Simplicial Complexes From Graphs Toward Graph Persistence

Persistent homology is a branch of computational topology which uses geometry and topology for shape description and analysis. This dissertation is an introductory study to link persistent homology and graph theory, the connection being represented by various methods to build simplicial complexes from a graph. The methods we consider are the complex of cliques, of independent sets, of neighbours, of enclaveless sets and complexes from acyclic subgraphs, each revealing several properties of the underlying graph. Moreover, we apply the core ideas of persistence theory in the new context of graph theory, we define the persistent block number and the persistent edge-block number.