Estimation of the maximum multiplicity of an eigenvalue in terms of the vertex degrees of the graph of a matrix (Q2782047)

Let \(T\) be a tree on \(n\) vertices whose degrees are \(d_1\geq\dots\geq d_k\geq 3 >d_{k+1}\geq\dots\geq d_n\). Let \(m(T)\) be the maximum multiplicity of an eigenvalue among matrices whose graph is \(T\). There are some formulas to compute \(m(T)\), but neither of these formulas is expressible in terms of overt parameters of \(T\). In this paper, the best possible lower and upper bounds and characterization of the cases of equality in these bounds are given: NEWLINE\[NEWLINE1+\sum_{i=1}^k(d_i-2)-e\leq m(T)\leq 1+\sum_{i=1}^k(d_i-2)NEWLINE\]NEWLINE where \(e=e(H)\) is the number of edges present in \(H\), \(H=H(T)\) is the subgraph of \(T\) induced by the \(k\) vertices of degree at least 3 in \(T\). A by-product is a sequential algorithm to calculate the exact maximum multiplicity by simple counting.