The first general Zagreb index of graphs and their line graphs

Let α be an arbitrary real number. The first general Zagreb index M_α (G) of a graph G is equal to the sum of the α th powers of the degrees of the vertices of G. Let α be a real number and G, a graph of order n. In this paper, we show that (1) if α≥ 1 and G is connected that is neither a path nor a star, then M_α (G)≤ M_α (L(G)) ; (2) if 0<α <1 and δ (G)≥ 2 , then M_α (G)≤ M_α (L(G)) with equality if and only if G≅ C_n ; (3) if α≤ -1 and G is a connected graph of size m≤ n , then M_α (L(G))≤ M_α (G) with equality if and only if G≅ C_n .