  In this article, we introduce a relation including ideals of an evolution
algebra and hereditary subsets of vertices of its associated graph and
establish some properties among them. This relation allows us to determine
maximal ideals and ideals having the absorption property of an evolution
algebra in terms of its associated graph. We also define a couple of
order-preserving maps, one from the sets of ideals of an evolution algebra to
that of hereditary subsets of the corresponding graph, and the other in the
reverse direction. Conveniently restricted to the set of absorption ideals and
to the set of hereditary saturated subsets, this is a monotone Galois
connection. According to the graph, we characterize arbitrary dimensional
finitely-generated (as algebras) evolution algebras under certain restrictions
of its graph. Furthermore, the simplicity of finitely-generated perfect
evolution algebras is described on the basis of the simplicity of the graph.
