The Hyper-Wiener Index of Unicyclic graph with Given Diameter
Received:April 19, 2019  Revised:April 14, 2020
Key Word: Hyper-Wiener index   Unicyclic graph   Diameter  
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Gai-Xiang Cai Anqing Normal University caigaixiang@sina.com 
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Abstract:
      The hyper-Wiener index is a kind of extension of the Wiener index, used for predicting physicochemical properties of organic compounds. The hyper-Wiener index $WW(G)$ is defined as $WW(G)=\frac{1}{2}\sum\limits_{u,v\in V(G)}(d_G(u,v)+d^2_G(u,v))$ with the summation going over all pairs of vertices in $G$, $d_G(u,v)$ denotes the distance of the two vertices $u$ and $v$ in the graph $G$. In this paper, we study the minimum hyper-Wiener indices among all the unicyclic graph with $n$ vertices and diameter $d$, and characterize the corresponding extremal graphs.
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