In the original article ``Wiener index of Eulerian graphs'' [Discrete Applied Mathematics Volume 162, 10 January 2014, Pages 247-250], the authors state that the Wiener index (total distance) of an Eulerian graph is maximized by the cycle. We explain that the initial proof contains a flaw and note that it is a corollary of a result by Plesník, since an Eulerian graph is $2$-edge-connected. The same incorrect proof is used in two referencing papers, ``Zagreb Indices and Multiplicative Zagreb Indices of Eulerian Graphs'' [Bull. Malays. Math. Sci. Soc. (2019) 42:67-78] and ``Harary index of Eulerian graphs'' [J. Math. Chem., 59(5):1378-1394, 2021]. We give proofs of the main results of those papers and the $2$-edge-connected analogues.