Spectra, signless Laplacian and Laplacian spectra of complementary prisms of graphs

The complementary prism GG‾ of a graph G is obtained from the disjoint union of G and its complement G‾ by adding an edge for each pair of vertices (v,v′), where v is in G and its copy v′ is in G‾. The Petersen graph C5C5‾ and, for n≥2, the corona product of Kn and K1 which is KnKn‾ are examples of complementary prisms. This paper is devoted to the computation of eigenpairs of the adjacency, signless Laplacian and Laplacian matrices of a complementary prism GG‾ in terms of the eigenpairs of the corresponding matrices of G. Particular attention is given to the complementary prisms of regular graphs. Furthermore, Petersen graph is shown to be the unique complementary prism which is a strongly regular graph.