Some properties of Cayley signed graphs on finite Abelian groups - Iranmanesh, Mohammad A.; Moghaddami, Nasrin (Electronic Journal of Graph Theory and Applications (EJGTA), 2025)
This paper establishes explicit combinatorial characterizations for fundamental structural properties of Cayley signed graphs defined on finite Abelian groups. We derive precise necessary and sufficient conditions for balance, clusterability, and sign-compatibility of both these graphs and their line graphs. By leveraging the prime factorization structure of the underlying group G, we prove that the signed graph Σ is balanced precisely when 2 appears among the prime factors of G. Furthermore, we demonstrate that the line graph L(Σ) is balanced if and only if G ≅ ℤ2 × ℤ2α for α ∈ {1, 2}.