I want to know what is the significance or importance of the largest and smallest eigenvalue in graph analysis (social network).
I understood how eigenvectors helps to find the centrality of the nodes in graph. But I'm wondering how the smallest and largest eigenvalues are useful in graph analysis.
Eigenvalues and eigenvectors of Laplacian of a graph can give a complete description of graph topology (it is the same as knowing the weight matrix.).
For instance, $0$ eigenvalues show the number of isolated subgraphs and eigenvectors of these $0$ eigenvalues are constant through the subgraphs.
As eigenvalues increase corresponding eigenvectors (localized to different subgraphs) start to have more high-frequency components.
In fact, Graph Signal Processing people do use eigenvectors as a Graph Fourier Transform basis.
In short, eigenvectors corresponding to small eigenvalues tend to include low-frequency components of the graph topology while large eigenvalues tend to correspond to fast-changing components of the topology.