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Regular graphs and stability

Published online by Cambridge University Press:  09 April 2009

D. A. Holton  and
Douglas D. Grant
D. A. Holton
Affiliation:
University of MelbourneParkville Victoria 3052, Australia.
Douglas D. Grant
Affiliation:
University of MelbourneParkville Victoria 3052, Australia.
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Abstract

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We show that a graph G is semi-stable at vertex v if and only if the set of vertices of G adjacent to v is fixed by the automorphism group of Gv, the subgraph of G obtained by deleting v and its incident edges. This result leads to a neat proof that regular graphs are semi-stable at each vertex. We then investigate stable regular graphs, concentrating mainly on stable vertex-transitive graphs. We conjecture that if G is a non-trivial vertex-transitive graph, then G is stable if and only if γ(G) contains a transposition, offering some evidence for its truth.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 20 , Issue 3 , November 1975 , pp. 377 - 384
Copyright
Copyright © Australian Mathematical Society 1975

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