Some results on graph spectra Original Research Article

This paper presents a variety of results on graph spectra. The number of main eigenvalues of a graph is shown to be equal to the rank of an associated matrix. We establish a condition for a graph to have exactly two main eigenvalues and then show how to evaluate them and their associated eigenvectors. It is shown that the main eigenvalues and corresponding eigenvectors of a graph determine those of its complement. We generalize to any eigenvalue a condition for 0 and −1 to be eigenvalues of a graph and its complement, respectively. Finally, we generalize to non-simple eigenvalues a result on the components of an eigenvector associated with a simple eigenvalue.