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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Kim, IJ., Shader, B.L. Smith Normal Form and acyclic matrices. J Algebr Comb 29, 63–80 (2009). https://doi.org/10.1007/s10801-008-0121-8
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DOI: https://doi.org/10.1007/s10801-008-0121-8