Abstract
Hyper-star graph HS(2n,n) was introduced to be a competitive model to both hypercubes and star graphs. In this paper, we study its properties by giving a closed form solution to the surface area of HS(2n,n) and discussing its Hamiltonicity by establishing an isomorphism between the graph and the well known middle levels problem. We also develop a single-port optimal neighbourhood broadcasting algorithm for HS(2n,n).
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Akers, S.B., Harel, D., Krishnamurthy, B.: The Star Graph: an Attractive Alternative to the n-Cube. In: Proceedings of the International Conference on Parallel Processing, pp. 393–400 (1987)
Akers, S.B., Krishnamurthy, B.: A Group Theoretic Model for Symmetric Interconnection Networks. IEEE Trans. on Compu. c-38 (4), 555–566 (1989)
Al-Ayyoub, A.-E., Day, K.: The hyperstar Interconnection Network. Journal of Parallel and Distributed Computing 48, 175–199 (1998)
Barry, P.: On Integer-Sequence-Based Constructions of Generalized Pascal Triangles. Journal of Integer Sequences 9, 1-34 (2006)
Bermond, J.C., Ferreira, A., Perennes, S., Peters, J.G.: Neighbourhood Broadcasting in Hypercubes. SIAM Journal on Discrete Mathematics 21(4), 823–843 (2007)
Cheng, E., Lipták, L.: Structural Properties of Hyper-stars. Ars Combinatoria 80, 65–73 (2006)
Cheng, E., Qiu, K., Shen, Z.: On the Surface Areas and Average Distances of Meshes and Tori. Parallel Processing Letters 21(1), 61–75 (2011)
Cheng, E., Qiu, K., Shen, Z.: On the Surface Area of the Augmented Cubes. Journal of Supercomputing 61, 856–868 (2012)
Cheng, E., Shah, M.: A Strong Structural Theorem for Hyper-stars. Congressus Numerantium 179, 181–191 (2006)
Cosnard, M., Ferreira, A.: On the Real Power of Loosely Coupled Parallel Architectures. Parallel Processing Letters 1(2), 103–111 (1991)
Fertin, G., Raspaud, A.: k-Neighbourhood Broadcasting. In: Proceedings of the 8th International Colloquium on Structural Information and Communication Complexity (SIROCCO 2001), pp. 133–146 (2001)
Fujita, S.: Optimal neighborhood Broadcast in Star Graphs. Journal of Interconnection Networks 4(4), 419–428 (2003)
Imani, N., Sarbazi-Azad, H., Akl, S.G.: Some Topological Properties of Star Graphs: the Surface Area and Volume. Discrete Mathematics 309(3), 560–569 (2009)
Johnson, S.L., Ho, C.T.: Optimal Broadcasting and Personalized Communication in Hypercubes. IEEE Trans. Computers 38(9), 1249–1268 (1989)
Kim, J.-S., Oh, E., Lee, H.-O., Heo, Y.-N.: Topological and Communication Aspects of Hyper-Star Graphs. In: Yazıcı, A., Şener, C. (eds.) ISCIS 2003. LNCS, vol. 2869, pp. 51–58. Springer, Heidelberg (2003)
Kim, J.S., Cheng, E., Lipták, L., Lee, H.O.: Embedding Hypercubes, Rings and Odd Graphs into Hyper-stars. International Journal of Computer Mathematics 86(5), 771–778 (2009)
Kim, J.S., Cheng, E., Lipták, L., Lee, H.O.: A Note on Embedding among Folded Hypercubes, Even Graphs and Odd Graphs. International Journal of Computer Mathematics 8(5), 882–891 (2011)
Kim, J.S., Kim, S.W., Cheng, E., Lipták, L.: Topological Properties of Folded Hyper-star Networks. Journal of Supercomputing 59, 1336–1347 (2012)
Lee, H.O., Kim, J.S., Oh, E., Lim, H.S.: Hyper-Star Graph: A New Interconnection Network Improving the Network Cost of the Hypercube. In: Shafazand, H., Tjoa, A.M. (eds.) EurAsia-ICT 2002. LNCS, vol. 2510, pp. 858–865. Springer, Heidelberg (2002)
Leighton, T.: Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. Morgan Kaufman, San Mateo (1992)
Lipták, L., Cheng, E., Kim, J.S., Kim, S.W.: One-to-Many Node Disjoint Paths of Hyper-Star Networks. Discrete Applied Mathematics 160, 2006–2014 (2012)
Madabhushi, S., Lakshmivarahan, S., Dhall, S.: Analysis of the Modified Even Networks. Technical Report, School of Electrical Engineering and Computer Science, University of Oklahoma (1990)
Portier, F., Vaughan, T.: Whitney Numbers of the Second Kind for the Star Poset. European Journal of Combinatorics 11(3), 277–288 (1990)
Qiu, K.: On a Unified Neighbourhood Broadcasting Scheme for Interconnection Networks. Parallel Processing Letters 17(4), 425–437 (2007)
Sampels, M.: Vertex-Symmetric Generalized Moore graphs. Discrete Applied Mathematics 138, 195–202 (2004)
Savage, C.D., Winkler, P.: Monotone Gray code and the Middle Levels Problem. Journal of Combinatorial Theory, Series A 70, 230–248 (1995)
Shields, I., Shields, B.J., Savage, C.D.: An Update on the Middle Levels Problem. Discrete Mathamatics 309, 5271–5277 (2009)
Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences, http://oeis.orgo
Wang, L., Subramanian, S., Latifi, S., Srimani, P.K.: Distance Distribution of Nodes in Star Graphs. Applied Mathematics Letters 19(8), 780–784 (2006)
West, D.B.: Revolving Door (Middle Levels) Conjecture, http://www.math.uiuc.edu/ west/openp/revolving
Yang, J.S., Chang, J.M.: Independent Spanning Trees on Folded Hyper-stars. Networks 56(4), 272–281 (2010)
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Zhang, F., Qiu, K., Kim, J.S. (2014). Hyper-Star Graphs: Some Topological Properties and an Optimal Neighbourhood Broadcasting Algorithm. In: Sun, Xh., et al. Algorithms and Architectures for Parallel Processing. ICA3PP 2014. Lecture Notes in Computer Science, vol 8630. Springer, Cham. https://doi.org/10.1007/978-3-319-11197-1_40
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DOI: https://doi.org/10.1007/978-3-319-11197-1_40
Publisher Name: Springer, Cham
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