Branch duplication for the construction of multiple eigenvalues in an Hermitian matrix whose graph is a tree

Suppose that the eigenvalues of an Hermitian matrix $A$ whose graph is a tree $T$ are known, as well as the eigenvalues of the principal submatrix of $A$ corresponding to a certain branch of $T$. A method for constructing a larger tree $T'$, in which the branch is ``duplicated'', and an Hermitian matrix $A'$ whose graph is $T'$ is described. The eigenvalues of $A'$ are all of those of $A$, together with those corresponding to the branch, including multiplicities.

This idea is applied (1) to give a solution to the inverse eigenvalue problem for stars, (2) to prove that the known diameter lower bound, for the minimum number of distinct eigenvalues among Hermitian matrices with a given graph, is best possible for trees of bounded diameter, and (3) to increase the list of trees for which all possible lists for the possible spectra are know. A generalization of the basic branch duplication method is presented.