Persistent homology is a branch of computational topology which uses geometry and topology for shape description and analysis. There are two links between persistent homology and graph theory. The first is represented by the various methods to build simplicial complexes from a weighted graph in order to study those simplicial complexes through persistent homology. The second is the application of the core ideas of persistence theory using invariants from graph theory. For example we studied blocks and edge-blocks along filtrations of a graph, in the way generators are studied in persistent homology.