  Considering a graph $H$ of order $p$, a generalized $H$-join operation of a
family of graphs $G_1,..., G_p$, constrained by a family of vertex subsets $S_i
\subseteq V(G_i)$, $i=1,..., p,$ is introduced. When each vertex subset $S_i$
is $(k_i,\tau_i)$-regular, it is deduced that all non-main adjacency
eigenvalues of $G_i$, different from $k_i-\tau_i$, for $i=1,..., p,$ remain as
eigenvalues of the graph $G$ obtained by the above mentioned operation.
Furthermore, if each graph $G_i$ of the family is $k_i$-regular, for $i=1,...,
p$, and all the vertex subsets are such that $S_i=V(G_i)$, the $H$-generalized
join operation constrained by these vertex subsets coincides with the
$H$-generalized join operation. Some applications on the spread of graphs are
presented. Namely, new lower and upper bounds are deduced and a infinity family
of non regular graphs of order $n$ with spread equals $n$ is introduced.
