On the multiplicity of the adjacency eigenvalues of graphs

Let G be a simple graph with the adjacency matrix A ( G ) . A well-known result of Cvetkovic 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 , ± 1 .