The Hyper-Wiener Index of Unicyclic Graph with Given Diameter
Received:April 19, 2019  Revised:April 21, 2020
Key Word: hyper-Wiener 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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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 $WW(G)=\frac{1}{2}\sum_{u,v\in V(G)}(d_G(u,v)+d^2_G(u,v))$ 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.