Some Results on Prime and Distinct Prime Distance Labeling of Graphs
Parthiban A.1, Samdanielthompson G.2, Ajaz Ahmad Pir3, Ihsan Ul Haq4
1Parthiban A, Department of Mathematics, Lovely Professional University, Phagwara (Punjab), India.
2Samdanielthompson G, Department of Mathematics, Hindustan College of Arts and Science, Padur, Kelambakkam, Chennai (Tamil Nadu), India.
3Ajaz Ahmad Pir, Department of Mathematics, Lovely Professional University, Phagwara (Punjab), India.
4Ihsan Ul Haq, Department of Mathematics, Lovely Professional University, Phagwara (Punjab), India.
Manuscript received on 25 November 2019 | Revised Manuscript received on 19 December 2019 | Manuscript Published on 30 December 2019 | PP: 1007-1009 | Volume-9 Issue-1S4 December 2019 | Retrieval Number: A12161291S419/19©BEIESP | DOI: 10.35940/ijeat.A1216.1291S419
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© The Authors. Blue Eyes Intelligence Engineering and Sciences Publication (BEIESP). This is an open access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Abstract: Let be the set of all integers. A graph is a prime distance graph if there exists an injective function such that for any two adjacent vertices and , the integer is a prime. So is a prime distance graph if and only if there exists a prime distance labeling of . If the edge labels of are also distinct, then is called a distinct prime distance labeling of and is called a distinct prime distance graph. The generalized Petersen graphs are defined to be a graph on vertices with and , subscripts modulo . In this paper, we show that the generalized Petersen graphs permit a prime distance labeling for all even and conjecture that and admit a prime distance labeling for any and all odd , respectively. We also prove that the cycle admits a distinct prime distance labeling for all , besides establishing the prime distance labeling for some graphs.
Keywords: Prime Distance Graphs, Prime Distance Labeling, Distinct Prime Distance Labeling, Generalized Petersen Graphs.
Scope of the Article: Cryptography and Applied Mathematics