Smith Normal Form and acyclic matrices |
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Authors: | In-Jae Kim Bryan L Shader |
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Institution: | (1) Department of Mathematics and Statistics, Minnesota State University, Mankato, MN 56001, USA;(2) Department of Mathematics, University of Wyoming, Laramie, WY 82071, USA |
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Abstract: | 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 maximal multiplicity MaxMult(T) occurring for an eigenvalue of a symmetric matrix whose graph is T (resp. the minimal 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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Keywords: | Smith Normal Form Acyclic matrix Graph spectra |
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