On the spectrum of the normalized graph Laplacian
نویسندگان
چکیده
The spectrum of the normalized graph Laplacian yields a very comprehensive set of invariants of a graph. In order to understand the information contained in those invariants better, we systematically investigate the behavior of this spectrum under local and global operations like motif doubling, graph joining or splitting. The eigenvalue 1 plays a particular role, and we therefore emphasize those constructions that change its multiplicity in a controlled manner, like the iterated duplication of nodes. Let Γ be a finite and connected graph with N vertices. Two vertices i, j ∈ Γ are called neighbors, i ∼ j, when they are connected by an edge of Γ. For a vertex i ∈ Γ, let ni be its degree, that is, the number of its neighbors. For functions v from the vertices of Γ to R, we define the (normalized) Laplacian as ∆v(i) := v(i)− 1 ni ∑
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