Computation of the eigenvalues of complete signed graphs

Pirzada, Shariefuddin (University of Kashmir)
Ul Rashid, Mir Riyaz (University of Kashmir)
Rehman, Amir
Baskoro, Edy Tri (Institut Teknologi Bandung)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 14, No 1 (2026): Electronic Journal of Graph Theory and Applications

A signed graph Σ is the ordered pair (G,σ), where G=(V,E) is a finite simple graph, called the underlying graph, and σ: E(G) → {+1, -1} is a sign function or a signature of Σ. Let (K_n,σ) be a complete signed graph with n vertices. In this paper, we give a complete description of the adjacency, Laplacian and net Laplacian spectrum of a complete signed graph (K_n,σ) whenever its negative edges induce either a complete tripartite graph or a friendship graph. This is an addition to the class of complete signed graphs whose spectra is completely known.