The structure of graphs with extremal hyper-Wiener index

Abstract

The hyper-Wiener index of a graph $G$ is defined as $WW(G)=\frac{1}{2}\sum\limits_{\{u,v\}\subseteq V(G)}(d_{G}^{2}(u,v)+d_{G}(u,v))$, where $d_{G}(u,v)$ denotes the distance between $u$ and $v$ in $G$.
In this paper, we determine the maximum hyper-Wiener index of 2-connected graphs and 2-edge-connected graphs, which extends the result of Plesnik [On the sum of all distances in a graph or digraph, J. Graph Theory {8} (1984) 1-21].
Then based on the above results, we characterize the first two maximum graphs among the graphs with two vertices of odd degree, the minimum graphs among the graphs with $2k$ ($0\leq k\leq \lfloor \frac{n}{2} \rfloor$) vertices of odd degree, which extends the result of Hou, Chen and Zhang [Hyper-Wiener index of Eulerian graphs, Appl. Math. J. Chin. Univ. {31} (2016) 248-252].