Comparing the Zagreb indices for connected bicyclic graphs

The first Zagreb index M-1(G) and the second Zagreb index M-2(G) of a (molecule) graph G are defined as M-1(G) = Sigma(u is an element of V(G)) (d(u))(2) and M-2(G) = Sigma(u nu is an element of E(G)) d(u)d(v), where d(u) denotes the degree of a vertex u in G. The AutoGraphiX system [1] [41 [5] conjectured M-1/n <= M-2/m (where n = vertical bar V(G)vertical bar and m = vertical bar E(G)vertical bar) for simple connected graphs. Hansen and Vukicevic [11] proved it is true for chemical graphs and it does not hold for all general graphs. Vukicevic and Graovac [22] proved that it is also true for trees. Liu [15] proved that it is true for unicyclic graphs. In this paper, we show that M-1/n <= M-2/m, holds for connected bicyclic graphs except one class and characterize the extremal graph. Additionally, we construct the counterexamples of connected bicyclic graphs from the the class we exclude.