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It is known that each eigenvalue of a real symmetric, irreducible, tridiagonal matrix has multiplicity 1. The graph of such a matrix is a path. In this paper, we extend the result by classifying those trees for which each of the associated acyclic matrices has distinct eigenvalues whenever the diagonal entries are distinct.

CLASSIFICATION OF TREES EACH OF WHOSE ASSOCIATED ACYCLIC MATRICES WITH DISTINCT DIAGONAL ENTRIES HAS DISTINCT EIGENVALUES