  Given a graph $G=(V,E)$, an integer $k$, and a function $f_G:V^k \times V^k
\to {0,1}$, the $k^{th}$ graph product of $G$ w.r.t $f_G$ is the graph with
vertex set $V^k$, and an edge between two vertices $x=(x_1,...,x_k)$ and
$y=(y_1,...,y_k)$ iff $f_G(x,y)=1$. Graph products are a basic combinatorial
object, widely studied and used in different areas such as hardness of
approximation, information theory, etc. We study graph products for functions
$f_G$ of the form $f_G(x,y)=1$ iff there are at least $t$ indices $i \in [k]$
s.t. $(x_i,y_i)\in E$, where $t \in [k]$ is a fixed parameter in $f_G$. This
framework generalizes the well-known graph tensor-product (obtained for $t=k$)
and the graph or-product (obtained for $t=1$). The property that interests us
is the edge distribution in such graphs. We show that if $G$ has a spectral
gap, then the number of edges connecting "large-enough" sets in $G^k$ is
"well-behaved", namely, it is close to the expected value, had the sets been
random. We extend our results to bi-partite graph products as well. For a
bi-partite graph $G=(X,Y,E)$, the $k^{th}$ bi-partite graph product of $G$
w.r.t $f_G$ is the bi-partite graph with vertex sets $X^k$ and $Y^k$ and edges
between $x \in X^k$ and $y \in Y^k$ iff $f_G(x,y)=1$. Finally, for both types
of graph products, optimality is asserted using the "Converse to the Expander
Mixing Lemma" obtained by Bilu and Linial in 2006. A byproduct of our proof
technique is a new explicit construction of a family of co-spectral graphs.
