Star complements and exceptional graphs

Abstract

Let G be a finite graph of order n with an eigenvalue μ of multiplicity k. (Thus the μ-eigenspace of a (0, 1)-adjacency matrix of G has dimension k.) A star complement for μ in G is an induced subgraph G - X of G such that | X | = k and G - X does not have μ as an eigenvalue. An exceptional graph is a connected graph, other than a generalized line graph, whose eigenvalues lie in [- 2, ∞). We establish some properties of star complements, and of eigenvectors, of exceptional graphs with least eigenvalue -2.