%0 Journal Article
%T Eigenvalue -1 and triangle-free graphs
%J Journal of Advanced Mathematical Modeling
%I Shahid Chamran University of Ahvaz
%Z 2251-8088
%A Esmailian, Hossein
%A Ghorbani, Ebrahim
%D 2022
%\ 03/21/2022
%V 12
%N 1
%P 81-89
%! Eigenvalue -1 and triangle-free graphs
%K Triangle-free graphs
%K Graph eigenvalue
%K Star complement technique
%R 10.22055/jamm.2022.38047.1947
%X 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.
%U https://jamm.scu.ac.ir/article_17406_6552881e2aa03d13f489f41a48b95e8a.pdf