More on Comparison Between First Geometric-Arithmetic Index and Atom-Bond Connectivity Index

The first geometric-arithmetic (GA) index and atom-bond connectivity (ABC) index are molecular structure descriptors which play a significant role in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies. Das and Trinajsti\'{c} [\textit{Chem. Phys. Lett.} \textbf{497} (2010) 149-151] showed that $GA$ index is greater than $ABC$ index for all those graphs (except $K_{1,4}$ and $T^{*}$, see Figure 1) in which the difference between maximum and minimum degree is less than or equal to 3. In this note, it is proved that $GA$ index is greater than $ABC$ index for line graphs of molecular graphs, for general graphs in which the difference between maximum and minimum degree is less than or equal to $(2\delta-1)^{2}$ (where $\delta$ is the minimum degree and $\delta\geq2$) and for some families of trees. Thereby, a partial solution to an open problem proposed by Das and Trinajsti\'{c} is given.Comment: 10 pages, 2 tables, 1 figure, revised versio

On the Difference of Atom-Bond Sum-Connectivity and Atom-Bond-Connectivity Indices

The atom-bond-connectivity (ABC) index is one of the well-investigated degree-based topological indices. The atom-bond sum-connectivity (ABS) index is a modified version of the ABC index, which was introduced recently. The primary goal of the present paper is to investigate the difference between the aforementioned two indices, namely $ABS-ABC$. It is shown that the difference $ABS-ABC$ is positive for all graphs of minimum degree at least $2$ as well as for all line graphs of those graphs of order at least $5$ that are different from the path and cycle graphs. By means of computer search, the difference $ABS-ABC$ is also calculated for all trees of order at most $15$.Comment: 16 pages and 5 figure

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 was introduced in the chemical graph theory very recently. The eccentric connectivity index is a distance-based molecular structure descriptor that was used for mathematical modeling of biological activities of diverse nature. The second geometric-arithmetic index 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, which yielded promising results when compared to analogous descriptors. In this note we prove that for chemical trees T. For connected graph G of order n with maximum degree, it is proved that if and, otherwise. Moreover, we show that for paths and some class of bipartite graphs. This work is licensed under a Creative Commons Attribution 4.0 International License