Wenshui Lin

Let $\lambda_1(T)$ and $\Delta(T)$ respectively denote the largest eigenvalue and the maximum degree of a tree $T$. The set of trees on $n$ vertices and the set of trees with perfect matchings on $2k$ vertices are denoted by ${\cal F}_n$ and ${\cal T}_{2k}$, respectively. In the paper titled "Ordering trees by their largest eigenvalues" (submitted to LAA, in press), we showed that $\lambda_1(T)$ monotonously increases with $\Delta(T)$ for $T\in {\cal F}_n$ when $n\geq7$ and $\Delta(T)\geq\lfloor\frac{2n}{3}\rfloor-1$. Here an analogous result on the trees in ${\cal T}_{2k}$ is obtained. Let ${\cal T}_{2k} ^{(\Delta)}=\{T\in {\cal T}_{2k} | \Delta(T)=\Delta \}$ and $T^{(i)} \in {\cal T}_{2k}^{(i)}$, $i=k,k-1,\ldots,2$. It is shown that, $\lambda_1(T^{(k)})>\lambda_1(T^{(k-1)})> \cdots> \lambda_1(T^{(\lfloor\frac{2k}{3}\rfloor+1)})> \lambda_1(T^{(l)})$, where $k\geq 6$ and $l \leq \lfloor\frac{2k}{3}\rfloor$.