Undirected graphs of Hermitian matrices that admit only two distinct eigenvalues

We consider the problem of determining those undirected n-vertex graphs with a corresponding Hermitian matrix that admits only two distinct eigenvalues, with multiplicities k   and n−kn−k. After giving some general algebraic characterizations of these dual multiplicity graphs, we then prove two major graph theoretic necessary conditions on such graphs. Construction techniques are then developed, and these lead to a characterization of dual multiplicity graphs for which the lesser multiplicity is two.