## Abstract

Let G be a connected graph and d(x,y) be the distance between the vertices x and y. A subset of vertices W = {w_{1},w_{2},...,w_{k}} is called a resolving set for G if for every two distinct vertices x,y ∈ V(G), there is a vertex w_{i} ∈ W such that d(x,w_{i}) ≠ d(y,w_{i}). A resolving set containing a minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension denoted by dim(G). Let F be a family of connected graphs G_{n}: F = (G_{n})_{n≥1} depending on n as follows: the order | V(G) | = φ(n) and lim n→ ∞ φ(n) = ∞. If there exists a constant C > 0 such that dim(G) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension; otherwise F has unbounded metric dimension. If all graphs in F have the same metric dimension (which does not depend on n), then F is called a family with constant metric dimension. In this paper, we study the metric dimension of some zig-zag chain graphs generated by regular hexagons. We determine the exact value of their metric dimension and prove that these zig-zag chain graphs have unbounded metric dimension. It is an interesting and classical problem to classify the graphs families with respect to the nature of their metric dimension.

Original language | English |
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Pages (from-to) | 411-421 |

Number of pages | 11 |

Journal | Utilitas Mathematica |

Volume | 100 |

Publication status | Published - 2016 |

Externally published | Yes |

## Keywords

- Basis
- Metric dimension
- Resolving set
- Unbounded
- Zig-zag chain graphs

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty
- Applied Mathematics