Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

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

• 83 Accesses

• 1 Citations

## 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.

This is a preview of subscription content, log in to check access.

## 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 $$A_{n, n-1} \equiv S_n.$$

## References

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

## 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.

## Author information

Correspondence to Zhizhang Shen.

## Rights and permissions

Reprints and Permissions

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