We present two distinct applications of an inequality relating the multiplicity of an eigenvalue of a graph to a certain subgraph. The first is related to a recent classification, established by Kim and Shader, for the class of those trees for which each of the associated matrices have distinct eigenvalues whenever the diagonal entries are distinct. We analyze the minimum number of distinct diagonal entries and the corresponding location, in order to preserve such multiplicity characterization. The second application involves a new property of a star set of a graph due to P. Rowlinson.