×

Computing the upper bounds for the metric dimension of cellulose network. (English) Zbl 1431.05057

Summary: Let \(G=(V,E)\) be a connected graph and \(d(x,y)\) be the distance between the vertices \(x\) and \(y\) in \(G\). A set of vertices \(W\) resolves a graph \(G\) if every vertex is uniquely determined by its vector of distances to the vertices in \(W\). A metric dimension of \(G\) is the minimum cardinality of a resolving set of \(G\) and is denoted by \(\dim(G)\). In this paper we study three dimensional chemical structure of cellulose network and then we converted it into planar chemical structure, consequently we obtained cellulose network graphs denoted by \(CL^k_n\). We prove that \(\dim(CL^k_n)\leq 4\) in certain cases.

MSC:

05C12 Distance in graphs
05C62 Graph representations (geometric and intersection representations, etc.)
05C07 Vertex degrees
05C10 Planar graphs; geometric and topological aspects of graph theory
05C90 Applications of graph theory
05C92 Chemical graph theory
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
PDFBibTeX XMLCite
Full Text: Link

References:

[1] P. S. Buczkowski, G. Chartrand, C. Poisson and P. Zhang, On k-dimensional graphs and their bases, Perioddica Math. Hung., 46(2003), 9-15. · Zbl 1026.05033
[2] J. C. Bermound, F. Comellas and D. F. Hsu, Distributed loop computer networks: survey, J. Parallel Distrib. Comput., 24(1995), 2-10.
[3] P. J. Cameron and J. H. Van, Lint, Designs, Graphs, Codes and their Links, in London Mathematical Society Student Texts, vol.22, Cambridge University Press, Cambridge, 1991. · Zbl 0743.05004
[4] G. Chartrand, L. Eroh, M. A. Johnson and O. R. Oellermann, Resolvability in graphs and metric dimension of a graph, Disc. Appl. Math., 105(2000), 99-113. · Zbl 0958.05042
[5] T. Y. Feng, A survey of interconnection networks, IEEE Comput., (1981), 12-27.
[6] W. D. Hills, The Connection Machine. Cambridge, MA: MIT Press, (1985).
[7] K. Hwang and J. Ghosh, Hypernet: A Communication-Efficient, Architecture for constructing Massively parallel computers, IEEE Trans. on Comput., 36(1987), 1450-1466.
[8] M. Imran, A. Q. Baig, S. A. Bokhary and I. Javaid, On the metric dimension of circulant graphs, Applied Mathematics Letters., 25(2012), 320-325. · Zbl 1243.05072
[9] M. Imran, M. K. Siddiqui and R. Naeem, On the Metric Dimension of Generalized Petersen Multigraphs, IEEE Access, 6(2018), 74328-74338.
[10] S. Imran, M. K. Siddiqui, M. Imran, M. Hussain, H. M. Bilal and I. Z.Cheema, A. Tabraiz, Z. Saleem, Computing the metric dimension of gear graphs, Symmetry, 10(2018), 1-12. · Zbl 1423.05056
[11] S. Imran, M. K. Siddiqui, M. Imran and M. Hussain, On metric dimensions of symmetric graphs obtained by rooted product, Mathematics, 6(2018), 1-12. · Zbl 1515.05067
[12] S. Khuller and B. Raghavachari, A. Rosenfeld, Localization in Graphs, Technical report CS-TR-3326, University of Maryland at College Park, 1994. · Zbl 0865.68090
[13] P. Manuel, B. Rajan, I. Rajasingh and C. Monica, On minimum metric dimension of honeycom networks, Journal of Discrete Algorithms, 6(2008), 20-27. · Zbl 1159.05308
[14] A. Seb¨o and E. Tannier, On metric generators of graphs, Math. Oper. Res., 29(2004), 383-393. · Zbl 1082.05032
[15] P. J. Slater, Dominating and references sets in graphs, J. Math. Phys. Sci., 22(1988), 445-455. · Zbl 0656.05057
[16] H. M. A. Siddiqui and M. Imran, Computing the metric dimension of wheel related graphs, Appl. Math. Comp., 242(2014), 624-632. · Zbl 1334.05133
[17] P. J. Slater, Leaves of trees, Congress. Number, 14(1975), 549-559.
[18] I. Tomescu and M. Imran, metric dimension and R-Sets of connected graph Graphs and Combinatorics, 27(2011), 585-591. · Zbl 1235.05046
[19] I. Tomescu and I. Javaid, On the metric dimension of the jahangir graph, Bull. Math. Soc. Sci. Math. Roumanie, 50(2007), 371-376. · Zbl 1164.05019
[20] R. S. Wilkov, Analysis and design of reliable compute networks, IEEE Trans. on commun, 20(1972), 660-678.
[21] C. · Zbl 0444.94048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.