Application of an identity for subtrees with a given eigenvalue

Abstract

For an Hermitian matrix whose graph is a tree and for a given eigenvalue having Parter vertices, the possibilities for the multiplicity are considered. If V = {v1,…, vk} is a fragme-nting Parter set in a tree relative to the eigenvalue λ, and Ti+1 is the component of T−{v1, v2,…, vi} in which vi+1 lies, it is shown that (Formula Presented), in which Ni is the number of components of Ti−vi in which ƛ is an eigenvalue. This identity is applied to make several observations, including about when a set of strong Parter vertices leaves only 3 components with ƛ and about multiplicities in binary trees. Furthermore, it is shown that one can construct an Hermitian matrix whose graph is a tree that has a strong Parter set V such that |V | = k for each k in 1 ≤ k ≤ m − 1 for given multiplicity m ≥ 2 of an eigenvalue λ. Finally, some examples are given, in which the notion of a fragmenting Parter set is used.