Sanming Zhou - On 4-valent Frobenius circulant graphs

dmtcs:582 - Discrete Mathematics & Theoretical Computer Science, October 31, 2012, Vol. 14 no. 2 - https://doi.org/10.46298/dmtcs.582
On 4-valent Frobenius circulant graphsArticle

Authors: Sanming Zhou ORCID1

  • 1 Department of Mathematics and Statistics [Melbourne]

A 4-valent first-kind Frobenius circulant graph is a connected Cayley graph DLn(1, h) = Cay(Zn, H) on the additive group of integers modulo n, where each prime factor of n is congruent to 1 modulo 4 and H = {[1], [h], −[1], −[h]} with h a solution to the congruence equation x 2 + 1 ≡ 0 (mod n). In [A. Thomson and S. Zhou, Frobenius circulant graphs of valency four, J. Austral. Math. Soc. 85 (2008), 269-282] it was proved that such graphs admit 'perfect ' routing and gossiping schemes in some sense, making them attractive candidates for modelling interconnection networks. In the present paper we prove that DLn(1, h) has the smallest possible broadcasting time, namely its diameter plus two, and we explicitly give an optimal broadcasting in DLn(1, h). Using number theory we prove that it is possible to recursively construct larger 4-valent first-kind Frobenius circulants from smaller ones, and we give a methodology for such a construction. These and existing results suggest that, among all 4-valent circulant graphs, 4-valent first-kind Frobenius circulants are extremely efficient in terms of routing, gossiping, broadcasting and recursive construction.


Volume: Vol. 14 no. 2
Section: Graph Theory
Published on: October 31, 2012
Accepted on: June 9, 2015
Submitted on: August 17, 2011
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Expander graphs, isoperimetric numbers, and forwarding indices; Code: FT110100629

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