g50-107\"></use></g></svg> in <span><svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.02496 8.8423\" width=\"9.02496pt\" 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=\"#g113-72\"></use></g></svg>.</span> Establishing relationships between two topological indices holds significant importance for researchers. Some implicit inequality relationships between <svg height=\"11.927pt\" style=\"vertical-align:-3.291101pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 14.0301 11.927\" width=\"14.0301pt\" 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=\"#g113-83\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,8.086,3.132)\"><use xlink:href=\"#g50-223\"></use></g></svg> and <svg height=\"11.927pt\" style=\"vertical-align:-3.291101pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 15.2698 11.927\" width=\"15.2698pt\" 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=\"#g113-82\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,9.321,3.132)\"><use xlink:href=\"#g50-223\"></use></g></svg> have been derived so far. In this paper, we establish explicit inequality relationships between <svg height=\"11.927pt\" style=\"vertical-align:-3.291101pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 14.0301 11.927\" width=\"14.0301pt\" 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=\"#g113-83\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,8.086,3.132)\"><use xlink:href=\"#g50-223\"></use></g></svg> and <span><svg height=\"11.927pt\" style=\"vertical-align:-3.291101pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 15.2698 11.927\" width=\"15.2698pt\" 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=\"#g113-82\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,9.321,3.132)\"><use xlink:href=\"#g50-223\"></use></g></svg>.</span> Also, we determine linear inequality relationships between these graph invariants. Moreover, we obtain some new inequalities for various vertex-degree-based topological indices by the appropriate choice of <span><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>.</span>","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"10 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-02-20","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/10.1155/2024/8204742","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
The Randić-type graph invariants are extensively investigated vertex-degree-based topological indices and have gained much prominence in recent years. The general Randić and zeroth-order general Randić indices are Randić-type graph invariants and are defined for a graph with vertex set as and , respectively, where is an arbitrary real number, denotes the degree of a vertex , and represents the adjacency of vertices and in . Establishing relationships between two topological indices holds significant importance for researchers. Some implicit inequality relationships between and have been derived so far. In this paper, we establish explicit inequality relationships between and . Also, we determine linear inequality relationships between these graph invariants. Moreover, we obtain some new inequalities for various vertex-degree-based topological indices by the appropriate choice of .
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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