Consider a connected simple graph with vertex set X of diameter d. Define Ri X2 by (x, y) Ri whenever x and y have graph distance

Primitive and imprimitive graphs

Distance-Regular Graphs

The base size of a permutation group, and the metric dimension of a graph, are two of a number of related parameters of groups, graphs, coherent configurations and association schemes. They have been repeatedly re-defined with different terminology in various different areas, including computational group theory and the graph isomorphism problem. We survey results on these parameters in their many incarnations, and propose a consistent terminology for them. We also present some new results, including on the base sizes of wreath products in the product action, and on the metric dimension of Johnson and Kneser graphs.

background

/pdf/base-size-metric-dimension-and-other-invariants-of-groups-ndmpltrqpe.pdf

Base size, metric dimension and other invariants of groups and graphs

We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms G and are either locally (G, s)-arc transitive for s > 2 or G-locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of G. Given a normal subgroup N which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of N preserves both local primitivity and local s-arc transitivity and leads us to study graphs where G acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for G in these two cases and give new constructions of examples for each possible type. The analysis raises several open problems which are discussed in the final section.

/pdf/analysing-finite-locally-s-arc-transitive-graphs-3vyblmoapc.pdf

Analysing Finite Locally s-arc Transitive Graphs

The theory of vertex-transitive graphs has developed in parallel with the theory of transitive permutation groups. The chapter explores the way the two theories have influenced each other. Examples are drawn from the enumeration of vertex-transitive graphs of small order, the classification problem for finite distance transitive graphs, and the investigations of finite 2-arc transitive graphs, finite primitive and quasiprimitive permutation groups, and finite locally primitive graphs. The nature of the group theoretic techniques used range from elementary ones to some involving the finite simple group classification. In particular the theorem of O’Nan and Scott for finite primitive permutation groups, and a generalisation of it for finite quasiprimitive permutation groups is discussed.

/pdf/finite-transitive-permutation-groups-and-bipartite-vertex-4alj43rm5h.pdf

Primitive and imprimitive graphs

Citations

Distance-Regular Graphs

Base size, metric dimension and other invariants of groups and graphs

Analysing Finite Locally s-arc Transitive Graphs

Finite Transitive Permutation Groups and Bipartite Vertex-Transitive Graphs

Automorphisms of graphs and coverings

Related Papers (5)

Distance-Regular Graphs

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