【摘要】 设G是无向图,A是加法交换群,且A*=A-0.如果G有一个定向D（G）,对任何满足∑_v∈V（G）b（v）=0的函数b:V（G）→A,都存在函数f:E（G）→A*使得在每个顶点v∈V（G）,从v发出的所有边上的f总值减去进入v的所有边上的f总值恰等于b（v）,则称G是A-连通的.群连通数为:A_g（G）=min{n:对任何满足|A|≥n的群A,G是A-连通的}.令q是正整数,广义q-树的定义是按下列递推形式给出的:最小的广义q-树是阶为q的完全图K_q;阶为n+1的广义q-树是由阶为n的广义q-树通过增加一个新顶点和连接此点与阶为n的广义q-树中任意给定的q个顶点得到的.本文对广义q-树G（q≥2）考察Λ_g（G）,证明了如果G是阶n≥3的广义2-树或阶为n∈{3,4}的广义3-树或阶为4的广义4-树,则Λ_g（G）=4;如果G是阶为n≥5的广义q-树（q≥3）,则A_g（G）=3.更多还原

【Abstract】 Let G be an undirected graph,A be an（additive） abelian group and A*= A-0.A graph G is A-connected if G has an orientation D（G） such that for every function b:V（G）→A satisfying∑_v∈V（G）b（v） = 0,there is a function f:E（G）→A* such that at each vertex v∈V（G）,the amount of f values on the edges directed out from v minus the amount of f values on the edges directed into v equals b（v）.The group connectivity number A₉（G） = min{n:G is A-connected for every abelian group A with |A|> n}.Let q be a positive integer.The generalized q-trees are defined by recursion:the smallest generalized q-tree is the complete graph K_q with order q,and a generalized q-tree with order n + 1 where n≥q is obtained by adding a new vertex adjacent to q arbitrarily selected vertices of a generalized q-tree with order n.In this paper,we investigate A_g（G） for generalized q-trees G with q > 2.We show that if G is a generalized 2-tree with order n > 3 or generalized 3-tree with order n∈{3,4} or generalized 4-tree with order 4,then A_g（G） = 4,and if G is a generalized q-tree for q≥3 with order n≥5,then A_g（G） = 3.更多还原