This chapter presents some examples of the spectra of complex networks, which we have tried to interpret or to understand using the theory of previous chapters. In contrast to the mathematical rigor of the other chapters, this chapter is more intuitively oriented and it touches topics that are not yet understood or that lack maturity. Nevertheless, the examples may give a flavor of how real-world complex networks are analyzed as a sequence of small and partial steps towards (hopefully) complete understanding.

Simple observations

When we visualize the density function fλ(x) of the eigenvalues of the adjacency matrix of a graph, defined in art. 121, peaks at x = 0, x = −1 and x = −2 are often observed. The occurrence of adjacency eigenvalue at those integer values has a physical explanation.

A graph with eigenvalue λ (A) = 0

A matrix has a zero eigenvalue if its determinant is zero (art. 138). A determinant is zero if two rows are identical or if some of the rows are linearly dependent. For example, two rows are identical resulting in λ (A) = 0, if two not mutually interconnected nodes are connected to a same set of nodes. Since the elements aij of an adjacency matrix A are only 0 or 1, linear dependence of rows occurs every time the sum of a set of rows equals another row in the adjacency matrix.