  Eigenvalues of a graph are the eigenvalues of the corresponding
(0,1)-adjacency matrix. The second largest eigenvalue lambda_2 provides
significant information on characteristics and structure of graphs. Therefore,
finding bounds for lambda_2 is a topic of interest in many fields. So far we
have studied the graphs with the property lambda_2 is less or equal to 2,
so-called reflexive graphs. The original RS-theorem is about them. In this
paper we generalize that concept and introduce the arbitrary bounds. The
Generalized RS-theorem gives us an answer whether the second largest eigenvalue
of a graph is greater than, less than, or equal to a, a>0, within some classes
of connected graphs with a cut-vertex. After removing the cut-vertex u of the
given graph G, we examine the indices of the components of G-u. The information
on these indices is used to make conclusions about the second largest
eigenvalue of the graph G. In the RS-theorem a=2. Here, we state and prove the
Generalized RS-theorem, one corollary, and, also, give some useful lemmas.
