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 ${\mathrm{\Gamma }}_{nv}(G)$. In this graph, the vertices represent the non-linear irreducible characters of $G$. There is an edge connecting two distinct vertices ${\chi }_1$ and ${\chi }_2$ if there is a vanishing element $g\in G$ such that ${\chi }_1(g){\chi }_2(g)\mathrm{\ne }0$. A finite group $G$ is described as a strongly ${\mathcal{H}}_1^{\mathrm{\prime }}$-group if the set of vanishing elements of each of its non-linear irreducible character is equal to the set of vanishing elements of $G$. In this paper, we prove Conjecture 1 from \cite{ourself}, which states that if ${\mathrm{\Gamma }}_{nv}(G)$ is null (has no edge), then $G$ must either be a strongly ${\mathcal{H}}_1^{\mathrm{\prime }}$-group or a Frobenius group, whose Frobenius complement is isomorphic to $Q_8$. Additionally, we show that if there are no triangles in ${\mathrm{\Gamma }}_{nv}(G)$, this implies that the group is solvable. We also look at the situation where the common non-zero graph of a finite group $G$ contains only one edge. This leads us to conclude that $G$ is either isomorphic to $S_4$; or that the single edge in ${\mathrm{\Gamma }}_{nv}(G)$ is of the form $\{\theta ,\zeta \theta \}$ for some $\zeta \in \text{Irr}(G\mathrm{/}{G}^{\mathrm{\prime }})$ and $\theta \in \text{Irr}(G\mathrm{\mid }{G}^{\mathrm{\prime }})$, $G$ is a $p$-nilpotent group for some prime $p$ and for all characters $\chi \in \text{Irr}(G\mathrm{\mid }{G}^{\mathrm{\prime }})-\{\theta ,\zeta \theta \}$, we find $\text{Van}(\chi )=\text{Van}(G)$. Lastly, our study also explores groups whose common non-zero graph forms a star shape, concluding that such configurations can only occur if $G$ has exactly one non-linear irreducible character.