Wiener index of unicycle graphs with given number of even degree vertices

The Wiener index of a connected graph is the sum of the distance of all pairs of distinct vertices. It was introduced by Wiener in 1947 to analyze some aspects of branching by fitting experimental data for several properties of alkane compounds. Denote by ${\mathcal{U}}_{n,r}$ the set of unicyclic graphs with $n$ vertices and $r$ vertices of even degree. In this paper we present a structural result on the graphs in ${\mathcal{U}}_{n,r}$ with minimum Wiener index and completely characterize such graphs when $r\le \frac{n+3}{2}$.