On the eigenvalues of signed complete bipartite graphs

Abstract: Let $\Gamma=(G,\sigma)$ be a signed graph, where $\sigma$ is the sign function on the edges of $G$. The adjacency matrix of $\Gamma=(G, \sigma)$ is a square matrix $A(\Gamma)=A(G, \sigma)=\left(a_{i j}^{\sigma}\right)$, where $a_{i j}^{\sigma}=\sigma\left(v_{i} v_{j}\right) a_{i j}$. In this paper, we determine the eigenvalues of the signed complete bipartite graphs. Let $(K_{p, q},\sigma)$, $p\leq q$, be a signed complete bipartite graph with bipartition $(U_p, V_q)$, where $U_p={u_1,u_2,\ldots,u_p}$ and $V_q={v_1,v_2,\ldots,v_q}$. Let $(K_{p, q},\sigma)[U_r\cup V_s]$, $r\leq p$ and $s\leq q $, be an induced signed subgraph on minimum vertices $r+s$, which contains all negative edges of the signed graph $(K_{p, q},\sigma)$. We show that the multiplicity of eigenvalue $0$ in $(K_{p, q},\sigma)$ is at least $ p+q-2k-2$, where $k=min(r,s)$. We determine the spectrum of signed complete bipartite graph whose negative edges induce disjoint complete bipartite subgraphs and path. We obtain the spectrum of signed complete bipartite graph whose negative edges (positive edges) induce an $r-$ regular subgraph $H$. We find a relation between the eigenvalues of this signed complete bipartite graph and the non-negative eigenvalues of $H$.