  The Wiener index of a connected graph is the sum of the distances between all
unordered pairs of vertices. A connected graph is Eulerian if its vertex
degrees are all even. In [Gutman, Cruz, Rada, Wiener index of Eulerian Graphs,
Discrete Applied Mathematics 132 (2014), 247-250] the authors proved that the
cycle is the unique graph maximising the Wiener index among all Eulerian graphs
of given order. They also conjectured that for Eulerian graphs of order $n \geq
26$ the graph consisting of a cycle on $n-2$ vertices and a triangle that share
a vertex is the unique Eulerian graph with second largest Wiener index. The
conjecture is known to hold for all $n\leq 25$ with exception of six values. In
this paper we prove the conjecture.
