The first general Zagreb index of graphs and their line graphs

Let alpha be an arbitrary real number. The first general Zagreb index M-alpha(G) of a graph G is equal to the sum of the alpha th powers of the degrees of the vertices of G. Let G be a real number and G, a graph of order n. In this paper, we show that (1) if alpha >= 1 and G is connected that is neither a path nor a star, then M-alpha(G) <= M-alpha(L(G)); (2) if 0< alpha <1 and delta(G)>= 2, then M-alpha(G) <= M-alpha(L(G)) with equality if and only if G congruent to C-n; (3) if alpha <=-1 and G is a connected graph of size alpha <= -1, then M-alpha(L(G)) <= M-alpha(G) with equality if and only if G congruent to C-n.