【摘要】 <正> 1 引言设G是有限阶简单图。以V(G)和E(G)分别表示G的顶点集与棱集。若S是V(G)或E(G)的子集,则以G—S表示从G中删去S后所得到的图(当S={x}V(G)时,将G—S记作G-x)。如果G是连通的,而G—S不连通或者是平凡图,则称S是G的断集(当SV(G))或截集(当SE(G))。最小断集或截集的基数称为G的连通度或棱连通度,分别用K(G)和λ(G)来表示。当G为平凡图或不连通时,我们约定其连通度与棱—更多还原

【Abstract】 Let G be a connected graph and denote the vertex connectivity and edge connectivity of G by K(G) and λ(G) respectively. δ(G) represents the minimum degree of G. In [1] J. Akiyama et al. introduced the concept of the cohesiveness of a vertex of a graph and obtained the following theorem: Theorem A If the cohesiveness of vertexx C_G(x)=K(G)-K(G-x)<0, then the vertex set adjacent to x is the unique minimum vertex cut in G. In this short paper, for the edge connectivity a result similar to theorem A is obtained. Our result runs as follows: Theorem A’ If the edge-cohesiveness of vertex x C’_G(x)=λ(G)-λ(G-x)<0, then all the edges incident with x is the unique minimum edge cut of G. Corollary If C’_G(x)<0, then d_G(x)=λ(G)=λ(G), and for any y∈V(G)-{x) we have d_G(y)>d_G(x),C’_G(y)≥0.更多还原