Unordered multiplicity lists of a class of binary trees

Associated with a given graph G on n vertices 1,2,…,n, is the set S(G) of all n×n real symmetric matrices A=[aij] whose off-diagonal entries are placed according to the edges of G, i.e., for i≠j, aij≠0 if and only if vertices i and j are adjacent. In this paper we study spectral properties of matrices in S(G) for a class of binary trees G. We first show that a matrix A in S(G) has no eigenvalues of multiplicity 4 or more, at most one eigenvalue of multiplicity 3, and at least three simple eigenvalues. We then completely determine the unordered multiplicity lists of these binary trees. As a consequence, it is shown that the minimum number of distinct eigenvalues of a matrix in S(G) is one more than the diameter of G.