Abstract
The Wiener index of a connected graph is the sum of distances between all pairs of vertices. Let \(\mathbb{G}(n,2k)\) be the set of all graphs on \(n\) vertices with exactly \(2k\) vertices of odd degree. \(\mathbb{G}(n,0)\) is just the set of all Eulerian
graphs on \(n\) vertices. In Gutman et al. (2014) and Dankelmann (2021), the authors characterized the graphs \(\mathbb{G}(n,0)\) with the first four minimum Wiener index and the first two maximum Wiener index. In the paper, we characterize
the graph in \(\mathbb{G}(n,2k)\) with the minimum Wiener index for all \(0\leq k\leq\frac{n}{2}\), and the graph in \(\mathbb{G}(n,2)\) with the first-maximum and second-maximum Wiener index.
