  We study the vanishing ideal of the parametrized algebraic toric associated
to the complete multipartite graph $\G=\mathcal{K}_{\alpha_1,...,\alpha_r}$
over a finite field of order $q$. We give an explicit family of binomial
generators for this lattice ideal, consisting of the generators of the ideal of
the torus, (referred to as type I generators), a set of quadratic binomials
corresponding to the cycles of length 4 in $\G$ and which generate the
\emph{toric algebra of $\G$} (type II generators) and a set of binomials of
degree $q-1$ obtained combinatorially from $\G$ (type III generators). Using
this explicit family of generators of the ideal, we show that its
Castelnuovo--Mumford regularity is equal to
$\max\set{\alpha_1(q-2),...,\alpha_r(q-2), \lceil (n-1)(q-2)/2\rceil}$, where
$n=\alpha_1+... + \alpha_r$.
