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

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    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 GAGA index is greater than ABCABC index for all those graphs (except K1,4K_{1,4} and T∗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 GAGA index is greater than ABCABC 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 (2δ−1)2(2\delta-1)^{2} (where δ\delta is the minimum degree and δ≥2\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

    Quasi-isometric classification of non-geometric 3-manifold groups

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    We describe the quasi-isometric classification of fundamental groups of irreducible non-geometric 3-manifolds which do not have "too many" arithmetic hyperbolic geometric components, thus completing the quasi-isometric classification of 3--manifold groups in all but a few exceptional cases.Comment: Minor revision (added footnote in the Introduction
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