**17 Spectral properties, transport, diffusion and dynamics**

In this chapter we discuss the spectral properties of networks, and their relation to dynamical properties such as diffusion. There are two main characteristic matrices for a graph, the adjacency matrix and the Laplacian. We discuss both of them and explain the relation to the dynamical properties. A good summary of results on the spectrum of networks can be found in [Chu97]. For a nice survey of random walks on graphs see [Lov96].

**17.1 The spectrum of the adjacency matrix**

As described in Chapter 1, the adjacency matrix, *A*, is an *N*× *N* matrix (where *N* is the number of nodes) whose entries ...

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