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Some Resolving Parameters in a Class of Cayley Graphs

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  • Jia-Bao Liu
  • Ali Zafari

Abstract

Resolving parameters are a fundamental area of combinatorics with applications not only to many branches of combinatorics but also to other sciences. In this study, we construct a class of Toeplitz graphs and will be denoted by T2n(W) so that they are Cayley graphs. First, we review some of the features of this class of graphs. In fact, this class of graphs is vertex transitive, and by calculating the spectrum of the adjacency matrix related with them, we show that this class of graphs cannot be edge transitive. Moreover, we show that this class of graphs cannot be distance regular, and because of the difficulty of the computing resolving parameters of a class of graphs which are not distance regular, we regard this as justification for our focus on some resolving parameters. In particular, we determine the minimal resolving set, doubly resolving set, and strong metric dimension for this class of graphs.

Suggested Citation

  • Jia-Bao Liu & Ali Zafari, 2022. "Some Resolving Parameters in a Class of Cayley Graphs," Journal of Mathematics, John Wiley & Sons, vol. 2022(1).
    • Handle: RePEc:wly:jjmath:v:2022:y:2022:i:1:n:9444579
    DOI: 10.1155/2022/9444579
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    References listed on IDEAS

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    1. András Sebő & Eric Tannier, 2004. "On Metric Generators of Graphs," Mathematics of Operations Research, INFORMS, vol. 29(2), pages 383-393, May.
    2. Jia-Bao Liu & Ali Zafari, 2020. "Computing Minimal Doubly Resolving Sets and the Strong Metric Dimension of the Layer Sun Graph and the Line Graph of the Layer Sun Graph," Complexity, Hindawi, vol. 2020, pages 1-8, September.
    3. Jia-Bao Liu & Ali Zafari & Hassan Zarei, 2020. "Metric Dimension, Minimal Doubly Resolving Sets, and the Strong Metric Dimension for Jellyfish Graph and Cocktail Party Graph," Complexity, Hindawi, vol. 2020, pages 1-7, May.
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