Spectra and optimal partitions of weighted graphs

The notion of the Laplacian of weighted graphs will be introduced, the eigenvectors belonging to k consecutive eigen-values of which define optimal k-dimensional Euclidean representation of the vertices. By means of these spectral techniques some combinatorial problems concerning minimal (k+ 1)-cuts of weighted graphs can be handled easily with linear algebraic tools. (Here k is an arbitrary integer between 1 and the number of vertices.) The (k+ 1)-variance of the optimal k-dimensional representatives is estimated from above by the k smallest positive eigenvalues and by the gap in the spectrum between the kth and (k+ 1)th positive eigenvalues in increasing order.