Comparing Zagreb Indices of Cyclic Graphs

Given a graph G = (V, E), the first Zagreb index M-1 is the sum of its vertices squared degrees and the second Zagreb index M-2 is the stun of its edges products of degrees. Recently, there has been much interest. in comparing M-1 and M-2 (and generalizations of them). The case of trees was handled in [19]. In this paper, we consider the case of cyclic graphs and provide two best possible lower hounds on M-2-M-1 in terms of order and cyclicity of G. On the basis of some experiments with the system AutoGraphiX, it was conjectured that M-1/n <= M-2/m. This was disproved in [6] both for disconnected and for connected graphs. However, it is true for chemical graphs. We show here that it still holds for unicyclic graphs but is not true in general for graphs with a. larger number of independent cycles.