Common non-zero graphs of groups with few edges

@misc{akhlaghi2026common,
  title = {{Common non-zero graphs of groups with few edges}},
  author = {Zeinab Akhlaghi and Sesuai Yash Madanha},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.39290},
  eprint = {2609.39290},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39290},
}

TY  - UNPB
TI  - Common non-zero graphs of groups with few edges
AU  - Zeinab Akhlaghi
AU  - Sesuai Yash Madanha
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.39290
UR  - https://arxiv.org/abs/2609.39290
AB  - 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 ∈ G such that χ_1(g) χ_2(g) ≠ 0. A finite group G is described as a strongly 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 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∈ Irr(G/G') and θ∈ Irr(G|G'), G is a p-nilpotent group for some prime p and for all characters χ∈ 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.
ER  -