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

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Abstract

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. 1
Fig. 2

Notes

  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 \(A_{n, n-1} \equiv S_n.\)

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Acknowledgments

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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Keywords

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