Some properties of Cayley signed graphs on finite Abelian groups

Mohammad A. Iranmanesh, Nasrin Moghaddami

Abstract


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}.


Keywords


Cayley signed graphs; graph balance; clusterability; sign-compatibility; line graphs; inite Abelian groups

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DOI: http://dx.doi.org/10.5614/ejgta.2025.13.2.13

References

M. Acharya and D. Sinha, A characterization of sigraphs whose line sigraphs and jump sigraphs are switching equivalent, Graph Theory Notes N. Y. 44 (2003), 30–34.

M. Behzad and G.T. Chartrand, Line coloring of signed graphs, Elem. Math. 24 (1969), 49–52.

N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, 1993.

J.A. Davis, Clustering and structural balance in graphs, Human Relations 20 (1967), 181–187.

C. Godsil and G. Royle, Algebraic Graph Theory, Springer-Verlag, New York, 2001.

F. Harary, On the notion of balance of a signed graph, Michigan Math. J. 2 (1953), 143–146.

M.A. Iranmanesh and N. Moghaddami, Domination parameters and diameter of Abelian Cayley graphs, Facta Univ. Ser. Math. Inform. 36 (2021), no. 4, 695–715.

M.A. Iranmanesh and N. Moghaddami, Domination number of Cayley graphs on finite Abelian groups, Iran. J. Sci. Technol. Trans. A Sci. 43 (2019), no. 5, 2523–2530.

D. Sinha and A. Dhama, Sign-compatibility of some derived signed graphs, Indian J. Math. 55 (2013), no. 1, 95–107.

D. Sinha and P. Garg, On the unitary Cayley signed graphs, Electron. J. Combin. 18 (2011), no. 1, Paper 133, 12 pp.

T. Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1982), no. 1, 47–74.

T. Zaslavsky, A mathematical bibliography of signed and gain graphs and allied areas, Electron. J. Combin. (1998), Dynamic Surveys, DS8.

T. Zaslavsky, Matrices in the theory of signed simple graphs, Advances in Discrete Mathematics and Applications, 207–229, Ramanujan Math. Soc. Lect. Notes Ser., 15, Mysore, 2008.


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