图的最大拉普拉斯特征值的上界

The Upper Bound for the Largest Laplacian Eigenvalue of Graphs

设G=(V,E)是n阶简单连通图,D(G)和A(G)分别表示图的度对角矩阵和邻接矩阵,L(G)=D(G)-A(G)则称为图G的拉普拉斯矩阵。利用图的顶点度和平均二次度结合非负矩阵谱理论给出了图的最大拉普拉斯特征值的新上界,同时给出了达到上界的极图,并且通过举例与已有的上界作了比较,说明在一定程度上优于已有结果。

Let G = （V,E）be a simple and connected graph with n vertices,D（G） and A（G） be the diagonal matrix of vertex degrees and the adjacency matrix of G,respectively. Then the matrix L（G） =D（G） -A（G） is called as the Laplacian matrix of G. In this paper, the spectral theory of nonnegative matrices was used to present a new upper bound on the largest Laplacian Eigenvalue of graphs in terms of the vertex degree and the average 2-degree. Moreover, the extremal graph which achieves the upper bound was determined. Besides, a example is given to illus- trate that the above result is better than the earlier and recent ones in some sense.