The H-join of arbitrary families of graphs - Cardoso, Domingos M. (2021)
The H-join of a family of graphs G = {G1, . . . , Gp}, also called the generalized composition, H[G1, . . . , Gp], where all graphs are undirected, simple and finite, is the graph obtained from the graph H replacing each vertex i of H by Gi and adding to the edges of all graphs in G the edges of the join Gi ∨ Gj , for every edge ij of H. Some well known graph operations are particular cases of the H-join of a family of graphs G as it is the case of the lexicographic product (also called composition) of two graphs H and G, H[G], which coincides with the H-join of family of graphs G where all the graphs in G are isomorphic to a fixed graph G. So far, 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 are limited to families of regular graphs. In this paper, we extend such a determination to families of arbitrary graphs.