  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\leq \frac{n+3}{2}$.

Combinatorics