Abstract
The Sombor index of a simple graph \(G\) is defined as \(SO(G)=\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2}\), where \(d_u\) is the degree of the vertex \(u\). In this paper we study the mean value of the Sombor index of graphs. Let \({\cal{F}}_n\) be the set of all labeled graphs on \(n\) vertices \(v_1,\ldots,v_n\). We obtain some explicit formulas for \(\sum_{{G\in{\cal{F}}_n}}SO(G)\). As a consequence we find that for large enough \(n\),
\(\sum_{{G\in{\cal{F}}_n}}SO(G)\simeq(\sqrt{2})^{n^2}\).
