Rational realization of maximum eigenvalue multiplicity of symmetric tree sign patterns

A sign pattern is a matrix whose entries are elements of {+, −, 0}; it describes the set of real matrices whose entries have the signs in the pattern. A graph (that allows loops but not multiple edges) describes the set of symmetric matrices having a zero-nonzero pattern of entries determined by the absence or presence of edges in the graph. DeAlba et al. [L.M. DeAlba, T.L. Hardy, I.R. Hentzel, L. Hogben, A. Wangsness, Minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns, Linear Algebra Appl., in press, doi:10.1016/j.laa.2006.02.018.] gave algorithms for the computation of maximum multiplicity and minimum rank of matrices associated with a tree sign pattern or tree, and an algorithm to obtain an integer matrix realizing minimum rank. We extend these results by giving algorithms to obtain a symmetric rational matrix realizing the maximum multiplicity of a rational eigenvalue among symmetric matrices associated with a symmetric tree sign pattern or tree.