Metric dimension of fullerene graphs
Abstract
A resolving set W is a set of vertices of a graph G(V, E) such that for every pair of distinct vertices u, v ∈ V(G), there exists a vertex w ∈ W satisfying d(u, w) ≠ d(v, w). A resolving set with minimum number of vertices is called metric basis of G. The metric dimension of G, denoted by dim(G), is the minimum cardinality of a resolving set of G. In this paper, we consider (3, 6)-fullerene and (4, 6)-fullerene graphs and compute the metric dimension for these fullerene graphs. We also give conjecture on the metric dimension of (3, 6)-fullerene and (4, 6)-fullerene graphs.
Keywords
Full Text:
PDFDOI: http://dx.doi.org/10.5614/ejgta.2019.7.1.7
References
A.R. Ashrafi and Z. Mehranian, Topological study of (3, 6)− and (4, 6)−fullerenes, Topo- logical Modelling of Nanostructures and Extended Systems, Springer Netherlands, (2013), 487–510.
R.C. Brigham, G. Chartrand, R.D. Dutton and P. Zhang, Resolving domination in graphs, Math. Bohem. 128 (1) (2003), 25–36.
G. Chartrand, L. Eroh, M.A. Johnson and O.R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math. 105 (2000), 99–113.
G. Chartrand, C. Poisson and P. Zhang, Resolvability and the upper dimension of graphs, Comput. Math. Appl. 39 (2000), 19–28.
G. Chartrand, E. Salehi and P. Zhang, The partition dimension of a graph, Aequationes Math. 59 (2000), 45–54.
P.W. Fowler and D.E. Manolopoulos, An Atlas of Fullerenes, Oxford Univ Press Oxford, (1995).
F. Harary and R.A. Melter, On the metric dimension of a graph, Ars Combin. 2 (1976), 191– 195.
H.W. Kroto, J.R. Heath, S.C. OBrien, R.F. Curl and R.E. Smalley, C60: buckminsterfullerene, Nature 318 (1985), 162–163.
S. Khuller, B. Raghavachari and A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math. 70 (3) (1996), 217–229.
F. Koorepazan-Moftakhar, A.R. Ashrafi and Z. Mehranian, Automorphism group and fixing number of (3, 6)- and (4, 6)−fullerene graphs, Electron. Notes Discrete Math. 45 (2014), 113–120.
F. Okamoto, B. Phinezyn and P. Zhang, The local metric dimension of a graph, Math. Bohem. 135 (3) (2010), 239–255.
B. Rajan, I. Rajasingh, M.C. Monica and P. Manuel, Metric dimension of enhanced hyper- cube networks, J. Combin. Math. Combin. Comput. 67 (2008), 5–15.
A. Sebo ̈ and E. Tannier, On metric generators of graphs, Math. Oper. Res. 29 (2) (2004), 383–393.
H.M.A. Siddiqui and M. Imran, Computation of metric dimension and partition dimension of Nanotubes, Journal of Computational and Theoretical Nanoscience 12 (2) (2015), 199–203.
H.M.A. Siddiqui and M. Imran, Computing metric and partition dimension of 2−Dimensional lattices of certain Nanotubes, Journal of Computational and Theoretical Nanoscience 11 (12) (2014), 2419–2423.
P.J. Slater, Leaves of trees, Congr. Numer. 14 (1975), 549–559.
Refbacks
- There are currently no refbacks.
ISSN: 2338-2287
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.