2 research outputs found
Nonpositive Eigenvalues of the Adjacency Matrix and Lower Bounds for Laplacian Eigenvalues
Let be the smallest number such that the adjacency matrix of any
undirected graph with vertices or more has at least nonpositive
eigenvalues. We show that is well-defined and prove that the values of
for are respectively. In addition, we
prove that for all , , in which
is the Ramsey number for and , and is the triangular
number. This implies new lower bounds for eigenvalues of Laplacian matrices:
the -th largest eigenvalue is bounded from below by the -th largest
degree, which generalizes some prior results.Comment: 23 pages, 12 figure