  Graphs are a basic tool for the representation of modern data. The richness
of the topological information contained in a graph goes far beyond its mere
interpretation as a one-dimensional simplicial complex. We show how topological
constructions can be used to gain information otherwise concealed by the
low-dimensional nature of graphs. We do that by extending previous work of
other researchers in homological persistence, by proposing novel
graph-theoretical constructions. Beyond cliques, we use independent sets,
neighborhoods, enclaveless sets and a Ramsey-inspired extended persistence.
