Matrix analysis
The Parter-Wiener Theorem: Refinement and Generalization
SIAM Journal on Matrix Analysis and Applications
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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.