Honam Mathematical Journal (호남수학학술지)

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DISTINGUISHING NUMBER AND DISTINGUISHING INDEX OF STRONG PRODUCT OF TWO GRAPHS

  • Received : 2019.01.28
  • Accepted : 2020.10.21
  • Published : 2020.12.25
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Abstract

The distinguishing number (index) D(G) (D'(G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. The strong product G ☒ H of two graphs G and H is the graph with vertex set V (G) × V (H) and edge set {{(x1, x2),(y1, y2)}|xiyi ∈ E(Gi) or xi = yi for each 1 ≤ i ≤ 2.}. In this paper we study the distinguishing number and the distinguishing index of strong product of two graphs. We prove that for every k ≥ 2, the k-th strong power of a connected S-thin graph G has distinguishing index equal two.

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Acknowledgement

The authors would like to express their gratitude to the referees for their careful reading and helpful comments.

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