Some inequalities for the first general zagreb index of graphs and line graphs

Abstract

The first general Zagreb index M1 alpha(G) of a graph G is equal to the sum of the alpha th powers of the vertex degrees of G. For alpha >= 0 and k >= 1, we obtain the lower and upper bounds for M1 alpha(G) and M1 alpha(L(G)) in terms of order, size, minimum/maximum vertex degrees and minimal non-pendant vertex degree using some classical inequalities and majorization technique, where L(G) is the line graph of G. Also, we obtain some bounds and exact values of M1 alpha(J(G)) and M1 alpha(Lk(G)), where J(G) is a jump graph (complement of a line graph) and Lk(G) is an iterated line graph of a graph G.