Spectra of M-rooted product of graphs

In this paper, we define a new graph operation, namely, M-rooted product of graphs which generalizes the existing rooted product of graphs. We obtain its generalized characteristic polynomial, and as a consequence we deduce the characteristic polynomial of its adjacency, Laplacian and signless Laplacian matrices. Using these results, we derive the L-spectrum of several families of M-rooted product of graphs and deduce several existing results on the spectra of the rooted product of graphs in the literature. As applications, we obtain infinitely many L-cospectral, A-cospectral and Q-cospectral graphs, and construct A-integral graphs and Q-integral graphs.