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

Abstract

The first general Zagreb index $M^{\alpha }_{1}(G)$ of a graph G is equal to the sum of the $\alpha$th powers of the vertex degrees of G. For $\alpha \ge 0$ and $k \ge 1$, we obtain the lower and upper bounds for $M^{\alpha }_{1}(G)$ and $M^{\alpha }_{1}(L(G))$ 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 $M^{\alpha }_{1}(J(G))$ and $M^{\alpha }_{1}(L^{k}(G))$, where J(G) is a jump graph (complement of a line graph) and $L^{k}(G)$ is an iterated line graph of a graph G.