Common non-zero graphs of groups with few edges

Let \( G \) be a finite group, and consider its common non-zero graph, which we denote by \( Γ_{nv}(G) \). In this graph, the vertices represent the non-linear irreducible characters of \( G \). There is an edge connecting two distinct vertices \( χ_1 \) and \( χ_2 \) if there is a vanishing element