On the largest eigenvalues of trees with perfect matchings

摘要

Let λ1 (G) and Δ (G), respectively, denote the largest eigenvalue and the maximum degree of a graph G. Let $mathcal {T}_{2m}$ be the set of trees with perfect matchings on 2m vertices, and $mathcal {T}_{2m}^{(Delta )} ={Tin mathcal {T}_{2m}vert Delta (T)=Delta }$ . Among the trees in $mathcal {T}_{2m}^{(Delta )} (mge 2)$ , we characterize the tree which alone minimizes the largest eigenvalue, as well as the tree which alone maximizes the largest eigenvalue when $leftlceil {frac{m}{2}} rightrceil + 1le Delta le m$ . Furthermore, it is proved that, for two trees T 1 and T 2 in $mathcal {T}_{2m}$ (m≥ 4), if $leftlceil {frac{2m}{3}} rightrceil le Delta (T_1 )le m$ and Δ (T 1) > Δ (T 2), then λ1 (T 1) > λ1 (T 2).