Some Upper Bounds on the First General Zagreb Index

Abstract

The first general Zagreb index $M^{\gamma }\left(G\right)$ or zeroth-order general Randić index of a graph G is defined as $M^{\gamma }\left(G\right)={{\displaystyle \sum }}_{v\in V}{d\left(v\right)}^{\gamma }$ where $\gamma$ is any nonzero real number, $d\left(v\right)$ is the degree of the vertex $v$ and $\gamma =2$ gives the classical first Zagreb index. The researchers investigated some sharp upper and lower bounds on zeroth-order general Randić index (for $\gamma <0$ ) in terms of connectivity, minimum degree, and independent number. In this paper, we put sharp upper bounds on the first general Zagreb index in terms of independent number, minimum degree, and connectivity for $\gamma$ . Furthermore, extremal graphs are also investigated which attained the upper bounds.