Comparison between the Szeged index and the eccentric connectivity index.

Let S z ( G ) and ξ c ( G ) be the Szeged index and the eccentric connectivity index of a graph G , respectively. In this paper we obtain a lower bound on S z ( T ) − ξ c ( T ) by double counting on some matrix and characterize the extremal graphs. From this result we compare the Szeged index and the eccentricity connectivity index of trees. For bipartite graphs we also compare the Szeged index and the eccentricity connectivity index. Moreover, we show that S z ( G ) − ξ c ( G ) ≥ − 4 for bipartite graphs and this result is not true in the general case. Finally, we classify the bipartite graphs G in which S z ( G ) − ξ c ( G ) ∈ { − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 } . [ABSTRACT FROM AUTHOR]