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

Abstract

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.