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Deriving length two path centered surface area for the arrangement graph: a generating function approach

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We discuss the notion of \(H\)-centered surface area of a graph \(G\), where \(H\) is a subgraph of \(G\), i.e., the number of vertices in \(G\) at a certain distance from \(H\), and focus on the special case when \(H\) is a length two path to derive an explicit formula for the length two path centered surface area of the general and scalable arrangement graph, following a generating function approach.

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Fig. 2


  1. 1.

    A cycle is trivial if it contains exactly one symbol, called a fixed point. It is non-trivial otherwise.

  2. 2.

    When 1 occurs in an external cycle, we need to consider an additional symbol, i.e., the external symbol associated with the external cycle where 1 occurs.

  3. 3.

    Notice that in this case, \(\mathcal {C}(u)\) contains at least four symbols, and at least 3 external cycles, thus, its distance to \(e_k\), and indeed, to \(p_2(v, w, x),\) is at least 1 by Eq. 1. To be more precise, although both \(E_2\) and \(E_3\) may be trivial, we can include them into the non-trivial \(g_{E}\) term and the \(b\) term of Eq. 1, since if they are trivial, the 1 as counted in the \(b\) term will be canceled out with the 1 in the \(g_{E}\) term, thus having no impact on the distance result. We also notice that the internal symbols as included in trivial internal cycles play no role in calculating the distance between \(u\) and \(e_k.\)

  4. 4.

    We notice that \(p_2^a,\) which swaps two internal symbols with one external symbol in \(A_{n, k},\,k \in [2, n-1],\) is the only viable length two path allowed for \(A_{n, n-1} \equiv S_n.\)


  1. 1.

    Akers SB, Krishnamurthy B (1989) A group theoretic model for symmetric interconnection networks. IEEE Trans Comput 38:555–566

  2. 2.

    Bae M, Bose B (1997) Resource placement in torus-based networks. IEEE Trans Comput 46:1083–1092

  3. 3.

    Benoumhani M (1996) On Whitney numbers of Dowling lattices. Discret Math 159:13–33

  4. 4.

    Cheng E, Qiu K, Shen Z (2009) A short note on the surface area of star graphs. Parallel Process Lett 19:19–22

  5. 5.

    Cheng E, Qiu K, Shen Z (2010) On deriving explicit formulas of the surface areas for the arrangement graphs and some of the related graphs. Int J Comput Math 87:2903–2914

  6. 6.

    Cheng E, Qiu K, Shen Z (2012) Length two path centered surface area for bipartite graphs, preprint. J Comb Math Comb Comput

  7. 7.

    Cheng E, Qiu K, Shen Z (2013) Length two path centered surface area for the arrangement graph. Int J Comput Math. doi:10.1080/00207160.2013.829215

  8. 8.

    Cheng E, Lipták L, Qiu K, Shen Z (2012) On deriving conditional diagnosability of interconnection networks. Inf Process Lett 112:674–677

  9. 9.

    Day K, Tripathi A (1992) Arrangement graphs: a class of generalized star graphs. Inf Process Lett 42:235–241

  10. 10.

    Fertin G, Raspaud A (2001) \(k\)-neighbourhood broadcasting. In: Proceedings of the 8th International Colloquium on Structural Information and Communication, Complexity (SIROCCO’01), pp 133–146

  11. 11.

    Flajolet P, Sedgewick R (2009) Analytic combinatorics. Cambridge University Press, UK

  12. 12.

    Graham R, Knuth D, Patashnik O (1989) Concrete mathematics. Addison-Wesley, Reading

  13. 13.

    Imani N, Sarbazi-Azad H, Akl SG (2009) Some combinatorial properties of the star graphs: the surface area and volume. Discret Math 309:560–569

  14. 14.

    Jwo JS, Lakshmivarahan S, Dhall SK (1993) A new class of interconnection networks based on the alternating graph. Networks 23:315–326

  15. 15.

    Lai P-L, Tan JJM, Chang C-P, Hsu L-H (2005) Conditional diagnosability measures for large multiprocessor systems. IEEE Trans Comput 54:165–175

  16. 16.

    Lin CK, Tan JM, Hsu LH, Cheng E, Lipták L (2008) Conditional diagnosability of Cayley graphs generated by transposition trees under the comparison diagnosis model. J Interconnect Netw 9:83–97

  17. 17.

    Maeng J, Malek M (1981) A comparison connection assignment for self-diagnosis of multiprocessor systems. In: Proceedings of the 11th International Symposium on Fault-Tolerant, Computing, pp 173–175

  18. 18.

    Portier F, Vaughan T (1990) Whitney numbers of the second kind for the star poset. Eur J Comb 11:277–288

  19. 19.

    Sampels M (2004) Vertex-symmetric generalized Moore graphs. Discret Appl Math 138:195–202

  20. 20.

    Sarbazi-Azad H, Ould-Khaoua M, Mackenzie L, Akl SG (2004) On the combinatorial properties of \(k\)- ary \(n\)- cubes. J Interconnect Netw 5:79–91

  21. 21.

    Stewart I (2012) A general technique to establish the asymptotic conditional diagnosability of interconnection networks. Theor Comput Sci 452:132–137

  22. 22.

    Wang L, Subrammanian S, Latifi S, Srimani PK (2006) Distance distribution of nodes in star graphs. Appl Math Lett 19:780–784

  23. 23.

    Wilf H (1994) Generatingfunctionology, 2nd edn. Academic Press, San Diego

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We would like to thank three anonymous referees for a number of helpful comments and suggestions which led to an improved presentation of this paper.

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Correspondence to Zhizhang Shen.

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Cheng, E., Qiu, K. & Shen, Z. Deriving length two path centered surface area for the arrangement graph: a generating function approach. J Supercomput 68, 1241–1264 (2014). https://doi.org/10.1007/s11227-014-1085-1

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  • Length two path centered surface area
  • Arrangement graph
  • Generating function
  • Interconnection networks
  • Parallel computing