Research article

On partition dimension of fullerene graphs

  • Received: 18 April 2018 Accepted: 17 July 2018 Published: 24 July 2018
  • 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.

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

    Related Papers:

  • 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.


    加载中
    [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. Rodríguez-Velázquez, I. G. Yero, M. Lemanska, On the partition dimension of trees, Disc. Appl. Math., 166 (2014), 204–209.
    [15] J. A. Rodríguez-Velázquez, 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.
  • Reader Comments
  • © 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3531) PDF downloads(834) Cited by(32)

Article outline

Figures and Tables

Figures(4)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog