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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, the signless Laplacian and the 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.

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An eigenvalue of the adjacency matrix of a graph is said to be main if the all-1 vector is non-orthogonal to the associated eigenspace. This paper explores some new aspects of the study of main eigenvalues of graphs, investigating specifically cones over strongly regular graphs and graphs for which the least eigenvalue is non-main. In this case, we characterize paths and trees with diameter-3 satisfying the property. We may note that the importance of least eigenvalues of graphs for the equilibria of social and economic networks was recently uncovered in literature.