Comparison Between Zagreb Eccentricity Indices and the Eccentric Connectivity Index, the Second Geometric-arithmetic Index and the Graovac-Ghorbani Index.

The concept of Zagreb eccentricity indices (E1 and E2) was introduced in the chemical graph theory very recently. The eccentric connectivity index (ξc) is a distance-based molecular structure descriptor that was used for mathematical modeling of biological activities of diverse nature. The second geometric-arithmetic index (GA2) was introduced in 2010, is found to be useful tool in QSPR and QSAR studies. In 2010 Graovac and Ghorbani introduced a distance-based analog of the atom-bond connectivity index, the Graovac-Ghorbani index (ABCGG), which yielded promising results when compared to analogous descriptors. In this note we prove that E1(T)>ξc(T) for chemical trees T. For connected graph G of order n with maximum degree Δ , it is proved that ξc(G)>E2(G) if Δ=n-1 and ξc(G)>E2(G) otherwise. Moreover, we show that GA2>ABCGGfor paths and some class of bipartite graphs. [ABSTRACT FROM AUTHOR]