TY - JOUR

T1 - Finite locally-quasiprimitive graphs

AU - Li, Cai-Heng

AU - Praeger, Cheryl

AU - Venkatesh, A.

AU - Zhou, S.

PY - 2002

Y1 - 2002

N2 - A finite graph is said to be locally-quasiprimitive relative to a subgroup G of automorphisms if, for all vertices alpha, the stabiliser in G of alpha is quasiprimitive on the set of vertices adjacent to alpha, (A permutation group is said to be quasiprimitive if all of its non-trivial normal subgroups are transitive.) The graph theoretic condition of local quasiprimitivity is strictly weaker than the conditions of local primitivity and 2-arc transitivity which have been studied previously. It is shown that the family of locally-quasiprimitive graphs is closed under the formation of a certain kind of quotient graph, called a normal quotient, induced by a normal subgroup. Moreover, a locally-quasiprimitive graph is proved to be a multicover of each of its normal quotients. Thus finite locally-quasiprimitive graphs which are minimal in the sense that they have no non-trivial proper normal quotients form an important sub-family, since each finite locally-quasiprimitive graph has at least one such graph as a normal quotient. These minimal graphs in the family are called "basic" locally-quasiprimitive graphs, and their structure is analysed. The process of constructing locally-quasiprimitive graphs with a given locally-quasiprimitive graph Sigma as a normal quotient is then considered. It turns out that this can be viewed as a problem of constructing covering graphs of certain multigraphs associated with Sigma. Further, it is shown that, under certain conditions, a locally-quasiprimitive graph can be reconstructed from knowledge of two of its normal quotients. Finally a series of open problems is presented. (C) 2002 Elsevier Science B.V. All rights reserved.

AB - A finite graph is said to be locally-quasiprimitive relative to a subgroup G of automorphisms if, for all vertices alpha, the stabiliser in G of alpha is quasiprimitive on the set of vertices adjacent to alpha, (A permutation group is said to be quasiprimitive if all of its non-trivial normal subgroups are transitive.) The graph theoretic condition of local quasiprimitivity is strictly weaker than the conditions of local primitivity and 2-arc transitivity which have been studied previously. It is shown that the family of locally-quasiprimitive graphs is closed under the formation of a certain kind of quotient graph, called a normal quotient, induced by a normal subgroup. Moreover, a locally-quasiprimitive graph is proved to be a multicover of each of its normal quotients. Thus finite locally-quasiprimitive graphs which are minimal in the sense that they have no non-trivial proper normal quotients form an important sub-family, since each finite locally-quasiprimitive graph has at least one such graph as a normal quotient. These minimal graphs in the family are called "basic" locally-quasiprimitive graphs, and their structure is analysed. The process of constructing locally-quasiprimitive graphs with a given locally-quasiprimitive graph Sigma as a normal quotient is then considered. It turns out that this can be viewed as a problem of constructing covering graphs of certain multigraphs associated with Sigma. Further, it is shown that, under certain conditions, a locally-quasiprimitive graph can be reconstructed from knowledge of two of its normal quotients. Finally a series of open problems is presented. (C) 2002 Elsevier Science B.V. All rights reserved.

U2 - 10.1016/S0012-365X(01)00258-8

DO - 10.1016/S0012-365X(01)00258-8

M3 - Article

SN - 0012-365X

VL - 246

SP - 197

EP - 218

JO - Discrete Mathematics

JF - Discrete Mathematics

ER -