The minimum number of multiplicity 1 eigenvalues among real symmetric matrices whose graph is a nonlinear tree (Q2076168)

Let \(T\) be a nonlinear tree, that is, a tree without a path with all its vertices of degree greater than two. Let \(\mathcal{S}(T)\) be the set of the symmetric matrices associated with \(T\). This paper deals with the minimum number of eigenvalues with multiplicity one for \(A\in\mathcal{S}(T)\). This number is denoted by \(U(T)\). In Section 3 the authors study the nonlinear trees using the diameter. Section 4, the most important section of this paper, contains the main results about the number \(U(T)\). Theorems 4.2 and 4.3 show that \(U(T)=2\) when \(T\) has diameter 5 or 6. In this section there are also two conjectures (Conjectures 4.9 and 4.13). In my opinion, the results of this paper will help new researchers to solve this challenging problem.