Given a graph G on n vertices and \(z\in R^ n\), we say that a vertex of G is positive, nonnegative, null, etc. if the corresponding entry of z has that property. For z such that Az\(\geq \alpha z\) (A is the adjacency matrix of G) the number of components of the subgraph induced by positive vertices is bounded. Inequalities for several related quantities are derived provided z is an eigenvector. The paper reflects a recent trend in the theory of graph spectra: studying eigenspaces of graphs.