Eigenvalues of a H-generalized join graph operation constrained by vertex subsets

A generalized H-join operation of a family of graphs G1,…,Gp, where H has order p, constrained by a family of vertex subsets Si⊆V(Gi), i=1,…,p, is introduced. When each vertex subset Si is (ki,τi)-regular, it is deduced that all non-main adjacency eigenvalues of Gi, different from ki-τi, remain as eigenvalues of the graph G obtained by this operation. If each Gi is ki-regular and all the vertex subsets are such that Si=V(Gi), the H-generalized join constrained by these vertex subsets coincides with the H-join operation. Furthermore, some applications on the spread of graphs are presented.