Shahid Chamran University of AhvazJournal of Advanced Mathematical Modeling2251-808812120220321Eigenvalue -1 and triangle-free graphsEigenvalue -1 and triangle-free graphs81891740610.22055/jamm.2022.38047.1947FAHosseinEsmailianDepartment of Mathematics, K. N. Toosi University of Technology, P. O. Box 16765-3381, Tehran, IranEbrahimGhorbaniDepartment of Mathematics, K. N. Toosi University of Technology, P. O. Box 16765-3381, Tehran, IranJournal Article19700101Determining 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.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.https://jamm.scu.ac.ir/article_17406_6552881e2aa03d13f489f41a48b95e8a.pdf