Graphs with [formula omitted] main eigenvalues

Abstract

An eigenvalue μ of a graph G of order n is a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector e. Characterizing graphs with s(2≤s≤n) number of main eigenvalues is a long-standing problem in algebraic and graph theory. For graphs with n−1 main eigenvalues, we show that the unique non-main eigenvalue and the corresponding eigenvector can be obtained from its walk matrix W without solving the characteristic equation. We construct infinitely many graphs with n−1 main eigenvalues by the union and join, and the corona operations. We also determine the automorphism group of such graphs. We enumerate all connected graphs with n≤8 vertices and n−1 main eigenvalues, up to isomorphism. For the case n=7 we state the spectrum, the complement and the structure of the Galois group of the graph.