## Spectra and optimal partitions of weighted graphs.(English)Zbl 0796.05066

Authors’ abstract: The notion of the Laplacian of weighted graphs will be introduced, the eigenvectors belonging to $$k$$ consecutive eigenvalues which define an 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 $$k$$-th and $$(k+1)$$-th positive eigenvalues in increasing order.

### MSC:

 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)

### Keywords:

Laplacian; weighted graphs; eigenvectors; eigenvalues; spectrum
Full Text:

### References:

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