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 ( 1 E and 2 E ) was introduced in the chemical graph theory very recently. The eccentric connectivity index ( ) c ξ is a distance-based molecular structure descriptor that was used for mathematical modeling of biological activities of diverse nature. The second geometric-arithmetic index 2 ( ) GA 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 ( ) GG ABC , which yielded promising results when compared to analogous descriptors. In this note we prove that  1( ) ( ) c E T ξ T for chemical trees T. For connected graph G of order n with maximum degree Δ , it is proved that  2 ( ) ( ) c ξ G E G if   Δ 1 n and  2 ( ) ( ) c ξ G E G , otherwise. Moreover, we show that  2 GG GA ABC for paths and some class of bipartite graphs.