  In the first part of this paper, we survey results that are associated with
three types of Laplacian matrices:difference, normalized, and signless. We
derive eigenvalue and eigenvector formulaes for paths and cycles using
circulant matrices and present an alternative proof for finding eigenvalues of
the adjacency matrix of paths and cycles using Chebyshev polynomials. Even
though each results is separately well known, we unite them, and provide
uniform proofs in a simple manner. The main objective of this study is to solve
the problem of finding graphs, on which spectral clustering methods and
normalized cuts produce different partitions. First, we derive a formula for a
minimum normalized cut for graph classes such as paths, cycles, complete
graphs, double-trees, cycle cross paths, and some complex graphs like lollipop
graph $LP_{n,m}$, roach type graph $R_{n,k}$, and weighted path $P_{n,k}$.
Next, we provide characteristic polynomials of the normalized Laplacian
matrices ${\mathcal L}(P_{n,k})$ and ${\mathcal L}(R_{n,k})$. Then, we present
counter example graphs based on $R_{n,k}$, on which spectral methods and
normalized cuts produce different clusters.
