Eigenvalues of Symmetrical Matrices and Graphs Original Research Article

We investigate the class image of integral domains A having the property that each totally real integral element over A is an eigenvalue of a symmetric matrix over A. image is shown to contain all Dedekind domains and any domain in which −1 is a sum of squares. In the case A = image, these results imply that any totally real algebraic integer is the eigenvalue of the adjacency matrix of some regular graph, thus affirming a conjecture of Hoffman.