Graph components of prime spectra

The prime spectrum, Spec(A), of a ring A is a T0-space and it is partially ordered by inclusion between prime ideals. The partial order makes Spec(A) into a graph – the vertices are the prime ideals, and there is an edge between two vertices if there is a containment relation between them. The graph and the topological space Spec(A) both have connected components, which are called graph components and topological components. Every topological component is a union of graph components. The paper is devoted to a study of the graph components. The main question is how properties of the graph components of the prime spectrum correspond to arithmetic properties of a ring. Given a property P that graph components may or may not have, let R(P) be the class of rings A such that every graph component of Spec(A) has property P. For which properties P is it true that R(P) is an elementary class of rings in the sense of model theory?