设 G 是一个简单无向图.V(G),E(G)分别表示 G 的顶点集和边集.(?)表示 G 的补图.我们以 S_(?) 表示 n+1阶星图 k_(1,n-1).称 G 是(p,p—k)图,如果|E(G)|=|V(G)|—k.称|V(G)|为图 G 的阶.设 G_1,G_2是同阶图,(?)_1是 V(G_1)到 V(G_2)的...设 G 是一个简单无向图.V(G),E(G)分别表示 G 的顶点集和边集.(?)表示 G 的补图.我们以 S_(?) 表示 n+1阶星图 k_(1,n-1).称 G 是(p,p—k)图,如果|E(G)|=|V(G)|—k.称|V(G)|为图 G 的阶.设 G_1,G_2是同阶图,(?)_1是 V(G_1)到 V(G_2)的一个双射,(?)_2是 V(G_2)上的一个置换,我们用(?)_2(?)_1表示 V(G_1)到 V(G_2)的双射。展开更多
设 G 是有限群,S 为 G 的一个非空子集,e 是 G 中的单位元,如果 e(?)S,则称 S 为 G的一个 Gayley-子集.定义 Cayley 有向图 X=X(G,S)如下:V(X)=G,E(X)={(a,b)|a,b∈G,ba^(-1)∈S}.当 S=S^(-1)时 X 是无向图,简称 Cayley 图.若 X 有 Hami...设 G 是有限群,S 为 G 的一个非空子集,e 是 G 中的单位元,如果 e(?)S,则称 S 为 G的一个 Gayley-子集.定义 Cayley 有向图 X=X(G,S)如下:V(X)=G,E(X)={(a,b)|a,b∈G,ba^(-1)∈S}.当 S=S^(-1)时 X 是无向图,简称 Cayley 图.若 X 有 Hamiltonian 圈(简记为 H-圈),也称 X 是-H-图.继 Lovasz 提出“仅有有限个顶点传递的连通图是非 H-图”的猜想后,Parsons 等猜测“连通 Cayley 图是 H-图”.但由于要一般性地解决这个问题极其困难。展开更多
A graph Γ is said to be G-locally primitive, whereG is a subgroup of automorphisms of Γ, if the stabiliser Gα of a vertex α acts primitively on the set Γ(α) of vertices of Γ adjacent to α. For a finite non-abe...A graph Γ is said to be G-locally primitive, whereG is a subgroup of automorphisms of Γ, if the stabiliser Gα of a vertex α acts primitively on the set Γ(α) of vertices of Γ adjacent to α. For a finite non-abelian simple group L and a Cayley subset S of L, suppose that LG≤Aut(L), and the Cayley graph Γ=Cay (L, S) is G-locally primitive. In this paper we prove that L is a simple group of Lie type, and either the valency of Γ is an add prine divisor of ｜Out(L)｜, or L=PΩ+8 (q) and Γ has valency 4. In either cases, it is proved that the full automorphism group of Γ is also almost simple with the same socle L.展开更多
Given a simple undirected graph G. We denote the sets of its vertices and edges by V（G） and E（G） respectively. G is called a （P, P—K） graph （or G=（P, P—K）） if |V（G）|=P and |E（G）|=|V（G）|—K. Let {G1, ...Given a simple undirected graph G. We denote the sets of its vertices and edges by V（G） and E（G） respectively. G is called a （P, P—K） graph （or G=（P, P—K）） if |V（G）|=P and |E（G）|=|V（G）|—K. Let {G1, G2} be a pair of graphs of the same order. If G1 is isomorphic to a subgraph of （?）2, where （?）2 is the com-展开更多
Let G be a finite group and H a nonempty subset of G.We call H Cayley subset if the identity element e of G is not in H.For each Cayley subset H we define the Cayley digraph X=X(G, H)
Let G be a finite group and S a nonempty subset of G. We call S a Cayley subset if the identity element e of G is not in S. For each Cayley subset S, we define the Cayley digraph X=X( G, S)
Let G be a finite group, and S a subset of G1 with S=S-1. We use X=Cay(G,S) to denote the Cayley graph of G with respect to S. We call S a CI-subset of G, if for any isomorphism Cay(G,S)Cay(G,T) there is an α∈Aut(G)...Let G be a finite group, and S a subset of G1 with S=S-1. We use X=Cay(G,S) to denote the Cayley graph of G with respect to S. We call S a CI-subset of G, if for any isomorphism Cay(G,S)Cay(G,T) there is an α∈Aut(G) such that Sα=T. Assume that m is a positive integer. G is called an m-CI-group if every subset S of G with S=S-1 and |S|≤m is CI. In this paper we prove that the alternating group A5 is a 4-CI-group, which was a conjecture posed by Li and Praeger.展开更多
文摘设 G 是一个简单无向图.V(G),E(G)分别表示 G 的顶点集和边集.(?)表示 G 的补图.我们以 S_(?) 表示 n+1阶星图 k_(1,n-1).称 G 是(p,p—k)图,如果|E(G)|=|V(G)|—k.称|V(G)|为图 G 的阶.设 G_1,G_2是同阶图,(?)_1是 V(G_1)到 V(G_2)的一个双射,(?)_2是 V(G_2)上的一个置换,我们用(?)_2(?)_1表示 V(G_1)到 V(G_2)的双射。
基金This work was supported by the National Natural Science Foundation of China (Grant No. 69873002).
文摘A graph Γ is said to be G-locally primitive, whereG is a subgroup of automorphisms of Γ, if the stabiliser Gα of a vertex α acts primitively on the set Γ(α) of vertices of Γ adjacent to α. For a finite non-abelian simple group L and a Cayley subset S of L, suppose that LG≤Aut(L), and the Cayley graph Γ=Cay (L, S) is G-locally primitive. In this paper we prove that L is a simple group of Lie type, and either the valency of Γ is an add prine divisor of ｜Out(L)｜, or L=PΩ+8 (q) and Γ has valency 4. In either cases, it is proved that the full automorphism group of Γ is also almost simple with the same socle L.
文摘Given a simple undirected graph G. We denote the sets of its vertices and edges by V（G） and E（G） respectively. G is called a （P, P—K） graph （or G=（P, P—K）） if |V（G）|=P and |E（G）|=|V（G）|—K. Let {G1, G2} be a pair of graphs of the same order. If G1 is isomorphic to a subgraph of （?）2, where （?）2 is the com-
文摘Let G be a finite group and H a nonempty subset of G.We call H Cayley subset if the identity element e of G is not in H.For each Cayley subset H we define the Cayley digraph X=X(G, H)
文摘Let G be a finite group and S a nonempty subset of G. We call S a Cayley subset if the identity element e of G is not in S. For each Cayley subset S, we define the Cayley digraph X=X( G, S)
基金the National Natural Science Foundation of China (Grant Nos. 19831050 and69873002) and the Doctoral Program Foundation of Institutions of Higher Education of China (Grant No. 97000141) , and also by Korea Science and Engineering Foundation (Grant No. K
文摘Let G be a finite group, and S a subset of G1 with S=S-1. We use X=Cay(G,S) to denote the Cayley graph of G with respect to S. We call S a CI-subset of G, if for any isomorphism Cay(G,S)Cay(G,T) there is an α∈Aut(G) such that Sα=T. Assume that m is a positive integer. G is called an m-CI-group if every subset S of G with S=S-1 and |S|≤m is CI. In this paper we prove that the alternating group A5 is a 4-CI-group, which was a conjecture posed by Li and Praeger.
