On the multiplicity of the adjacency eigenvalues of graphs

Let G   be a simple graph with the adjacency matrix A(G)A(G). A well-known result of Cvetković and Gutman states that removing a pendant vertex and its neighbour, does not change the nullity of the graph. We generalize this theorem and some other theorems similar to it for an arbitrary eigenvalue of a graph. Also, for an arbitrary eigenvalue λ, we use a corresponding star set to delete some subgraphs and to determine the multiplicity of λ  . We use star sets to find some Parter–Wiener vertices, in trees. We state our results for real symmetric matrices (and so for weighted graphs) and as a corollary we use them for simple graphs. Furthermore, we use these methods to obtain some results for λ=0,±1λ=0,±1.