On the eigenvalues of complete bipartite signed graphs

Abstract

Let Γ = (G, σ) be a signed graph, where σ is the sign function on the edges of G. The adjacency matrix of Γ is defined canonically. Let (K_p, q, σ), p ≤ q, be a complete bipartite signed graph with bipartition (U_p, V_q), where U_p = {u₁, u₂, …, u_p} and V_q = {v₁, v₂, …, v_q}. Let (K_p, q, σ)[U_r ∪ V_s], r ≤ p and s ≤ q, be an induced signed subgraph on minimum vertices r + s, which contains all negative edges of the signed graph (K_p, q, σ). In this paper, we show that the nullity of the signed graph (K_p, q, σ) is at least p + q − 2k − 2, where k = min (r, s). The spectrum of a complete bipartite signed graph whose negative edges induce either a disjoint complete bipartite subgraphs or a path is determined. Finally, we obtain the spectrum of a complete bipartite signed graph whose negative edges (positive edges) induce a regular subgraph H. It turns out that there is a relationship between the eigenvalues of this complete bipartite signed graph and the non-negative eigenvalues of H.