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. It is proved that if G is a connected bipartite graph of order n ≥ 4, size m ≥ n, and if ` is the length of a longest isometric cycle of G, then Sz(G) − W (G) ≥ n(m − n + ` − 2) + (`/2) − ` + 2`. 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, Li, Liu, European J. Combin. 36 (2014) 237–246]. Several lower bounds on the difference Sz(G) −W (G) for general graphs G are also given without any condition on the girth.