An important problem is to characterize all possible eigenvalues over all symmetric matrices corresponding to a given tree $T$. The minimum number of distinct eigenvalues over this family of symmetric matrices, is denoted by $q(T)$. Using combinatorial properties of tree $T$, we are able to construct a lower bound for the parameter $q(T)$. A theorem developed by Parter and Wiener ensures the existence of a principal submatrix, in which the multiplicity of a non-simple eigenvalue is increased. This
could be used for a partial solution to the inverse eigenvalue problem for a tree and resolving possible ordered multiplicity lists. We will present the application of the obtained results on numerous examples throughout the work.
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