图的度平方和的下界

Lower bound of the sum of the squares of the degrees of a graph

对图的度平方和的下界进行了讨论.用G=(V,E)表示一个具有n个点e条边的简单图,并且点的度数分别为d1,d2,…,dn.利用均值不等式及图中度序列的关系,给出了图G的度平方和的两个下界,并确定了达到这两个下界的极图.同时也给出了度平方和下界的简单应用,用它们来确定一个图及其补图中三角形的总个数.

The paper has discussed the lower bound of the sum of the squares of the degrees of a graph. Let G = （V,E） be a simple graph with n vertices, e edges, and vertex degrees d1 ,d2 ,… ,dn, and presented two lower bounds for ∑i=1ndi^2 through the inequality of mean value and the relationship of the degree sequence of a graph and determined the extreme graphs which achieved the bounds. It applied the consequences to bounding the total number of triangles in a graph and its complement.