ENRU  
Volume 22, No 6, 2015, P. 29–42 UDC 519.176
Keywords: undirected circulant graph, Degree/Diameter problem, network design, genetic algorithm. DOI: 10.17377/daio.2015.22.509 Emilia A. Monakhova ^{1} Received 8 September 2015 References[1] V. V. Korneev, About macrostructure of homogeneous computing systems, in Vychislitel’nye sistemy (Computing Systems), Vol. 60, pp. 17–34, Izd. Inst. Mat., Novosibirsk, 1974.[2] E. A. Monakhova, Synthesis of optimal Diophantine structures, in Vychislitel’nye sistemy (Computing Systems), Vol. 80, pp. 18–35, Izd. Inst. Mat., Novosibirsk, 1979. [3] E. A. Monakhova, On an extremal family of circulant networks, Diskretn. Anal. Issled. Oper., 18, No. 1, 77–84, 2011. Translated in J. Appl. Ind. Math., 5, No. 4, 595–600, 2011. [4] E. A. Monakhova, Structural and communicative properties of circulant networks, Prikl. Diskretn. Mat., No. 3, 92–115, 2011. [5] E. A. Monakhova, A new attainable lower bound on the number of nodes in quadruple circulant networks, Diskretn. Anal. Issled. Oper., 20, No. 1, 37–44, 2013. [6] E. A. Monakhova, On synthesis of multidimensional circulant graphs of diameter two, Izv. TPU, 323, No. 2, 25–28, 2013. [7] J. C. Bermond, F. Comellas, and D. F. Hsu, Distributed loop computernetworks: A survey, J. Parallel Distrib. Comput., 24, No. 1, 2–10, 1995. [8] D. Bevan, G. Erskine, and R. Lewis, Large circulant graphs of fixed diameter and arbitrary degree, 2015 (Cornell Univ. Libr. ePrint Archive, arXiv:1506.04962). [9] S. Chen and X.D. Jia, Undirected loop networks, Networks, 23, No. 4, 257–260, 1993. [10] R. Dougherty and V. Faber, The degreediameter problem for several varieties of Cayley graphs I: The Abelian case, SIAM J. Discrete Math., 17, No. 3, 478–519, 2004. [11] B. Elspas, Topological constraints on interconnectionlimited logic, Proc. 5th Annu. Symp. Switch. Circuit Theory Log. Des., Princeton, NJ, USA, Nov. 11–13, 1964, pp. 133–137, IEEE, New York, 1964. [12] B. Elspas and J. Turner, Graphs with circulant adjacency matrices, J. Comb. Theory, 9, No. 3, 297–307, 1970. [13] R. FeriaPurrón, J. Ryan, and H. PérezRosés, Searching for large multiloop networks, Electron. Notes Discrete Math., 46, 233–240, 2014. [14] R. FeriaPurrón, H. PérezRosés, and J. Ryan, Searching for large circulant graphs, 2015 (Cornell Univ. Libr. ePrint Archive, arXiv:1503.07357). [15] F. K. Hwang, A survey on multiloop networks, Theor. Comput. Sci., 299, No. 1–3, 107–121, 2003. [16] R. R. Lewis, The degreediameter problem for circulant graphs of degree 8 and 9, Electron. J. Comb., 24, No. 4, P4.50, 1–19, 2014. [17] R. R. Lewis, Improved upper bounds for the order of some classes of Abelian Cayley and circulant graphs of diameter two, 2015 (Cornell Univ. Libr. ePrint Archive, arXiv:1506.02844). [18] H. Macbeth, J. Siagiová and J. Sirán, Cayley graphs of given degree and diameter for cyclic, Abelian, and metacyclic groups, Discrete Math., 312, No. 1, 94–99, 2012. [19] M. Miller and J. Sirán, Moore graphs and beyond: A survey of the degree/diameter problem, Electron. J. Comb., Dyn. Surv., DS14, 1–61, 2005. [20] E. A. Monakhova, Optimal triple loop networks with given transmission delay: Topological design and routing, Proc. Int. Network Optim. Conf., Évry/Paris, France, Oct. 27–29, 2003, pp. 410–415, INT, Paris, 2003. [21] E. A. Monakhova, A survey on undirected circulant graphs, Discrete Math. Algorithms Appl., 4, No. 1, 1250002, 1–30, 2012. [22] E. A. Monakhova, O. G. Monakhov, and E. V. Mukhoed, Genetic construction of optimal circulant network designs, in Evolutionary Image Analysis, Signal Processing and Telecommunications (Proc. 1st Eur. Workshops EvoIASP’99 EuroEcTel’99, Göteborg, Sweden, May 26–27, 1999), pp. 215–223, Springer, Heidelberg, 1999 (Lect. Notes Comput. Sci., Vol. 1596). [23] F. P. Muga II, Maximal order of 3 and 5regular circulant graphs,Matimyás Mat., 22, No. 3, 33–38, 1999. [24] H. PérezRosés, Algebraic and computerbased methods in the undirected degree/diameter problem  A brief survey, Electron. J. Graph Theory Appl., 2, No. 2, 166–190, 2014. [25] C. R. Reeves, Genetic algorithms for the operations researcher, INFORMS J. Comput., 9, No. 3, 231–250, 1997. [26] The Degree/Diameter Problem, in Combinatorics Wiki. Available at http://combinatoricswiki.org/wiki/The_Degree/Diameter_Problem. Accessed Nov. 2, 2015. [27] The Degree Diameter Problem for Circulant Graphs. in Combinatorics Wiki. Available at http://combinatoricswiki.org/wiki/The_Degree_Diameter_Problem_for_Circulant_Graphs. Accessed Nov. 2, 2015. [28] C. K. Wong and Don Coppersmith, A combinatorial problem related to multimodule memory organizations, J. Assoc. Comput. Mach., 21, No. 3, 392–402, 1974. [29] C. K. Wong and T. W. Maddocks, A generalized Pascal’s triangle, Fibonacci Q., 13, 134–136, 1975. 

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