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

Let W ( G ) and S z ( 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 S z ( G ) − W ( G ) ≥ n ( m − n + ℓ − 2 ) + ( ℓ / 2 ) 3 − ℓ 2 + 2 ℓ . It is also proved if G is a connected graph of order n ≥ 5 and girth g ≥ 5 , then S z ( G ) − W ( G ) ≥ P I v ( G ) − n ( n − 1 ) + ( n − g ) ( g − 3 ) + P ( g ) , where P I v ( 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 S z ( G ) − W ( G ) for general graphs G are also given without any condition on the girth.