In this paper, we describe the canonical partial order on the idempotent set of the strong endomorphism monoid of a graph, and using this we further characterize primitive idem potenes from the viewpoint of combinator...In this paper, we describe the canonical partial order on the idempotent set of the strong endomorphism monoid of a graph, and using this we further characterize primitive idem potenes from the viewpoint of combinatorics. The number of them is also given.展开更多
Let G be a p-group (p odd prime) and let X = Cay(G, S) be a 4-valent connected Cayley graph. It is shown that if G has nilpotent class 2, then the automorphism group Ant(X) of X is isomorphic to the semidirect product...Let G be a p-group (p odd prime) and let X = Cay(G, S) be a 4-valent connected Cayley graph. It is shown that if G has nilpotent class 2, then the automorphism group Ant(X) of X is isomorphic to the semidirect product GR x Ant(G,S), where GR is the right regular representation of G and Aut(G,S) is the subgroup of the automorphism group Aut(G) of G which fixes S setwise. However the result is not true if G has nilpotent class 3 and this paper provides a counterexample.展开更多
In this paper, the half-strong, the locally strong and the quasi-strong endomorphisms of a split graph are investigated. Let X be a split graph and let End（X）, hEnd（X）, 1End（X） and qEnd（X） be the endomorphism ...In this paper, the half-strong, the locally strong and the quasi-strong endomorphisms of a split graph are investigated. Let X be a split graph and let End（X）, hEnd（X）, 1End（X） and qEnd（X） be the endomorphism monoid, the set of all half-strong endomorphisms, the set of all locally strong endomorphisms and the set of all quasi-strong endomorphisms of X, respectively. The conditions under which hEnd（X） forms a submonoid of End（X） are given. It is shown that 1End（X） = qEnd（X） for any split graph X. The conditions under which 1End（X） （resp. qEnd（X）） forms a submonoid of End（X） are also given. In particular, if hEnd（X） forms a monoid, then 1End（X） （resp. qEnd（X）） forms a monoid too.展开更多
A graph is called edge-transitive if its full automorphism group acts transitively on its edge set.In this paper,by using classification of finite simple groups,we classify tetravalent edge-transitive graphs of order ...A graph is called edge-transitive if its full automorphism group acts transitively on its edge set.In this paper,by using classification of finite simple groups,we classify tetravalent edge-transitive graphs of order p2q with p,q distinct odd primes.The result generalizes certain previous results.In particular,it shows that such graphs are normal Cayley graphs with only a few exceptions of small orders.展开更多
A graph is symmetric or 1-regular if its automorphism group is transitive or regular on the arc set of the graph, respectively. We classify the connected pentavalent symmetric graphs of order 2p^3 for each prime p. Al...A graph is symmetric or 1-regular if its automorphism group is transitive or regular on the arc set of the graph, respectively. We classify the connected pentavalent symmetric graphs of order 2p^3 for each prime p. All those symmetric graphs appear as normal Cayley graphs on some groups of order 2p^3 and their automorphism groups are determined. For p = 3, no connected pentavalent symmetric graphs of order 2p^3 exist. However, for p = 2 or 5, such symmetric graph exists uniquely in each case. For p 7, the connected pentavalent symmetric graphs of order 2p^3 are all regular covers of the dipole Dip5 with covering transposition groups of order p^3, and they consist of seven infinite families; six of them are 1-regular and exist if and only if 5 |(p- 1), while the other one is 1-transitive but not 1-regular and exists if and only if 5 |(p ± 1). In the seven infinite families, each graph is unique for a given order.展开更多
In this paper, we investigate semisymmetric graphs of order 6p2 and of prime valency. First, we give a classification of the quasiprimitive permutation groups of degree dividing 3p2, and then, on the basis of the clas...In this paper, we investigate semisymmetric graphs of order 6p2 and of prime valency. First, we give a classification of the quasiprimitive permutation groups of degree dividing 3p2, and then, on the basis of the classification result, we prove that, for primes k and p, a connected graph Γ of order 6p2 and valency k is semisymmetric if and only if k = 3 and either Γ is the Gray graph, or p ≡ 1 (mod 6) and Γ is isomorphic to one known graph.展开更多
A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. An infinite family of cubic 1-regular graphs was constructed in [7] as cyclic coverings of the three-dimensional Hypercube, and a...A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. An infinite family of cubic 1-regular graphs was constructed in [7] as cyclic coverings of the three-dimensional Hypercube, and a classification of all s-regular cyclic coverings of the complete bipartite graph of order 6 was given in [8] for each s ≥ 1, whose fibre-preserving automorphism subgroups act arc-transitively. In this paper, the authors classify all s-regular dihedral coverings of the complete graph of order 4 for each s ≥ 1, whose fibre-preserving automorphism subgroups act arc-transitively. As a result, a new infinite family of cubic 1-regular graphs is constructed.展开更多
文摘In this paper, we describe the canonical partial order on the idempotent set of the strong endomorphism monoid of a graph, and using this we further characterize primitive idem potenes from the viewpoint of combinatorics. The number of them is also given.
