Skip to main content
SpringerLink
Log in

The Problem of a Symmetric Graph with a Maximum Number of Vertices and Minimum Diameter

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

The paper gives a solution for the problem of the topology of the communication subsystem graph for high-performance multi-core computing systems. In this graph, each vertex is connected to four neighbors, and the number of vertices is the maximum for a given graph diameter. The solution to this problem is the family of circulants \(C(2D(D+1)+1;1,2D+1)\), \(D\) is diameter. This graph is invariant under transforming its arbitrary vertex into any other, and its vertices are located densely in the vicinity of the root vertex, which determines its compliance with the diameter optimality criterion. All statements formulated during the solution of the problem are proven.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

REFERENCES

  1. L. Benini and L. G. De Micheli, ‘‘Networks on chips: A new SoC paradigm,’’ Computer 35, 70–78 (2019). https://doi.org/10.1109/2.976921

    Article  Google Scholar 

  2. R. Marculescu and P. Bogdan, ‘‘The chip is the network: Toward a science of network-on-chip design,’’ Found. Trends Electron. Des. Autom. 2, 371–461 (2009). https://doi.org/10.1561/1000000011

    Article  Google Scholar 

  3. J. Kim, W. J. Dally, S. Scott, and D. Abts, ‘‘Technology-driven, highly-scalable dragonfly topology,’’ ACM SIGARCH Comput. Archit. News 36 (3), 77–88 (2008). https://doi.org/10.1145/1394608.1382129

    Article  Google Scholar 

  4. J. Kim, J. Balfour, and W. Dally, ‘‘Flattened butterfly topology for on-chip networks,’’ in MICRO 2007—Proceedings of the 40th Annual IEEE/ACM International Symposium on Microarchitecture (2007), pp. 172–182.

  5. M. N. M. Ali, M. M. H. Rahman, R. M. Nor, D. K. Behera, T. M. T. Sembok, Y. Miura, and Y. Inoguchi, ‘‘SCCN: A time-effective hierarchical interconnection network for network-on-chip,’’ Mob. Networks Appl. 24, 1255–1264 (2019). https://doi.org/10.1007/s11036-019-01262-2

    Article  Google Scholar 

  6. K. Tatas, K. Siozios, D. Soudris, and A. Jantsch, ‘‘The spidergon STNoC,’’ in Designing 2D and 3D Network-on-Chip Architectures (Springer, 2014), pp. 161–190.

    Book  Google Scholar 

  7. E. A. Monakhova, ‘‘A survey on undirected circulant graphs,’’ Discret. Math. Algorithms Appl. 04, 17–47 (2012). https://doi.org/10.1142/S1793830912500024

    Article  MathSciNet  Google Scholar 

  8. M. Oehlers and B. Fabian, ‘‘Graph metrics for network robustness—a survey,’’ Mathematics 9, 895 (2021). https://doi.org/10.3390/math9080895

    Article  Google Scholar 

  9. J. M. Hernandez and P. van Mieghem, Classification of Graph Metrics (Delft Univ. Technol., Mekelweg, 2011), pp. 1–20.

  10. A. Sukhov and A. Romanov, Serendipity: When Research in One Area Leads to a Positive Result in Another. https://cacm.acm.org/blogs/blog-cacm/270457-serendipity-when-research-in-one-area- leads-to-a-positive-result-in-another/fulltext. Accessed 2023.

  11. X. Huang, A. F. Ramos, and Y. Deng, ‘‘Optimal circulant graphs as low-latency network topologies,’’ J. Supercomput. 78, 13491–13510 (2022). https://doi.org/10.1007/s11227-022-04396-5

    Article  Google Scholar 

  12. A. Romanov, ‘‘The dataset for optimal circulant topologies,’’ Big Data Cogn. Comput. 7, 80 (2023). https://doi.org/10.3390/bdcc7020080

    Article  Google Scholar 

  13. R. R. Lewis, ‘‘Analysis and construction of extremal circulant and other abelian cayley graphs,’’ Ph.D. Thesis (Open Univ., 2021). https://doi.org/10.21954/ou.ro.00013612

  14. H. Perez-Roses, M. Bras-Amoros, and J. M. Serradilla-Merinero, ‘‘Greedy routing in circulant networks,’’ Graphs Comb. 38, 86 (2022). https://doi.org/10.1007/s00373-022-02489-9

    Article  MathSciNet  Google Scholar 

  15. T. Dobravec, B. Robic, and B. Vilfan, ‘‘Dynamic shortest path routing in 2-circulants,’’ in Proceedings of the International Symposium on Computer and Information Sciences (2022), pp. 332–336.

  16. B. Robic, ‘‘Optimal routing in 2-jump circulant networks,’’ Tech. Rep. N397 (1996).

  17. T. Dobravec, J. Zerovnik, and B. Robic, ‘‘An optimal message routing algorithm for circulant networks,’’ J. Syst. Archit. 52, 298–306 (2006). https://doi.org/10.1016/j.sysarc.2005.12.003

    Article  Google Scholar 

  18. J. Zerovnik, B. Robic, and T. Dobravec, ‘‘Optimal permutation routing in 2-jump circulant networks,’’ in Proceedings of the 1st International Conference on Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing SNPD (2020), pp. 175–180.

  19. D. Gomez, J. Gutierrez, and A. Ibeas, ‘‘Optimal routing in double loop networks,’’ Theor. Comput. Sci. 381, 68–8 (2007). https://doi.org/10.1016/j.tcs.2007.04.002

    Article  MathSciNet  Google Scholar 

  20. B.-X. Chen, J.-X. Meng, and W.-J. Xiao, ‘‘A constant time optimal routing algorithm for undirected double-loop networks,’’ Lect. Notes Comput. Sci. 3794, 308–316 (2005).

    Article  Google Scholar 

  21. P. K. Jha, ‘‘Dimension-order routing algorithms for a family of minimal-diameter circulants,’’ J. Interconnect. Networks 14, 1350002 (2013). https://doi.org/10.1142/S0219265913500023

  22. A. Y. Romanov, ‘‘Development of routing algorithms in networks-on-chip based on ring circulant topologies,’’ Heliyon 5, e01516 (2019). https://doi.org/10.1016/j.heliyon.2019.e01516

Download references

ACKNOWLEDGMENTS

Dedicated to the memory of Professor B.M. Bredikhin, he instilled in us a love of number theory.

Funding

The results of this research were obtained within the RSF grant (project no. 22-29-00979).

Author information

Authors and Affiliations

  1. Sevastopol State University, 299053, Sevastopol, Russia

    A. M. Sukhov

  2. National Research University Higher School of Economics, 123458, Moscow, Russia

    A. Y. Romanov & A. A. Amerikanov

Authors
  1. A. M. Sukhov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Y. Romanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. A. Amerikanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. M. Sukhov, A. Y. Romanov or A. A. Amerikanov.

Ethics declarations

The authors of this work declare that they have no conflicts of interest.

Additional information

Publisher’s Note.

Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

(Submitted by A. M. Elizarov)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sukhov, A.M., Romanov, A.Y. & Amerikanov, A.A. The Problem of a Symmetric Graph with a Maximum Number of Vertices and Minimum Diameter. Lobachevskii J Math 44, 5453–5459 (2023). https://doi.org/10.1134/S1995080223120351

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080223120351

Keywords:

Search

Navigation