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运筹学学报（中英文） ›› 2025, Vol. 29 ›› Issue (1): 216-224.doi: 10.15960/j.cnki.issn.1007-6093.2025.01.018

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积图的Steiner k-hyper Wiener指标

王朝平1, 刘蒙蒙1,*()   

  1. 1. 兰州交通大学数理学院, 甘肃兰州 730070
  • 收稿日期:2021-12-06 出版日期:2025-03-15 发布日期:2025-03-08
  • 通讯作者: 刘蒙蒙 E-mail:liumm05@163.com
  • 基金资助:
    甘肃高等学校创新能力提升项目(2019A-37)

Steiner k-hyper Wiener index of graph products

Chaoping WANG1, Mengmeng LIU1,*()   

  1. 1. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China
  • Received:2021-12-06 Online:2025-03-15 Published:2025-03-08
  • Contact: Mengmeng LIU E-mail:liumm05@163.com

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摘要：

令图G是一个连通图。当$ 2\leqslant k\leqslant n-1$时, 图G的Steiner k-hyper Wiener指标定义为$ {\rm SWW}_{k}(G)=\frac{1}{2}\sum_{S\subseteq V (G), |S|=k}d_{G}(S)+\frac{1}{2}\sum_{S\subseteq V (G), |S|=k}d_{G}(S)^{2}$, 其中$ d_{G}(S)$表示图G中S的Steiner距离, 即连通图G中包含点集S的最小连通子图的边数。本文中我们确定了连图和字典积图的Steiner k-hyper Wiener指标的表达式, 给出了笛卡尔积图, 聚类图和冠状图的Steiner k-hyper Wiener指标的下限。

关键词: 积图, Steiner k-hyper Wiener指标, Steiner距离

Abstract:

Let G be a connected graph. For $ 2\leqslant k\leqslant n-1$, the Steiner k-hyper Wiener index $ {\rm SWW}_{k}(G)$ is defined as ${\rm SWW}_{k}(G)=\frac{1}{2}\sum_{S\subseteq V(G), |S|=k}d_{G}(S)+\frac{1}{2}\sum_{S\subseteq V(G), |S|=k}d_{G}(S)^{2} $, where $d_{G}(S) $ is the Steiner distance of S, means the minimum size of a connected subgraph which vertex set contains S. In this paper, we establish expressions for the Steiner k-hyper Wiener index on the join and lexicographical product of graphs and give lower bounds for the Steiner $k$-hyper Wiener index on cartesian, cluster and corona product of graphs.

Key words: product graph, Steinerk-hyper Wiener index, Steiner distance

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