Comparing Zagreb indices for connected graphs

It was conjectured that for each simple graph G=(V,E)G=(V,E) with n=|V(G)|n=|V(G)| vertices and m=|E(G)|m=|E(G)| edges, it holds M2(G)/m≥M1(G)/nM2(G)/m≥M1(G)/n, where M1M1 and M2M2 are the first and second Zagreb indices. Hansen and Vukičević proved that it is true for all chemical graphs and does not hold in general. Also the conjecture was proved for all trees, unicyclic graphs, and all bicyclic graphs except one class. In this paper, we show that for every positive integer kk, there exists a connected graph such that m−n=km−n=k and the conjecture does not hold. Moreover, by introducing some transformations, we show that M2/(m−1)>M1/nM2/(m−1)>M1/n for all bicyclic graphs and it does not hold for general graphs. Using these transformations we give new and shorter proofs of some known results.