paper

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

arXiv:2402.10557

Abstract

Fix $m \in N$ . 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.