Ordering Trees Having Small Reverse Wiener Indices

Rundan Xing, Bo Zhou

The Wiener index $W(G)$ of a connected graph $G$ is defined as the sum of distances between all unordered pairs of vertices of $G$. As a variation of the Wiener index, the reverse Wiener index of $G$ is defined as $\Lambda(G)=\frac{1}{2}n(n-1)d-W(G)$, where $n$ is the number of vertices, and $d$ is the diameter of $G$. It is known that the star is the unique $n$-vertex tree with the smallest reverse Wiener index. We now determine the second and the third smallest reverse Wiener indices of n-vertex trees, and characterize the trees whose reverse Wiener indices attain these values for $n\geq 5$.