In this thesis the characterization of graphs is studied by star complements technique. Let G be a graph of order n and let µ be its eigenvalue of multiplicity k. A star complement for µ in G, is an induced subgraph of G, H, of order n-k with no eigenvalue µ. The set of all vertices in G\H, X, is called a star set for µ in G. One main problem in the context of star complement is determination of all maximal graphs prescribing a given graph H as a star complement for a given eigenvalue µ. The reconstruction theorem states that for a given eigenvalue µ of G, knowledge of a star complement corresponding to µ, together with knowledge of the edge set betwee X and its complement ?X, is sufficient to reconstruct G. For a given star complement H the range of possible values for the corresponding eigenvalue µ is constrained by the condition that µ must be a simple eigenvalue of some one-vertex extension of H, and a double eigenvalue of some two-vertex extension of H. In this thesis we give some known characterizations of graphs by star complements technique. We give examples of graph Characterizations arising in the case where the star complement is a complete bipartite graph, a disjoint union of some isolated vertices with a given complete bipartite
