## Frobenius circulant graphs of valency four

##### Author

Thomson, Alison; ZHOU, SANMING##### Date

2008##### Source Title

Journal of the Australian Mathematical Society##### Publisher

Cambridge University Press##### Affiliation

Department of Mathematics and Statistics##### Metadata

Show full item record##### Document Type

Journal Article##### Citations

Thomson, A., & Zhou, S. (2008). Frobenius circulant graphs of valency four. Journal of the Australian Mathematical Society, 85(2), 269-282.##### Access Status

**Open Access**

##### Description

© 2008 Australian Mathematical Society. Online edition of the journal is available at http://journals.cambridge.org/JAZ

##### Abstract

A first kind Frobenius graph is a Cayley graph Cay.K; S/ on the Frobenius kernel of a Frobenius group K o H such that S D aH for some a 2 K with haH i D 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 graph; circulant graph; multi-loop network; double-loop network; Frobenius group; Frobenius graph; complete rotation; gossiping; minimum gossip time; routing; edgeforwarding; index; arc-forwarding indexExport Reference in RIS Format

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