Star complements and exceptional graphs

Let G be a finite graph of order n with an eigenvalue mu of multiplicity k. (Thus the mu-eigenspace of a (0, 1)-adjacency matrix of G has dimension k.) A star complement for mu in G is an induced subgraph G - X of G such that vertical bar X vertical bar = k and G - X does not have it as an eigenvalue. An exceptional graph is a connected graph, other than a generalized line graph, whose eigenvalues lie in [-2, infinity). We establish some properties of star complements, and of eigenvectors, of exceptional graphs with least eigenvalue -2. (C) 2007 Elsevier Inc. All rights reserved.