具有特征根1的树

On Trees With 1 as the Eigenvalue

研究了树是否具有特征值1的问题.利用引理1得到了两种具有特征根1的树Tm和T m,其中树Tm具有m-1重特征根;树T m具有m-1+t(t为图T-u中1的重数)重特征根.定义了K2平凡的树和非K2平凡的树,对K2平凡的树T,判断它是否含特征根1可化为判断比T更低阶的图的问题;对非K2平凡的树T,判断它是否含特征根1或化为判断比T更低阶的图或计算T的"1 出值".

This paper aims at exploring the issue of whether a tree can have 1 as its eigenvalue since very little is known about it. Based on Lemma 1 (by E. Heilbronner), the paper deducted two kinds of trees: T_(m) and T~*_(m). The former has (m-1)-fold eigenvalue and the latter's eighenvalue is (m-1+t)-fold (t is the fold-number of 1 in graph T-u). Afterwards, two general trees, K_(2)-trivial and non-K_(2)-trivial, are defined. Whether the K_(2)-trivial tree has 1 as its eigenvalue can be converted into the validation of the graph with fewer vertices. As for the non-K_(2)-trivial tree, to confirm whether it has 1 as its eigenvalue, one of the following two ways can be adopted: 1) converting it in the validation of the graph with fewer vertices; 2) calculating 1-exitvalue of the tree.