Distinct eigenvalues are realizable with generic eigenvectors

Motivated by applications in matrix constructions used in the inverse eigenvalue problem for graphs, we study a concept of genericity for eigenvectors associated with a given spectrum and a connected graph. This concept generalizes the established notion of a nowhere-zero eigenbasis. Given any simple graph G on n vertices and any spectrum with no multiple eigenvalues, we show that the family of eigenbases for symmetric matrices with this graph and spectrum is generic, strengthening a result of Monfared and Shader. We illustrate applications of this result by constructing new achievable ordered multiplicity lists for partial joins of graphs and providing several families of joins of graphs that are realizable by a matrix with only two distinct eigenvalues.