The Main Eigenvalues of a Graph: a Survey

Let G be a simple graph with vertex set V (G) = {1, 2, . . . , n} and (0, 1)adjacency matrix A. The eigenvalue μ of A is said to be a main eigenvalue of G if the eigenspace E(μ) is not orthogonal to the all-1 vector j. An eigenvector x is a main eigenvector if xj 6= 0. The main eigenvalues of the connected graphs of order ≤ 5 are listed in [12, Appendix B], and those of all the connected graphs on 6 vertices are given in [10]. In this section we introduce notation and survey the basic results concerning main eigenvalues and main angles (as defined below). In Section 2, we provide a general context for the investigation of the main eigenvectors of G and its complement G. We also extend the notion of star partition to a refined star partition that takes account of main eigenvalues. In Section 3, we discuss graphs with just two main eigenvalues in the context of measures of irregularity of a graph, and we note the connection with harmonic graphs. In Section 4, we deal with a simple instance of graphs with just three main eigenvalues. Let A have spectral decomposition