Some extremal properties of the multiplicatively weighted Harary index of a graph

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• 作者：Shuchao Li ; Huihui Zhang
• 关键词：Multiplicatively weighted Harary index ; Pendants ; Diameter ; Matching number ; Domination number ; Bipartition
• 刊名：Journal of Combinatorial Optimization
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：31
• 期：3
• 页码：961-978
• 全文大小：588 KB
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• 作者单位：Shuchao Li (1)
  Huihui Zhang (1)
  
  1. Faculty of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, People’s Republic of China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Convex and Discrete Geometry
  Mathematical Modeling and IndustrialMathematics
  Theory of Computation
  Optimization
  Operation Research and Decision Theory
  
• 出版者：Springer Netherlands
• ISSN：1573-2886

文摘

Let \(G=(V_G, E_G)\) be a simple connected graph. The multiplicatively weighted Harary index of \(G\) is defined as \(H_M(G)=\sum _{\{u,v\}\subseteq V_G}\delta _G(u)\delta _G(v)\frac{1}{d_G(u,v)},\) where \(\delta _G(u)\) is the vertex degree of \(u\) and \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G.\) This novel invariant is in fact the modification of the Harary index in which the contributions of vertex pairs are weighted by the product of their degrees. Deng et al. (J Comb Optim 2014, doi:10.​1007/​s10878-013-9698-5) determined the extremal values on \(H_M\) of graphs among \(n\)-vertex trees (resp. unicyclic graphs). In this paper, as a continuance of it, the monotonicity of \(H_M(G)\) under some graph transformations were studied. Using these nice mathematical properties, the extremal graphs among \(n\)-vertex trees with given graphic parameters, such as pendants, matching number, domination number, diameter, vertex bipartition, et al. are characterized, respectively. Some sharp upper bounds on the multiplicatively weighted Harary index of trees with given parameters are determined.