Comparison between the Szeged index and the eccentric connectivity index

Let Sz(G)Sz(G) and ξc(G)ξc(G) be the Szeged index and the eccentric connectivity index of a graph GG, respectively. In this paper we obtain a lower bound on Sz(T)−ξc(T)Sz(T)−ξc(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 Sz(G)−ξc(G)≥−4Sz(G)−ξc(G)≥−4 for bipartite graphs and this result is not true in the general case. Finally, we classify the bipartite graphs GG in which Sz(G)−ξc(G)∈{−4,−3,−2,−1,0,1,2}.