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 (σ_c) 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 (ABC_GG), which yielded promising results when compared to analogous descriptors. In this note we prove that E₁ (T) σ_c (T) for chemical trees T. For connected graph G of order n with maximum degree Δ , it is proved that 2 σ_c (G) E (G) if Δ= n-1 and σ_c (G) E₂ (G) , otherwise. Moreover, we show that GA₂ ABC_GG for paths and some class of bipartite graphs.