中文摘要：

假定Г是一个有限的、单的、无向的且无孤立点的图，G是Aut（Г）的一个子群,如果G在Г的边集合上传递，则称Г是G-边传递图,我们完全分类了当G为一个有循环的极大子群的素数幂阶群时的G-边传递图,结果为：设图Г含有一个阶为p^n（p是素数，n≥2）的自同构群，且G有一个极大子群循环，则Г是G-边传递的，当且仅当，同构于下列图之一 1）p^mK1,p^n-1-m,0≤m≤n-1; 2）p^mK1,p^n-m,0≤m≤n; 3）p^mKp,p^n-m-1,0≤m≤n-2; 4）p^n-mCpm,p^m≥3,m〈n; 5）2^n-2K1,1; 6）p^n-1-mCpm,p^m≥3,m≤n-1; 7）2p^n-mCpm,p^m≥3,m≤n-1; 8）2p^n-mK1,pm,0≤m≤n; 9）p^n-mK1,2pm,0≤m≤n; 10）p^n-mK2,pm,0≤m≤n; 11）C（2pn-m,1,p^m）; 12）p^kC（2p^m-k,1,p^n-m）,0〈k〈m,0〈m≤n; 13）（t-s,2^m）C（2m＋1/（t-s,2^m）,1,2^n-1-m）,其中0≤m≤n-1,2^n-2（s-1）≡0（mod 2^m）,t≡1（mod2）,≠t（mod2^m）,1≤s≤2^m,1≤t≤2^n-1; 14）p∪i=1C^i p^n-1,其中C^i p^n-1=Ca1a1＋[1＋（i-1）p^n-1]^a1＋2[1＋（i-1）p^n-2]…a1＋（p^n-1-1）[1＋（i-1）p^n-2]≈Cpn-1,i=1,1…,p; 15）2∪i=1C^i 2^n-1,其中C^i 2^n-1=Ca1a1＋[1＋（i-1）（2^n-2-1）^a1＋2[1＋（i-1）（2^n-2-1）]…a1＋（2n-1-1）[1＋（i-1）（2n-2-1）]≈C2n-1,i-1,2.

英文摘要：

Abstract Let Г be a finite simple undirected graph with no isolated vertices, G is a subgroup of Aut（Г）. The graph Г is said to be G-edge transitive if G is transitive on the set of edges of Г. We obtain a complete classification of G-edge transitive graphs, when G is a group of prime-power order with a cyclic maximal subgroup. This extends Sander＇s conclusion. Then Г is G-edge-transitive if and only if Г is one of following graphs：1）p^mK1,p^n-1-m,0≤m≤n-1; 2）p^mK1,p^n-m,0≤m≤n; 3）p^mKp,p^n-m-1,0≤m≤n-2; 4）p^n-mCpm,p^m≥3,m〈n; 5）2^n-2K1,1; 6）p^n-1-mCpm,p^m≥3,m≤n-1; 7）2p^n-mCpm,p^m≥3,m≤n-1; 8）2p^n-mK1,pm,0≤m≤n; 9）p^n-mK1,2pm,0≤m≤n; 10）p^n-mK2,pm,0≤m≤n; 11）C（2pn-m,1,p^m）; 12）p^kC（2p^m-k,1,p^n-m）,0〈k〈m,0〈m≤n; 13）（t-s,2^m）C（2m＋1/（t-s,2^m）,1,2^n-1-m）,其中0≤m≤n-1,2^n-2（s-1）≡0（mod 2^m）,t≡1（mod2）,≠t（mod2^m）,1≤s≤2^m,1≤t≤2^n-1; 14）p∪i=1C^i p^n-1,where C^i p^n-1=Ca1a1＋[1＋（i-1）p^n-1]^a1＋2[1＋（i-1）p^n-2]…a1＋（p^n-1-1）[1＋（i-1）p^n-2]≈Cpn-1,i=1,1…,p; 15）2∪i=1C^i 2^n-1,where C^i 2^n-1=Ca1a1＋[1＋（i-1）（2^n-2-1）^a1＋2[1＋（i-1）（2^n-2-1）]…a1＋（2n-1-1）[1＋（i-1）（2n-2-1）]≈C2n-1,i-1,2.