几类图的拉普拉斯特征值的前三项和的上界

Upper Bound of the Sum of the First 3 Laplacian Eigenvalues of Several Classes of Graphs

设G是一个顶点集为V(G),边集为E(G))的简单图.S_k(G)表示图G的拉普拉斯特征值的前k项部分和.Brouwer et al.给出如下猜想:S_k(G)≤e(G)+((k+1)/2),1≤k≤n.证明了当k=3时,对边数不少于n^2/4-n/4的图及有完美匹配或有6-匹配的图,猜想是正确的.

Let G be a simple graph with vertex set V（G） and edge set E（G）.Denote the sum of the first k Laplacian eigenvalues of graph G by Sk{G）.Brouwer et al.proposed a conjecture that Sk（G） ≤ e（G） ＋（（k＋1）/2）,where 1 ≤k≤ n,for any simple graph G.In this paper,when k = 3,we prove that the conjecture is true for several classes of graphs as follows：these graphs whose number of edges is greater than or equal to（n^2）/4-n/4,these graphs which have perfect matching and these graphs which have 6-mathing.