On the spectrum of generalized H-join operation constrained by indexing maps -- I

Fix m ∈ N. A new generalization of the H-join operation of a family of graphs ₁, G₂, , Gₖ\ constrained by indexing maps I₁,I₂,,Iₖ is introduced as Hₘ-join of graphs, where the maps Iᵢ:V(Gᵢ) to $[m]$. Various spectra, including adjacency, Laplacian, and signless Laplacian spectra, of any graph G, which is a Hₘ-join of graphs is obtained by introducing the concept of E-main eigenvalues. More precisely, we deduce that in the case of adjacency spectra, there is an associated matrix Eᵢ of the graph Gᵢ such that a Eᵢ-non-main eigenvalue of multiplicity mᵢ of A(Gᵢ) carry forward as an eigenvalue for $A(G)$ with the same multiplicity mᵢ, while an Eᵢ-main eigenvalue of multiplicity mᵢ carry forward as an eigenvalue of G with multiplicity at least mᵢ - m. As a corollary, the universal adjacency spectra of some families of graphs is obtained by realizing them as Hₘ-joins of graphs. As an application, infinite families of cospectral families of graphs are found.