AIMS Mathematics, 2018, 3(3): 343-352. doi: 10.3934/Math.2018.3.343.

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On partition dimension of fullerene graphs

School of Natural Sciences, National University of Sciences and Technology, H-12 Islamabad,Pakistan

Let $G=(V(G),E(G))$ be a connected graph and $\Pi=\{S_{1},S_2,\dots,S_{k}\}$ be a $k$-partition of $V(G)$.The representation $r(v|\Pi)$ of a vertex $v$ with respect to $\Pi$ is the vector $(d(v,S_{1}),d(v,S_2),\dots,d(v,S_{k}))$, where $d(v,S_{i})=\min\{d(v,s_{i})\mid s_{i}\in S_{i}\}$.The partition $\Pi$ is called a resolving partition of $G$ if $r(u|\Pi)\neq r(v|\Pi)$ for all distinct $u,v\in V(G)$.The partition dimension of $G$, denoted by $pd(G)$, is the cardinality of a minimum resolving partition of $G$.In this paper, we calculate the partition dimension of two $(4,6)$-fullerene graphs. We also give conjectures on the partition dimension of two $(3,6)$-fullerene graphs.
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Keywords partition dimension; fullerene graphs

Citation: Naila Mehreen, Rashid Farooq, Shehnaz Akhter. On partition dimension of fullerene graphs. AIMS Mathematics, 2018, 3(3): 343-352. doi: 10.3934/Math.2018.3.343

References

  • 1. A. R. Ashrafi, Z. Mehranian, Topological study of (3; 6)- and (4; 6)-fullerenes, In: topological modelling of nanostructures and extended systems, Springer Netherlands, (2013), 487–510.
  • 2. G. Chartrand, L. Eroh, M. A. Johnson, et al. Resolvability in graphs and the metric dimension of a graph, Disc. Appl. Math., 105 (2000), 99–113.
  • 3. G. Chartrand, E. Salehi, P. Zhang, The partition dimension of a graph, Aequationes Math., 59 (2000), 45–54.
  • 4. G. Chartrand, E. Salehi, P. Zhang, On the partition dimension of a graph, Congr. Numer., 131 (1998), 55–66.
  • 5. C. Grigorious, S. Stephen, B. Rajan, et al. On the partition dimension of circulant graphs, The Computer Journal, 60 (2016), 180–184.
  • 6. F. Harary, R. A. Melter, On the metric dimension of a graph, Ars Combin., 2 (1976), 191–195.
  • 7. I. Javaid, N. K. Raja, M. Salman, et al. The partition dimension of circulant graphs, World Applied Sciences Journal, 18 (2012), 1705–1717.
  • 8. F. Koorepazan-Moftakhar, A. R. Ashrafi, Z. Mehranian, Automorphism group and fixing number of (3; 6) and (4; 6)-fullerene graphs, Electron. Notes Discrete Math., 45 (2014), 113–120.
  • 9. H. W. Kroto, J. R. Heath, S. C. O'Brien, et al. C60: buckminsterfullerene, Nature, 318 (1985), 162–163.
  • 10. R. A. Melter, I. Tomescu, Metric bases in digital geometry, Computer vision, graphics, and image Processing, 25 (1984), 113–121.
  • 11. H. M. A. Siddiqui, M. Imran, Computation of metric dimension and partition dimension of Nanotubes, J. Comput. Theor. Nanosci., 12 (2015), 199–203.
  • 12. H. M. A. Siddiqui, M. Imran, Computing metric and partition dimension of 2-Dimensional lattices of certain Nanotubes, J. Comput. Theor. Nanosci., 11 (2014), 2419–2423.
  • 13. P. J. Slater, of trees, Congress. Numer., 14 (1975), 549–559.
  • 14. J. A. Rodr´ıguez-Vel´azquez, I. G. Yero, M. Lemanska, On the partition dimension of trees, Disc. Appl. Math., 166 (2014), 204–209.
  • 15. J. A. Rodr´ıguez-Vel´azquez, I. G. Yero, H. Fernau, On the partition dimension of unicyclic graphs, Bull. Math. Soc. Sci. Math. Roumanie, 57 (2014), 381–391.
  • 16. I. Tomescu, I. J. Slamin, On the partition dimension and connected partition dimension of wheels, Ars Combin., 84 (2007), 311–317.
  • 17. I. Tomescu, Discrepancies between metric dimension and partition dimension of a connected graph, Disc. Math., 308 (2008), 5026–5031.

 

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