Let $p, q > 0$ be integers and let $K_{p,q}$ be a complete bipartite graph. Let $A(K)$ denote the adjacency matrix of $K_{p,q}$ according to a convenient labeling of vertices , which is the $2 \times 2$ block matrix
$$\begin{bmatrix} 0_{p \times p} & 1_{p \times q}\\ 1_{q \times p} & 0_{q \times q} \end{bmatrix}$$
I know that the spectral radius of $A(K)$ is $\sqrt{pq}$ but couldn't prove it. I tried to find the determinant of the characteristic polynomial in order to find the spectrum but also couldn't. Any help would be appreciated.
The spectral radius of $A(K)$ is the square root of the spectral radius of $A(K)^2$. Note that $$ A(K)^2 = \begin{pmatrix} q\mathbf{1}_{p \times p} & 0\\ 0 & p\mathbf{1}_{q \times q} \end{pmatrix}. $$ Hence $$ \det(xI - A(K)^2) = \det(xI - q\mathbf{1}_{p \times p})\det(xI - p\mathbf{1}_{q \times q}) = (x - pq)^2x^{p + q - 2}. $$