  To some extent, graph evolutionary mechanisms can be explained by its
spectra. Here, we are interested in two graph operations, namely, motif
(subgraph) doubling and attachment that are biologically relevant. We
investigate how these two processes affect the spectrum of the normalized graph
Laplacian. A high (algebraic) multiplicity of the eigenvalues $1, 1\pm 0.5,
1\pm \sqrt{0.5}$ and others has been observed in the spectrum of many real
networks. We attempt to explain the production of distinct eigenvalues by motif
doubling and attachment. Results on the eigenvalue $1$ are discussed
separately.
