Finite locally-quasiprimitive graphs

Cai-Heng Li, Cheryl Praeger, A. Venkatesh, S. Zhou

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26 Citations (Web of Science)


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.
Original languageEnglish
Pages (from-to)197-218
JournalDiscrete Mathematics
Publication statusPublished - 2002


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