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

Let S z ⋆ ( G ) and W ( G ) be the revised Szeged index and the Wiener index of a graph G . Chen et al. (2014) proved that if G is a non-bipartite connected graph of order n ≥ 4 , then S z ⋆ ( G ) − W ( G ) ≥ ( n 2 + 4 n − 6 ) / 4 . Using a matrix method we prove that if G is a non-bipartite graph of order n , size m , and girth g , then S z ⋆ ( G ) − W ( G ) ≥ n ( m − 3 n 4 ) + P ( g ) , where P is a fixed cubic polynomial. Graphs that attain the equality are also described. If in addition g ≥ 5 , then S z ⋆ ( G ) − W ( G ) ≥ n ( m − 3 n 4 ) + ( n − g ) ( g − 3 ) + P ( g ) . These results extend the bound of Chen, Li, and Liu as soon as m ≥ n + 1 or g ≥ 5 . The remaining cases are treated separately.