The Bounds for the First General Zagreb Index of a Graph

Abstract

The first general Zagreb index of a graph $G$ is defined as the sum of the $\alpha$th powers of the vertex degrees of $G$, where $\alpha$ is a real number such that $\alpha \neq 0$ and $\alpha \neq 1$. In this note, for $\alpha > 1$, we present upper bounds involving chromatic and clique numbers for the first general Zagreb index of a graph; for an integer $\alpha \geq 2$, we present a lower bound involving the independence number for the first general Zagreb index of a graph.