{"title":"On the extremal cacti with minimum Sombor index","authors":"Qiaozhi Geng, Shengjie He, Rong-Xia Hao","doi":"10.3934/math.20231537","DOIUrl":null,"url":null,"abstract":"<abstract><p>Let $ H $ be a graph with edge set $ E_H $. The Sombor index and the reduced Sombor index of a graph $ H $ are defined as $ SO(H) = \\sum\\limits_{uv\\in E_H}\\sqrt{d_{H}(u)^{2}+d_{H}(v)^{2}} $ and $ SO_{red}(H) = \\sum\\limits_{uv\\in E_H}\\sqrt{(d_{H}(u)-1)^{2}+(d_{H}(v)-1)^{2}} $, respectively. Where $ d_{H}(u) $ and $ d_{H}(v) $ are the degrees of the vertices $ u $ and $ v $ in $ H $, respectively. A cactus is a connected graph in which any two cycles have at most one common vertex. Let $ \\mathcal{C}(n, k) $ be the class of cacti of order $ n $ with $ k $ cycles. In this paper, the lower bound for the Sombor index of the cacti in $ \\mathcal{C}(n, k) $ is obtained and the corresponding extremal cacti are characterized when $ n\\geq 4k-2 $ and $ k\\geq 2 $. Moreover, the lower bound of the reduced Sombor index of cacti is obtained by similar approach.</p></abstract>","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"19 1","pages":"0"},"PeriodicalIF":1.8000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"AIMS Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/math.20231537","RegionNum":3,"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
Let $ H $ be a graph with edge set $ E_H $. The Sombor index and the reduced Sombor index of a graph $ H $ are defined as $ SO(H) = \sum\limits_{uv\in E_H}\sqrt{d_{H}(u)^{2}+d_{H}(v)^{2}} $ and $ SO_{red}(H) = \sum\limits_{uv\in E_H}\sqrt{(d_{H}(u)-1)^{2}+(d_{H}(v)-1)^{2}} $, respectively. Where $ d_{H}(u) $ and $ d_{H}(v) $ are the degrees of the vertices $ u $ and $ v $ in $ H $, respectively. A cactus is a connected graph in which any two cycles have at most one common vertex. Let $ \mathcal{C}(n, k) $ be the class of cacti of order $ n $ with $ k $ cycles. In this paper, the lower bound for the Sombor index of the cacti in $ \mathcal{C}(n, k) $ is obtained and the corresponding extremal cacti are characterized when $ n\geq 4k-2 $ and $ k\geq 2 $. Moreover, the lower bound of the reduced Sombor index of cacti is obtained by similar approach.
期刊介绍:
AIMS Mathematics is an international Open Access journal devoted to publishing peer-reviewed, high quality, original papers in all fields of mathematics. We publish the following article types: original research articles, reviews, editorials, letters, and conference reports.