Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders

Let G be a connected graph and u and v two vertices of G. The hyper-Wiener index of graph G is WW(G)=12∑u,v∈V(G)(dG(u,v)+dG2(u,v))$\begin{array}{} WW(G)=\frac{1}{2}\sum\limits_{u,v\in V(G)}(d_{G}(u,v)+d^{2}_{G}(u,v)) \end{array}$, where dG(u, v) is the distance between u and v. In this paper, we first give the recurrence formulae for computing the hyper-Wiener indices of polyphenyl chains and polyphenyl spiders. We then obtain the sharp upper and lower bounds for the hyper-Wiener index among polyphenyl chains and polyphenyl spiders, respectively. Moreover, the corresponding extremal graphs are determined.