On the geometric-arithmetic index by decompositions-CMMSE

The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. There are many papers studying different kinds of indices (as Wiener, hyper-Wiener, detour, hyper-detour, Szeged, edge-Szeged, PI, vertex-PI and eccentric connectivity indices) under particular cases of decompositions. The main aim of this paper is to show that the computation of the geometric-arithmetic index of a graph G is essentially reduced to the computation of the geometric-arithmetic indices of the so-called primary subgraphs obtained by a general decomposition of G. Furthermore, using these results, we obtain formulas for the geometric-arithmetic indices of bridge graphs and other classes of graphs, like bouquet of graphs and circle graphs. These results are applied to the computation of the geometric-arithmetic index of Spiro chain of hexagons, polyphenylenes and polyethene.