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 = {u 1 , u 2 , . . . , u p } and V q = {v 1 , v 2 , . . . , 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.