Discrete Applied Mathematics
Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree we focus upon M(2), the maximum value of the sum of the two largest multiplicities. The corresponding M(1) is already understood. The notion of assignment (of eigenvalues to subtrees) is formalized and applied. Using these ideas, simple upper and lower bounds are given for M(2) (in terms of simple graph theoretic parameters), cases of equality are indicated, and a combinatorial algorithm is given to compute M(2) precisely. In the process, several techniques are developed that likely have more general uses. (C) 2009 Elsevier B.V. All rights reserved.
Johnson, C. R., Jordan-Squire, C., & Sher, D. A. (2010). Eigenvalue assignments and the two largest multiplicities in a Hermitian matrix whose graph is a tree. Discrete Applied Mathematics, 158(6), 681-691.