The Ḣ-join operation of signed graphs constrained by indexing maps

Let H be a graph of order k and let F = {G_1, G_2, ..., G_k} be a family of k graphs. Then the H-join of the family F is obtained by replacing each vertex v_i of H with the graph G_i of F and respecting the adjacencies existing in H. To generalise this graph operation we consider a signed variant with the addition of fixing m in N and introducing indexing maps to define the H_m-join. Once having done so we can determine the characteristic polynomials and spectra of the compound graphs produced as pertaining to many graph matrices, such as the adjacency, Laplacian, universal adjacency, net Laplacian, and A_alpha. Furthermore we show that the H_m-join remains stable under switching of H.