ON THE MINIMUM NUMBER OF DISTINCT EIGENVALUES FOR A SYMMETRIC MATRIX WHOSE GRAPH IS A GIVEN TREE

By Onio Leal-Duarte, Charles R. Johnson. 2002

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SPECTRAL GRAPH THEORY AND THE INVERSE EIGENVALUE PROBLEM OF A GRAPH (2005, background): “[25] If T is a tree, for any B ∈ S(T ), the diameter of T is less than the number of distinct eigenvalues of B.”
GRAPHS WHOSE MINIMAL RANK IS TWO (2004, unknown): “Several additional results have been obtained for the multiplicities of eigenvalues of a matrix in S(G) in the case G is a tree ([10], [9], [11]).”
On the possible multiplicities of the eigenvalues of a Hermitian matrix whose graph is a tree (2002, unknown): “In [6], the minimum number of distinct eigenvalues among matrices in H(T ) , T a tree, was studied ( q(T ) = the fewest parts in a partition in L(T ) ) and related to the diameter of T , d(T ) = the number of vertices in a longest path in T (for trees).”
The minimum rank of symmetric matrices described by a graph: A survey☆ (2007, unknown): “In [46] it was shown that q(T ) ≥ diam(T )+1.”
Minimum number of distinct eigenvalues of graphs (2013, background): “This result was generalized to the case of trees, by observing that q(A) ≥ diam(G)+ 1, whenever A ∈ S(G) is an entry-wise nonnegative matrix (see [6]).”