On distance-balanced graphs

It is shown that the graphs for which the Szeged index equals ‖ G ‖ ⋅ ∣ G ∣ 2 4 are precisely connected, bipartite, distance-balanced graphs. This enables us to disprove a conjecture proposed in [M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, S.G. Wagner, Some new results on distance-based graph invariants, European J. Combin. 30 (2009) 1149–1163]. Infinite families of counterexamples are based on the Handa graph, the Folkman graph, and the Cartesian product of graph. Infinite families of distance-balanced, non-regular graphs that are prime with respect to the Cartesian product are also constructed.