The Minimum Number of Multiplicity 1 Eigenvalues among Real Symmetric Matrices Whose Graph is a Linear Tree

Let T be a linear tree, and let S(T) denote the set of real symmetric matrices whose graph is T. If U(T) is the minimum number of eigenvalues with multiplicity 1 among matrices in S(T) and T′ is a linear tree resulting from the addition of one vertex to T, we show that |U(T′)−U(T)|⩽1. We also determine the exact set of possible values of U(T)−U(T′), depending upon the manner in which the vertex is added to T to get T′. These results are then used to give a new bound for U(T), the diameter bound, and to improve an existing bound, 2+D2(T).