Eigenvalues of symmetrical matrices and graphs

Abstract

We investigate the class 풯 of integral domains A having the property that each totally real integral element over A is an eigenvalue of a symmetric matrix over A. 풯 is shown to contain all Dedekind domains and any domain in which -1 is a sum of squares. In the case A = 핫, 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. © 1994 Academic Press, Inc.