Smith Normal Form and Acyclic Matrics

Details Smith Normal Form and Acyclic Matrics Smith Normal Form and Acyclic Matrics

2005-08-15

arxiv.org/abs/math/0508265v1

Mathematics

Combinatorics

24 pages

Scientific paper

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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