由谱半径给树排序

Ordering Trees by Their Largest Eigenvalues

设Δ(T)和λ1(T)分别表示树Τ的最大度和谱半径,Tn表示有n个点的树且T(nΔ)={T∈Tn|Δ(T)=Δ},文章根据树的谱半径给Tnn-6(n≥18)中的树进行了排序并将结果扩大到第78棵树。

Let △（T） and λ（T） denote the maximum degree and the largest eigenvalue of a tree T , respectively. Let Tn be the set of trees on n vertices , and Tn^（△）=（T∈Tn｜△（T）=△}. In the present paper , among the trees in Tn^（△）（n≥4）, we characterize the tree which alone maximizes the largest eigenvalue when [n-2/2≤△≤n-1. Furthermore, it is proved that, for two trees T1 and T2 in Tn （n≥4）, if △（T1）≥[2n/3]-1 and △（T1）〉△（T2）,then λ1 （T1））〉λ1 （T2）. By applying this result, we extend the order of trees in T. by their largest eigenvalue to the 79th tree when n ≥18.