Eigenvalue assignments and the two largest multiplicities in a Hermitian matrix whose graph is a tree

Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree we focus upon M2M2, the maximum value of the sum of the two largest multiplicities. The corresponding M1M1 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 M2M2 (in terms of simple graph theoretic parameters), cases of equality are indicated, and a combinatorial algorithm is given to compute M2M2 precisely. In the process, several techniques are developed that likely have more general uses.