Perron vector of the adjacency matrix of complete bipartite graph $K_{p,q}$. We know that the spectral radius of the adjacency matrix of complete bipartite graph $K_{p,q}$ is $\sqrt{pq}$. Also note that the perron vector corresponding to the spectral radius has positive entries and is of the form $$(a,a,\cdots,a,b,b,\cdots,b)$$ where $a$ is the entry for the partition with vertex set with cardinality $p$ and $b$ for $q$.
Is there any known form how $a$ and $b$ looks like?

I think that $a= \sqrt{q}$, $b= \sqrt{p}$ works:
\begin{align*}
A_{p,q}v_{\sqrt{pq}} &= \phantom{\sqrt{pq}} \left[\begin{array}{cccccc}
  0& \dots & 0 & 1 & \dots& 1 \\
  \vdots& &  & && \vdots \\
  0& \dots & 0 & 1 & \dots& 1 \\
  1& \dots & 1 & 0 & \dots& 0 \\
  \vdots& &  & && \vdots \\
  1& \dots & 1 & 0 & \dots& 0 \\
\end{array}\right]
\begin{bmatrix}
  \sqrt{q} \\
  \vdots \\
  \sqrt{q} \\
  \sqrt{p} \\
  \vdots \\
  \sqrt{p} \\
\end{bmatrix} \\
{} &= \phantom{\sqrt{pq}}\begin{bmatrix}
  q\sqrt{p} \\
  \vdots \\
  q\sqrt{p} \\
 p\sqrt{q} \\
  \vdots \\
  p\sqrt{q} \\
\end{bmatrix} \\
{} &= \sqrt{pq} \begin{bmatrix}
  \sqrt{q} \\
  \vdots \\
  \sqrt{q} \\
  \sqrt{p} \\
  \vdots \\
  \sqrt{p} \\
\end{bmatrix} \\
\end{align*}
