Some Inequalities for the first General Zagreb Index of Graphs and Line Graphs

Abstract

The first general Zagreb index Mα1(G) of a graph G is equal to the sum of the αth powers of the vertex degrees of G. For α≥0 and k≥1, we obtain the lower and upper bounds for Mα1(G) and Mα1(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 Mα1(J(G)) and Mα1(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