Application of an identity for subtrees with a given eigenvalue

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Kenji Toyonaga
Charles R. Johnson


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 = {v_1, . . . , v_k} is a fragmenting Parter set in a tree relative to the eigenvalue , and T_{i+1} is the component of Tâ{v_1, v_2, . . . , v_i} in which v_{i+1} lies, it is shown that \sum_{i}^K N_i=m_A(\lambda)+2kâ1, in which N_i is the number of components of T_iâv_i in which lambda 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 \lambda 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 â 1 for given multiplicity m >= 2 of an eigenvalue lambda. Finally, some examples are given, in which the notion of afragmenting Parter set is used.

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