Spectra of generalized corona of graphs constrained by vertex subsets

Abstract

In this paper, we introduce a generalization of corona of graphs. This construction generalizes the generalized corona of graphs (consequently, the corona of graphs), the cluster of graphs, the corona-vertex subdivision graph of graphs and the corona-edge subdivision graph of graphs. Further, it enables to get some more variants of corona of graphs as its particular cases. To determine the spectra of the adjacency, Laplacian and the signless Laplacian matrices of the above mentioned graphs, we define a notion namely, the coronal of a matrix constrained by an index set, which generalizes the coronal of a graph matrix. Then we prove several results pertain to the determination of this value. Then we determine the characteristic polynomials of the adjacency and the Laplacian matrices of this graph in terms of the characteristic polynomials of the adjacency and the Laplacian matrices of the constituent graphs and the coronal of some matrices related to the constituent graphs. Using these, we derive the characteristic polynomials of the adjacency and the Laplacian matrices of the above mentioned existing variants of corona of graphs, and some more variants of corona of graphs with some special constraints.