| This dissertation continues the study of the spectral properties of S(T), the set of all n by n symmetric matrices A = [aij] corresponding to a tree T on n vertices, where aij ≠ 0 for i ≠ j if and only if i -- j is an edge in T. The spectrum of S(T) is the set of all spectra realized by some matrix in S(T). For a given graph G, the problem of characterizing the spectrum of S(G) has led to the study of possible multiplicities of eigenvalues in the spectra of matrices in S(G). This dissertation studies the problem of characterizing the trees T for which there is an A ∈ S(T) whose eigenvalues have multiplicities m1, m2, ... ,mk. A new technique based on Smith Normal Form and Hamming distance is developed. This technique is used to characterize such matrices, called acyclic matrices, that have at most 5 distinct eigenvalues. Some results are given for acyclic matrices with larger diameter. Also, the acyclic matrices that have at most 2 multiple eigenvalues and whose sum of multiplicities is the maximum possible are characterized. |