基金the National Natural Science Foundation of China (No.10071002) andCom2MaC-KOSEF.
文摘Let G be a p-group (p odd prime) and let X = Cay(G, S) be a 4-valent connected Cayley graph. It is shown that if G has nilpotent class 2, then the automorphism group Ant(X) of X is isomorphic to the semidirect product GR x Ant(G,S), where GR is the right regular representation of G and Aut(G,S) is the subgroup of the automorphism group Aut(G) of G which fixes S setwise. However the result is not true if G has nilpotent class 3 and this paper provides a counterexample.
基金supported by National Natural Science Foundation of China(Grant Nos. 10571077,10971086)
文摘In this paper, the half-strong, the locally strong and the quasi-strong endomorphisms of a split graph are investigated. Let X be a split graph and let End（X）, hEnd（X）, 1End（X） and qEnd（X） be the endomorphism monoid, the set of all half-strong endomorphisms, the set of all locally strong endomorphisms and the set of all quasi-strong endomorphisms of X, respectively. The conditions under which hEnd（X） forms a submonoid of End（X） are given. It is shown that 1End（X） = qEnd（X） for any split graph X. The conditions under which 1End（X） （resp. qEnd（X）） forms a submonoid of End（X） are also given. In particular, if hEnd（X） forms a monoid, then 1End（X） （resp. qEnd（X）） forms a monoid too.
基金supported by National Natural Science Foundation of China (Grant Nos.11071210 and 11171292)
文摘A graph is called edge-transitive if its full automorphism group acts transitively on its edge set.In this paper,by using classification of finite simple groups,we classify tetravalent edge-transitive graphs of order p2q with p,q distinct odd primes.The result generalizes certain previous results.In particular,it shows that such graphs are normal Cayley graphs with only a few exceptions of small orders.
基金supported by National Natural Science Foundation of China (Grant Nos. 11571035 and 11231008)
文摘A graph is symmetric or 1-regular if its automorphism group is transitive or regular on the arc set of the graph, respectively. We classify the connected pentavalent symmetric graphs of order 2p^3 for each prime p. All those symmetric graphs appear as normal Cayley graphs on some groups of order 2p^3 and their automorphism groups are determined. For p = 3, no connected pentavalent symmetric graphs of order 2p^3 exist. However, for p = 2 or 5, such symmetric graph exists uniquely in each case. For p 7, the connected pentavalent symmetric graphs of order 2p^3 are all regular covers of the dipole Dip5 with covering transposition groups of order p^3, and they consist of seven infinite families; six of them are 1-regular and exist if and only if 5 |(p- 1), while the other one is 1-transitive but not 1-regular and exists if and only if 5 |(p ± 1). In the seven infinite families, each graph is unique for a given order.
文摘In this paper, we investigate semisymmetric graphs of order 6p2 and of prime valency. First, we give a classification of the quasiprimitive permutation groups of degree dividing 3p2, and then, on the basis of the classification result, we prove that, for primes k and p, a connected graph Γ of order 6p2 and valency k is semisymmetric if and only if k = 3 and either Γ is the Gray graph, or p ≡ 1 (mod 6) and Γ is isomorphic to one known graph.
基金the Excellent Young Teachers Program of the Ministry of Education of Chinathe National Natural Science Foundation of China+1 种基金 the Scientific Research Foundation for the Returned Overseas Chinese Scholars the Ministry of Education of China and the Com2MaC-KOSEF in Korea.
文摘A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. An infinite family of cubic 1-regular graphs was constructed in [7] as cyclic coverings of the three-dimensional Hypercube, and a classification of all s-regular cyclic coverings of the complete bipartite graph of order 6 was given in [8] for each s ≥ 1, whose fibre-preserving automorphism subgroups act arc-transitively. In this paper, the authors classify all s-regular dihedral coverings of the complete graph of order 4 for each s ≥ 1, whose fibre-preserving automorphism subgroups act arc-transitively. As a result, a new infinite family of cubic 1-regular graphs is constructed.
