Comparison between Topological Indices of Graphs

A topological representation of a molecule can be carried out through molecular graph. The descriptors are numerical values associated with chemical constitution for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. A topological index is the graph invariant number calculated from a graph representing a molecule. The most of the proposed topological indices are related either to a vertex adjacency relationship (atom-atom connectivity) in the graph G or to topological distances in G. In this talk we compare between popular topological indices in chemical graph theor , for example, Zagreb indices (M1 and M2); Geometric-arithmetic indices (GA1; GA2 and GA3); Atom-bond connectivity indices (ABC1 and ABC2); Zagreb eccentricity indices (E1 and E2); Szeged index (Sz) and Wiener index (W ); Eccentric connectivity index (ξc) and Wiener index, etc. In particular, we compare between different topological indices of graphs. Moreover, we obtain a lower bound on 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.