最好可能的森谱界

The Best Possible Bound on Forest Spectra

证明了森或树的任一正特征值λｑ－ｉ（ｉ＝ｑ－１，ｑ－２，…，０）满足λｑ－ｉ≥２ｃｏｓ［ｔｉπ／（２ｔｉ＋１）］（ｔｉ＝［［２ｑ／（ｉ＋１）］／２］），并指出这个下界对于边独立数为ｑ的森或者顶点数为ｎ、边独立数为ｑ的森是最好可能的；对于边独立数为ｑ的树或者顶点数为ｎ、边独立数为ｑ的树当ｉ＝ｑ－２，ｑ－３，…，ｑ－［（ｑ＋１）／２］或当ｉ＝ｑ－［（ｑ＋１）／２］－１，ｑ－［（ｑ＋１）／２］－２，…，１（ｑ０（ｍｏｄｉ＋１））时。

Let λ q-i (i=q-1, q-2, …, 0) be any positive eigenvalue of a forest or a tree, q the independence number, and the maximum integer no greater than x . It is proved thatλ q-1 ≥2 cos [t i π /(2t i+1)] where t i=[[2q/(i+1)]/2]and this lower bound is the best possible one for a forest with n vertices and an edge independence number of q , or for a tree with an edge independence number q . For such a tree, if i=q-2, q-3, …, q- or i=q-[(q+1)/2]-1, q--2, …, 1 and q0 (mod i+1) , this lower bound is the best possible one.