Eigenvalue bounds of the Kirchhoff Laplacian

We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a graph 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. The case k=2 is a case for the Schur-Horn inequality l(1)+l(2)+...+l(k) bounded above by d(1)+d(2)+...+d(k). The case k=n is a result of Anderson and Morley. Already the corollary l(k) less or equal to 2 d(k) is in this case stronger than the Gershgorin circle theorem assuring that every disk of radius d(k) centered at d(k) contains the eigenvalue of K. We prove our theorem using the Cauchy interlace theorem.