The change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the changes in edge values around a classified vertex in a tree (Q1994054)

Let \(S(T)\) be a set of real symmetric matrices whose graph of off-diagonal entries is a given tree \(T\), and for \(A\in S(T)\) let \(m_A(\lambda)\) denote the multiplicity of \(\lambda\) as an eigenvalue of \(A\). The results show that when values of the edges incident to a particular vertex are simultaneously changed, then the multiplicity of \(\lambda\) changes by at most one. In addition, when an edge \(e\) is removed from \(T\) then \(m_A(\lambda)-1\leq m_{A_e}(\lambda)\leq m_A(\lambda)+2\) by interlacing, where \(A_e\in S(T-e)\). The edge \(e\) is called 2-Parter, Parter, neutral or downer if \(m_{A_e}(\lambda)-m_A(\lambda)\) is equal to \(2\), \(1\), \(0\) or \(-1\), respectively. It turns out that 2-Parter, Parter and downer edges cannot be incident to each other and that, moreover, a downer edge must be at distance at least two from any Parter or 2-Parter edge.