The Number Of Blocks Of A Graph With Given Minimum Degree

Let G be a graph,a nonseparable maximal subgraph of graph G is called a block of graph G,we denote the number of blocks of graph G by b(G).Albertson gave an upper bound of the number of cut vertices of G with given minimum degree,and showed the upper bound is asymptotically best possible,however,the problem of the corresponding number of blocks has not been solved.In this paper,through the research on block tree,we show that for a connected graph of order n with minimum degree k?1,the number of blocks b(G)<2k-3/k2-k-1n,and the upper bound is asymptotically tight;also,for a connected cubic graph G of order n?14,we prove that b(G)?n/2-2,and the upper bound is tight,we provide a method to construct the extreme graph.