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CORES OF SYMMETRIC GRAPHS

Part of: Designs and configurations Graph theory

Published online by Cambridge University Press:  01 October 2008

PETER J. CAMERON  and
PRISCILA A. KAZANIDIS
PETER J. CAMERON*
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK (email: P.J.Cameron@qmul.ac.uk)
PRISCILA A. KAZANIDIS
Affiliation:
Institute of Mathematics, College of Science, University of the Philippines Diliman, Quezon City 1101, Philippines (email: cristy@math.upd.edu.ph)
*
✉For correspondence; e-mail: P.J.Cameron@qmul.ac.uk
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Abstract

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The core of a graph Γ is the smallest graph Δ that is homomorphically equivalent to Γ (that is, there exist homomorphisms in both directions). The core of Γ is unique up to isomorphism and is an induced subgraph of Γ. We give a construction in some sense dual to the core. The hull of a graph Γ is a graph containing Γ as a spanning subgraph, admitting all the endomorphisms of Γ, and having as core a complete graph of the same order as the core of Γ. This construction is related to the notion of a synchronizing permutation group, which arises in semigroup theory; we provide some more insight by characterizing these permutation groups in terms of graphs. It is known that the core of a vertex-transitive graph is vertex-transitive. In some cases we can make stronger statements: for example, if Γ is a non-edge-transitive graph, we show that either the core of Γ is complete, or Γ is its own core. Rank-three graphs are non-edge-transitive. We examine some families of these to decide which of the two alternatives for the core actually holds. We will see that this question is very difficult, being equivalent in some cases to unsolved questions in finite geometry (for example, about spreads, ovoids and partitions into ovoids in polar spaces).

Keywords

graph homomorphismcore graphrank three graph

MSC classification

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 05B25: Finite geometries
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 2 , October 2008 , pp. 145 - 154
Copyright
Copyright © 2008 Australian Mathematical Society

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