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

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