  The $H$-join of a family of graphs $\mathcal{G}=\{G_1, \dots, G_p\}$, also
called the generalized composition, $H[G_1, \dots, G_p]$, where all graphs are
undirected, simple and finite, is the graph obtained by replacing each vertex
$i$ of $H$ by $G_i$ and adding to the edges of all graphs in $\mathcal{G}$ the
edges of the join $G_i \vee G_j$, for every edge $ij$ of $H$. Some well known
graph operations are particular cases of the $H$-join of a family of graphs
$\mathcal{G}$ as it is the case of the lexicographic product (also called
composition) of two graphs $H$ and $G$, $H[G]$. During long time the known
expressions for the determination of the entire spectrum of the $H$-join in
terms of the spectra of its components and an associated matrix were limited to
families of regular graphs. In this work, we extend such a determination, as
well as the determination of the characteristic polynomial, to families of
arbitrary graphs. From the obtained results, the eigenvectors of the adjacency
matrix of the $H$-join can also be determined in terms of the adjacency
matrices of the components and an associated matrix.
