Bounds for the smallest positive eigenvalues of trees with perfect matchings (Q1203866)

The authors prove the following statement: If \(T\) is a tree on \(2k\) vertices with the positive \(k\)-th largest eigenvalue \(\lambda_ k(T)\) (tree with perfect matchings), then \(\lambda_ k(T)\leq\lambda_ k(T^*_{2k})\), where \(T^*_{2k}\) is the tree \[ \undersetbrace k \to {\overset{\textstyle\circ} {\overset\vrule height4ex {depth1ex} \circ} \hbox to 2em{\hrulefill} \overset{\textstyle\circ} {\overset\vrule height4ex {depth1ex} \circ} \hbox to 2em{\hrulefill} \overset{\textstyle\circ} {\overset\vrule height4ex {depth1ex} \circ} \hbox to 2em{\hfill\dots\hfill} \overset{\textstyle\circ} {\overset\vrule height4ex {depth1ex} \circ} \hbox to 2em{\hrulefill} \overset{\textstyle\circ} {\overset\vrule height4ex {depth1ex} \circ} \hbox to 2em{\hrulefill} \overset{\textstyle\circ} {\overset\vrule height4ex {depth1ex} \circ}} \] Equality holds if and only if \(T\) is isomorphic to \(T^*_{2k}\).