Among the n eigenvalues of an n-by-n matrix may be several repetitions (thenumber of which counts toward the total of n). For general matrices over a gen-eral 〠eld, these multiplicities may be algebraic (the number of appearances asa root of the characteristic polynomial) or geometric (the dimension of the cor-responding eigenspace). These multiplicities are quite important in the analysisof matrix structure because of numerical calculation, a variety of applications,and for theoretical interest. We are primarily concerned with geometric multi-plicities and, in particular but not exclusively, with real symmetric or complexHermitian matrices, for which the two notions of multiplicity coincide.It has been known for some time, and is not surprising, that the arrange-ment of nonzero entries of a matrix, conveniently described by the graphof the matrix, limits the possible geometric multiplicities of the eigenvalues.Much less limited by this information are either the algebraic multiplicitiesor the numerical values of the (distinct) eigenvalues. So, it is natural to studyexactly how the graph of a matrix limits the possible geometric eigenvaluemultiplicities.