  In this article we prove a local large deviation principle (LLDP) for the
critical multitype Galton-Watson process from spectral potential point. We
define the so-called a spectral potential $U_{\skrik}(\,\cdot,\,\pi)$ for the
Galton-Watson process, where $\pi$ is the normalized eigen vector corresponding
to the leading \emph{Perron-Frobenius eigen value } $\1$ of the transition
matrix $\skria(\cdot,\,\cdot)$ defined from ${\skrik},$ the transition kernel.
We show that the Kullback action or the deviation function, $J(\pi,\rho),$ with
respect to an empirical offspring measure, $\rho,$ is the Legendre dual of
$U_{\skrik}(\,\cdot,\,\pi).$ From the LLDP we deduce a conditional large
deviation principle and a weak variant of the classical McMillian Theorem for
the multitype Galton-Watson process. To be specific, given any empirical
offspring measure $\varpi,$ we show that the number of critical multitype
Galton-Watson processes on $n$ vertices is approximately $e^{n\langle
\skrih_{\varpi},\,\pi\rangle},$ where $\skrih_{\varpi}$ is a suitably defined
entropy.
