On the difference between the Szeged index and the Wiener index of cacti

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A cactus is a connected graph in which any two cycles have at most one common vertex. Let Cn be the set of all n-vertex cacti with circumference at least 4. In Klavžar et al. (2018), the authors proved that the minimum and the second minimum values on the difference between the Szeged index and the Wiener index of graphs among Cn are 2n−5 and 4n−10, respectively. In this paper, we give a counterexample to show that the second result is not true and we determine the second minimum value on the difference between the Szeged index and the Wiener index of graphs among Cn.