Some Properties of the Zagreb Indices

Abstract

Let $G= (V,E)$, $V=\{1,2,\ldots , n\}$, $E=\{e_1,e_2,\ldots , e_m\}$, be a simple graph with $n$ vertices and $m$ edges. Denote by $d_1\ge d_2 \ge \cdots \ge d_n>0$, and $d(e_1) \ge d(e_2) \ge \cdots \ge d(e_m)$, sequences of vertex and edge degrees, respectively. Graph invariants referred to as the first and the first reformulated Zagreb indices are defied as $M_1 =\sum_{i=1}^n d_i^2 $ and $EM_1 =\sum_{i=1}^m d(e_i)^2$, respectively. Lower bounds for invariants $M_1$ and $EM_1$ are obtained.