On the adjacency and net Laplacian spectra of some complete signed graphs

Source abstract

Given a signed graph Σ = (G, σ), let A(Σ) and N(Σ) denote its adjacency and net Laplacian matrices, respectively. This paper provides a complete characterization of the adjacency and net Laplacian spectra of complete signed graphs (Kn, σ), where the negative edges (resp. positive edges) induce either a simple negative (resp. positive) corona signed subgraph K− p ◦−K1 (resp. Kp ◦+K1) or a negative (resp. positive) Mycielskian signed subgraph μ−(K− p ) (resp. μ+(Kp)), for 1 ≤ p ≤ ⌊ n 2 ⌋. Additionally, we identify specific values of p for which the adjacency spectrum of (Kn, σ) is symmetric and contains exactly 3, 4, or 5 distinct eigenvalues for the adjacency and net Laplacian matrices. These results contribute to the spectral understanding of signed graphs and highlight intriguing structural properties tied to their spectra.