森谱的界(英文)

The Bound on Forest Spectra

设λ_k(F)是树或者森的第k大特征值,[x]是不超过x的最大整数,q是F的边独立数.本文证明了:对于1≤k≤[(q+1)/2]有λ_k(F)≥1,并且这个下界是最好可能的;对于1≤i≤[q/2],若q为偶数,则有λ[(q+1)/2]+i(F)≥2cos((2iπ)/(4i+1)),若q为奇数,则有λ_([(q+1)/2]+i)(F)≥2cos(((2i+1)π)/(4i+3)),

Let λ_k(F) be the kth eigenvalue of a tree or a forest, [x] the integer not greater than x and q the edge independence number of F. It is shown that for 1≤k≤[(q+1)/2], λ_k(F)≥1 and this lower bound is best possible. It is also shown that for 1≤i≤ [q/2] if q is an even number λ_(i+[(q+1)/2] (F)≥2cos(2iπ/(4i+1)) and if q is an odd number λ_(i+[(q+1)/2](F)≥2cos((2i+l)π/(4i+3)).