{"title":"The minimum exponential atom-bond connectivity energy of trees","authors":"Wei Gao","doi":"10.1515/spma-2023-0108","DOIUrl":null,"url":null,"abstract":"Abstract Let G = ( V ( G ) , E ( G ) ) G=\\left(V\\left(G),E\\left(G)) be a graph of order n n . The exponential atom-bond connectivity matrix A e ABC ( G ) {A}_{{e}^{{\\rm{ABC}}}}\\left(G) of G G is an n × n n\\times n matrix whose ( i , j ) \\left(i,j) -entry is equal to e d ( v i ) + d ( v j ) − 2 d ( v i ) d ( v j ) {e}^{\\sqrt{\\tfrac{d\\left({v}_{i})+d\\left({v}_{j})-2}{d\\left({v}_{i})d\\left({v}_{j})}}} if v i v j ∈ E ( G ) {v}_{i}{v}_{j}\\in E\\left(G) , and 0 otherwise. The exponential atom-bond connectivity energy of G G is the sum of the absolute values of all eigenvalues of the matrix A e ABC ( G ) {A}_{{e}^{{\\rm{ABC}}}}\\left(G) . It is proved that among all trees of order n n , the star S n {S}_{n} is the unique tree with the minimum exponential atom-bond connectivity energy.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"31 7","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Special Matrices","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/spma-2023-0108","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract Let G = ( V ( G ) , E ( G ) ) G=\left(V\left(G),E\left(G)) be a graph of order n n . The exponential atom-bond connectivity matrix A e ABC ( G ) {A}_{{e}^{{\rm{ABC}}}}\left(G) of G G is an n × n n\times n matrix whose ( i , j ) \left(i,j) -entry is equal to e d ( v i ) + d ( v j ) − 2 d ( v i ) d ( v j ) {e}^{\sqrt{\tfrac{d\left({v}_{i})+d\left({v}_{j})-2}{d\left({v}_{i})d\left({v}_{j})}}} if v i v j ∈ E ( G ) {v}_{i}{v}_{j}\in E\left(G) , and 0 otherwise. The exponential atom-bond connectivity energy of G G is the sum of the absolute values of all eigenvalues of the matrix A e ABC ( G ) {A}_{{e}^{{\rm{ABC}}}}\left(G) . It is proved that among all trees of order n n , the star S n {S}_{n} is the unique tree with the minimum exponential atom-bond connectivity energy.
摘要 让 G = ( V ( G ) , E ( G ) )G=left(V\left(G),E\left(G)) 是一个阶数为 n n 的图。G G 的指数原子键连通性矩阵 A e ABC ( G ) {A}_{{e}^{{rm\{ABC}}}}\left(G) 是一个 n × n 次 n 矩阵，其 ( i , j ) \left(i. i. j) - 条目等于 n × n 次 n 矩阵、j) -项等于 e d ( v i ) + d ( v j ) - 2 d ( v i ) d ( v j ) {e}^{sqrt{tfrac{d\left({v}_{i})+d\left({v}_{j})-2}{d\left({v}_{i})d\left({v}_{j})}}} 如果 v i v j ∈ E ( G ) {v}_{i}{v}_{j}\in E\left(G) 、否则为 0。G G 的指数原子键连通能是矩阵 A e ABC ( G ) {A}_{{e}^{rm{ABC}}}}\left(G) 的所有特征值的绝对值之和。实验证明，在所有阶数为 n n 的树中，星 S n {S}_{n} 是唯一具有最小指数原子键连接能的树。
期刊介绍：
Special Matrices publishes original articles of wide significance and originality in all areas of research involving structured matrices present in various branches of pure and applied mathematics and their noteworthy applications in physics, engineering, and other sciences. Special Matrices provides a hub for all researchers working across structured matrices to present their discoveries, and to be a forum for the discussion of the important issues in this vibrant area of matrix theory. Special Matrices brings together in one place major contributions to structured matrices and their applications. All the manuscripts are considered by originality, scientific importance and interest to a general mathematical audience. The journal also provides secure archiving by De Gruyter and the independent archiving service Portico.
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