Eigenvalue -1 and triangle-free graphs

Determining the maximum order of graphs whose adjacency matrices have an eigenvalue µ with multiplicity k, is a problem which has been studied by several authors. The situation of the problem is quite different for the eigenvalues −1, 0. In this paper, we investigate this problem for triangle-free graphs and for the eigenvalue µ = −1. As the main result of the paper, we prove that the order of graphs with maximum degree d and the eigenvalue −1 with multiplicity k > 1 is at most k + d + 1. We also characterize the graphs attainting the lower bound.