The H-join of arbitrary families of graphs: the universal adjacency spectrum

Abstract

The H-join of a family of graphs $\mathcal{G}$={G_1, \dots, G_p}, also called 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 (that is, graphs in $\mathcal{G}$) and an associated matrix, related with the main eigenvalues of the components and the graph H, were limited to families $\mathcal{G}$ of regular graphs. In this work, with an approach based on the walk-matrix, we extend such a determination, as well as the determination of the characteristic polynomial, to the universal adjacency matrix of the H-join of families of arbitrary graphs. From the obtained results, the eigenvectors of the universal adjacency matrix of the H-join can also be determined in terms of the eigenvectors of the universal adjacency matrices of the components and an associated matrix.