On a conjecture about the Szeged index

Khalifeh, Yousefi-Azari, Ashrafi and Wagner [M.K. 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] conjectured that for a connected graph G on n vertices and m edges with Szeged index S z , S z = m n 2 / 4 if and only if G is a regular bipartite graph. In this note, we disprove this conjecture and then prove a stronger result from which it follows that the equality holds if and only if G is a transmission-regular bipartite graph.