  Figueroa-Centeno et al. introduced the following product of digraphs: let $D$
be a digraph and let $\Gamma$ be a family of digraphs such that $V(F)=V$ for
every $F\in \Gamma$. Consider any function $h:E(D)\longrightarrow\Gamma $. Then
the product $D\otimes_{h} \Gamma$ is the digraph with vertex set $V(D)\times V$
and $((a,x),(b,y))\in E(D\otimes_h\Gamma)$ if and only if $(a,b)\in E(D)$ and
$(x,y)\in E(h (a,b))$.
  In this paper, we introduce the undirected version of the
$\otimes_h$-product, which is a generalization of the classical direct product
of graphs and, motivated by it, we also recover a generalization of the
classical lexicographic product of graphs that was introduced by Sabidussi en
1961. We study connectivity properties and other invariants in terms of the
factors. We also present a new intersection graph that emerges when we
characterize the connectivity of $\otimes_h$-product of graphs.
