Common non-zero graphs of groups with few edges

Zeinab Akhlaghi; Sesuai Yash Madanha

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arXiv · 2609.39290

Common non-zero graphs of groups with few edges

Zeinab Akhlaghi·Sesuai Yash Madanha

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Abstract

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 $ g \in G $ such that $ χ_1(g) χ_2(g) \neq 0 $. A finite group $ G $ is described as a strongly $ \mathcal{H}_1' $-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 $ Γ_{nv}(G) $ is null (has no edge), then $ G $ must either be a strongly $ \mathcal{H}_1' $-group or a Frobenius group, whose Frobenius complement is isomorphic to $ Q_8 $. Additionally, we show that if there are no triangles in $ Γ_{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 $ Γ_{nv}(G) $ is of the form $ \{θ, ζθ\} $ for some $ ζ\in \Irr(G/G') $ and $ θ\in \Irr(G|G') $, $G$ is a $p$-nilpotent group for some prime $p$ and for all characters $ χ\in \Irr(G|G') - \{θ, ζθ\} $, we find $ \Van(χ) = \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.

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Zeinab Akhlaghi, Sesuai Yash Madanha. 2026-10-02. Common non-zero graphs of groups with few edges. https://arxiv.org/abs/2609.39290

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