The Bounds for the First General Zagreb Index of a Graph

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