Abstract
Recently a new vertex-degree based molecular structure descriptor was defined as Sombor index. Let \( G \) be a simple graph. The Sombor index of \( G \), denoted by \( SO(G) \), is \( \sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2}, \) where \( d_v \) is the degree
of \( v \). In this paper we investigate the relation between the Sombor index of a graph and the Sombor index of its subgraphs. As a concequence we find some relations between \( SO(G) \) and \( SO(\overline{G}) \), where \( \overline{G}
\) is the complement graph of \( G \). In particular, we show that if \( G \) is a graph of order \( n \), then \( SO(G)+SO({\overline G})\leq\frac{n(n-1)^2}{\sqrt{2}} \) and the equality holds if and only if \( G\cong K_n \) or \( G\cong
\overline{K_n} \).
