ar X iv : m at h / 05 08 26 5 v 1 [ m at h . C O ] 1 5 A ug 2 00 5 Smith Normal Form and Acyclic Matrices
نویسندگان
چکیده
An approach, based on the Smith Normal Form, is introduced to study the spectra of symmetric matrices with a given graph. The approach serves well to explain how the path cover number (resp. diameter of a tree T) is related to the maximum multiplicity occurring for an eigenvalue of a symmetric matrix whose graph is T (resp. the minimum number q(T) of distinct eigenvalues over the symmetric matrices whose graphs are T). The approach is also applied to a more general class of connected graphs G, not necessarily trees, in order to establish a lower bound on q(G).
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