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

Fix $m \in \mathbb N$. A new generalization of the $H$-join operation of a family of graphs $\{G_1, G_2, \dots, G_k\}$ constrained by indexing maps $I_1,I_2,\dots,I_k$ is introduced as $H_m$-join of graphs, where the maps $I_i:V(G_i)$ to $[m]$. Various spectra, including adjacency, Laplacian, and signless Laplacian spectra, of any graph $G$, which is a $H_m$-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 $E_i$ of the graph $G_i$ such that a $E_i$-non-main eigenvalue of multiplicity $m_i$ of $A(G_i)$ carry forward as an eigenvalue for $A(G)$ with the same multiplicity $m_i$, while an $E_i$-main eigenvalue of multiplicity $m_i$ carry forward as an eigenvalue of $G$ with multiplicity at least $m_i - m$. As a corollary, the universal adjacency spectra of some families of graphs is obtained by realizing them as $H_m$-joins of graphs. As an application, infinite families of cospectral families of graphs are found.