# Eigenvalue -1 and triangle-free graphs

Document Type : Original Paper

Authors

Department of Mathematics, K. N. Toosi University of Technology, P. O. Box 16765-3381, Tehran, Iran

Abstract

Determining the maximum order of graphs whose adjacency matrices have an eigenvalue \$mu\$ 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 \$mu=-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.

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