ABSTRACT
In this chapter we give an introduction to graph theory and study algebraic properties and invariants of edge ideals using the combinatorial structure of graphs. We present classifications of the following families: unmixed bipartite graphs, Cohen-Macaulay bipartite graphs, Cohen-Macaulay trees, shellable bipartite graphs, sequentially Cohen-Macaulay bipartite graphs. The invariants examined here include regularity, depth, Krull dimension, and multiplicity. Edge ideals are shown to have the persistence property.
