Proof of a conjecture on the Wiener index of Eulerian graphs

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 et al. (2014) 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≥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≤25 with exception of six values. In this paper we prove the conjecture.