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Description
For a tree $T$ we consider the set ${\mathcal S}(T)$ of real symmetric matrices whose off-diagonal zero-nonzero pattern is equal to the pattern of the adjacency matrix of $T$. It is well known that the maximum multiplicity of an eigenvalue over matrices in ${\mathcal S}(T)$ is equal to the path cover number $P(T)$ of the tree $T$.
We present a novel decomposition of the tree into a set of vertices and paths, which serves as a tool for analysing matrices achieving maximum multiplicity $P(T)$. We define a parameter ${\rm MM}(T)$ to be the maximal positive integer $k$ for which there exists a matrix $A\in S(T)$ with $k$ eigenvalues achieving the multiplicity $P(T)$. We derive a combinatorial upper bound for ${\rm MM}(T)$ and identify some families of trees that achieve this bound.