The HyperWiener Index of Unicyclic Graph with Given Diameter 
Received:April 19, 2019 Revised:April 21, 2020 
Key Word:
hyperWiener index unicyclic graph diameter

Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11871077), the Natural Science Foundation of Anhui Province (Grant No.1808085MA04) and the Natural Science Foundation of Department of Education of Anhui Province (Grant No.KJ2017A362). 
Author Name  Affiliation  Gaixiang CAI  School of Mathematics and Physics, Anqing Normal University, Anhui 246133, P. R. China  Guidong YU  School of Mathematics and Physics, Anqing Normal University, Anhui 246133, P. R. China Hefei Preschool Education College, Anhui 230013, P. R. China  Peilin MEI  School of Mathematics and Physics, Anqing Normal University, Anhui 246133, P. R. China 

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Abstract: 
The hyperWiener index is a kind of extension of the Wiener index, used for predicting physicochemical properties of organic compounds. The hyperWiener index $WW(G)$ is defined as $WW(G)=\frac{1}{2}\sum_{u,v\in V(G)}(d_G(u,v)+d^2_G(u,v))$ 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 hyperWiener indices among all the unicyclic graph with $n$ vertices and diameter $d$, and characterize the corresponding extremal graphs. 
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