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FROBENIUS CIRCULANT GRAPHS OF VALENCY FOUR

Part of: Graph theory Computer system organization Operations research and management science

Published online by Cambridge University Press:  01 October 2008

ALISON THOMSON  and
SANMING ZHOU
ALISON THOMSON
Affiliation:
Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia (email: a.thomson@ms.unimelb.edu.au)
SANMING ZHOU*
Affiliation:
Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia (email: smzhou@ms.unimelb.edu.au)
*
✉For correspondence; e-mail: smzhou@ms.unimelb.edu.au
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Abstract

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A first kind Frobenius graph is a Cayley graph Cay(K,S) on the Frobenius kernel of a Frobenius group such that S=aH for some a∈K with 〈aH〉=K, where H is of even order or a is an involution. It is known that such graphs admit ‘perfect’ routing and gossiping schemes. A circulant graph is a Cayley graph on a cyclic group of order at least three. Since circulant graphs are widely used as models for interconnection networks, it is thus highly desirable to characterize those which are Frobenius of the first kind. In this paper we first give such a characterization for connected 4-valent circulant graphs, and then describe optimal routing and gossiping schemes for those which are first kind Frobenius graphs. Examples of such graphs include the 4-valent circulant graph with a given diameter and maximum possible order.

Keywords

Cayley graphcirculant graphmulti-loop networkdouble-loop networkFrobenius groupFrobenius graphcomplete rotationgossipingminimum gossip timeroutingedge-forwarding indexarc-forwarding index

MSC classification

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 68M10: Network design and communication 90B18: Communication networks
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 2 , October 2008 , pp. 269 - 282
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

Alison Thomson was supported by a Postgraduate Award of the Australian Government. Sanming Zhou was supported by a Discovery Project Grant of the Australian Research Council.

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