On the difference between the revised Szeged index and the Wiener index

Let Sz⋆(G)Sz⋆(G) and W(G)W(G) be the revised Szeged index and the Wiener index of a graph GG. Chen et al. (2014) proved that if GG is a non-bipartite connected graph of order n≥4n≥4, then Sz⋆(G)−W(G)≥(n2+4n−6)/4Sz⋆(G)−W(G)≥(n2+4n−6)/4. Using a matrix method we prove that if GG is a non-bipartite graph of order nn, size mm, and girth gg, then Sz⋆(G)−W(G)≥n(m−3n4)+P(g), where PP is a fixed cubic polynomial. Graphs that attain the equality are also described. If in addition g≥5g≥5, then Sz⋆(G)−W(G)≥n(m−3n4)+(n−g)(g−3)+P(g). These results extend the bound of Chen, Li, and Liu as soon as m≥n+1m≥n+1 or g≥5g≥5. The remaining cases are treated separately.