Spectra of (M,ℳ)-corona-join of graphs

Abstract

In this paper, we introduce the (M, ℳ)-corona-join of G and ℋk constrained by vertex subsets 𝒯, which is the union of two graphs: one is the M-generalized corona of a graph G and a family of graphs ℋk constrained by vertex subset 𝒯 of the graphs in ℋk, where M is a suitable matrix; and the other one is the ℳ -join of ℋk, where ℳ is a collection of matrices. We determine the spectra of the adjacency, the Laplacian, the signless Laplacian and the normalized Laplacian matrices of some special cases of the (M, ℳ)-corona-join of G and ℋk constrained by vertex subsets 𝒯. These results enable us to deduce the spectra of all the existing variants of extended corona of graphs. Further, by using this graph operation, we construct infinitely many graphs which are simultaneously cospectral with respect to the above mentioned four type of matrices.