Improved bounds on the difference between the Szeged index and the Wiener index of graphs

Let W(G) and Sz(G) be the Wiener index and the Szeged index of a connected graph G, respectively. It is proved that if G is a connected bipartite graph of order n≥4, size m≥n, and if ℓ is the length of a longest isometric cycle of G, then Sz(G)−W(G)≥n(m−n+ℓ−2)+(ℓ/2)3−ℓ2+2ℓ. It is also proved if G is a connected graph of order n≥5 and girth g≥5, then Sz(G)−W(G)≥PIv(G)−n(n−1)+(n−g)(g−3)+P(g), where PIv(G) is the vertex PI index of G and P is a cubic polynomial. These theorems extend related results from Chen et al. (2014). Several lower bounds on the difference Sz(G)−W(G) for general graphs G are also given without any condition on the girth.