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Generalized Recursive Circulant Graphs
January 2012 (vol. 23 no. 1)
pp. 87-93
Chien-Yi Li, Dept. of Inf. Manage., Nat. Taiwan Univ. of Sci. & Technol., Taipei, Taiwan
Yue-Li Wang, Dept. of Inf. Manage., Nat. Taiwan Univ. of Sci. & Technol., Taipei, Taiwan
Shyue-Ming Tang, Fu Hsing Kang Sch., Nat. Defense Univ., Taipei, Taiwan
In this paper, we propose a new class of graphs called generalized recursive circulant graphs which is an extension of recursive circulant graphs. While retaining attractive properties of recursive circulant graphs, the new class of graphs achieve more flexibility in varying the number of vertices. Some network properties of recursive circulant graphs, like degree, connectivity and diameter, are adapted to the new graph class with more concise expression. In particular, we use a multidimensional vertex labeling scheme in generalized recursive circulant graphs. Based on the labeling scheme, a shortest path routing algorithm for the graph class is proposed. The correctness of the routing algorithm is also proved in this paper.

[1] S.B. Akers and B. Krishnamurthy, “A Group-Theoretic Model for Symmetric Interconnection Networks,” IEEE Trans. Computers, vol. 38, no. 4, pp. 555-566, Apr. 1989.
[2] B. Alspach, S. Locke, and D. Witte, “The Hamilton Spaces of Cayley Graphs on Abelian Groups,” Discrete Math., vol. 82, pp. 113-126, 1990.
[3] T. Araki, “Edge-Pancyclicity of Recursive Circulants,” Information Processing Letters, vol. 88, pp. 287-292, 2003.
[4] J.-C. Bermond, F. Comellas, and D.-F. Hsu, “Distributed Loop Computer Networks: A Survey,” J. Parallel and Distributed Computing, vol. 24, pp. 2-10, 1995.
[5] N. Biggs, Algebraic Graph Theory, second ed. Cambridge Univ. Press, 1993.
[6] D.K. Biss, “Hamiltonian Decomposition of Recursive Circulant Graphs,” Discrete Math., vol. 214, pp. 89-99, 2000.
[7] F. Boesch and A. Felzer, “A General Class of Invulnerable Graphs,” Networks, vol. 2, pp. 261-283, 1972.
[8] F.T. Boesch and R. Tindell, “Circulants and Their Connectivities,” J. Graph Theory, vol. 8, pp. 487-499, 1984.
[9] Z.R. Bogdanowicz, “Pancyclicity of Connected Circulant Graphs,” J. Graph Theory, vol. 22, pp. 167-174, 1996.
[10] F. Buckley and F. Harary, Distance in Graphs, pp. 73-75. Addison-Wesley, 1990.
[11] I. Chung, “Construction of a Parallel and Shortest Routing Algorithm on Recursive Circulant Networks,” Proc. Fourth Int'l Conf. High Performance Computing in the Asia-Pacific Region, pp. 580-585, 2000.
[12] B. Elspas and J. Turner, “Graphs with Circulant Adjacency Matrices,” J. Combinatorial Theory, vol. 9, pp. 297-307, 1970.
[13] C. Kim, J. Choi, and H.-S. Lim, “Embedding Full Ternary Trees into Recursive Circulants,” Proc. First EurAsian Conf. Information and Comm. Technology, pp. 874-882, 2002.
[14] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. M. Kaufmann, 1992.
[15] H.-S. Lim, J.-H. Park, and K.-Y. Chwa, “Embedding Trees in Recursive Circulants,” Discrete Applied Math., vol. 69, pp. 83-99, 1996.
[16] B. Mans, “Optimal Distributed Algorithms in Unlabeled Tori and Chordal Rings,” J. Parallel and Distributed Computing, vol. 46, pp. 80-90, 1997.
[17] C. Micheneau, “Disjoint Hamiltonian Cycles in Recursive Circulant Graphs,” Information Processing Letters, vol. 61, pp. 259-264, 1997.
[18] M. Muzychuk, M. Klin, and R. Pöschel, “The Isomorphism Problem for Circulant Graphs via Schur Ring Theory. DIMACS Ser.,” Discrete Math. and Theoretical Computer Science, vol. 56, pp. 241-264, 2001.
[19] J.-H. Park and K.-Y. Chwa, “Recursive Circulant: A New Topology for Multicomputer Networks,” Proc. Int'l Symp. Parallel Architectures, Algorithms and Networks (ISPAN '94), pp. 73-80, 1994.
[20] J.-H. Park and K.-Y. Chwa, “Recursive Circulants and Their Embeddings among Hypercubes,” Theoretical Computer Science, vol. 244, pp. 35-62, 2000.
[21] J.-H. Park, “Strong Hamiltonicity of Recursive Circulants,” J. Korean Information Science Soc., vol. 28, pp. 742-744, 2001.
[22] I. Stojmenović, “Multiplicative Circulant Networks: Topological Properties and Communication Algorithms,” Discrete Applied Math., vol. 77, pp. 281-305, 1997.
[23] C.-H. Tsai, J.J.-M. Tan, Y.-C. Chuang, and L.-H. Hsu, “Hamiltonian Properties of Faulty Recursive Circulant Graphs,” J. Interconnection Networks, vol. 3, pp. 273-289, 2002.
[24] C.-H. Tsai, J.J.-M. Tan, and L.-H. Hsu, “The Super-Connected Property of Recursive Circulant Graphs,” Information Processing Letters, vol. 91, pp. 293-298, 2004.
[25] J.-S. Yang, J.-M. Chang, S.-M. Tang, and Y.-L. Wang, “On the Independent Spanning Trees of Recursive Circulant Graphs $G(cd^m, d)$ with $d > 2$ ,” Theoretical Computer Science, vol. 410, pp. 2001-2010, 2009.
[26] J.-S. Yang, J.-M. Chang, S.-M. Tang, and Y.-L. Wang, “Constructing Multiple Independent Spanning Trees on Recursive Circulant Graphs $G(2^m, 2)$ ,” Int'l J. Foundations of Computer Science, vol. 21, pp. 73-90, 2010.

Index Terms:
graph theory,routing algorithm,generalized recursive circulant graph,network properties,graph class,multidimensional vertex labeling scheme,shortest path routing,Routing,Labeling,Indexes,Electronic mail,Cities and towns,Transforms,Hypercubes,bipartite graphs.,Circulant graphs,recursive circulant graphs,generalized recursive circulant graphs,shortest path,routing algorithms,diameter
Chien-Yi Li, Yue-Li Wang, Shyue-Ming Tang, "Generalized Recursive Circulant Graphs," IEEE Transactions on Parallel and Distributed Systems, vol. 23, no. 1, pp. 87-93, Jan. 2012, doi:10.1109/TPDS.2011.109
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