  We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a graph or
quiver is bounded above by d(k)+d(k-1) for all k in {1,...,n}. Here
l(1),...,l(n) is a non-decreasing list of the eigenvalues of K and d(1),..,d(n)
is a non-decreasing list of vertex degrees with the additional assumption
d(0)=0. We also prove that in general the weak Brouwer-Haemers lower bound d(k)
+ (n-k) holds for all eigenvalues l(k) of the Kirchhoff matrix of a quiver.
