A Remark on the First Zagreb Index

Abstract

Let $G=(V,E)$ be a simple connected graph with $n\ge 3$ vertices, $m$ edges and vertex degree sequence $d_1\ge d_2\ge \cdots \ge d_n >0$. A graph invariant  referred to as the first Zagreb index is defined as $M_1 =M_1(G) =\sum_{i=1}^n d_i^2$. The upper and lower bounds of $M_1$ are determined.