#### Articles

# Prime labeling of special graphs

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**Authors:**

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M. D. M. C. P. Weerarathna ,

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University of Peradeniya, LK

##### About M. D. M. C. P.

Department of Mathematics, Faculty of Science

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T. R. D. S. M. Thennakoon,

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University of Peradeniya, LK

##### About T. R. D. S. M.

Department of Mathematics, Faculty of Science

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K. D. E. Dhananjaya,

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University of Peradeniya, LK

##### About K. D. E.

Department of Mathematics, Faculty of Science

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A. A. I. Perera

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University of Peradeniya, LK

##### About A. A. I.

Department of Mathematics, Faculty of Science

## Abstract

Prime labeling is the most interesting category of graph labeling with various applications. A graph *G= (V(G), E(G))* with |*v(G)*| vertices are said to have prime labeling if its vertices are labeled with distinct positive integers 1,2,3,……,|*v*| such that for each edge *uv **e E(G)* the labels assigned to *u* and *v* are relatively prime, where *V(G)* and *E(G) *are vertex set and edge set of *G*, respectively. Therefore, the graph *G* has a prime labeling whenever any of two adjacent vertices can be labeled as two relative prime numbers and is called a prime graph. In our work, we focus on the prime labeling method for newly constructed graphs obtained by replacing each edge of a star graph *K*_{1,n }by a complete tripartite graph *K*_{1,m,1} for *m* = 2,3,4, and 5, which are prime graphs. In addition to that, investigate another type of simple undirected finite graphs generalized by using circular ladder graphs. These new graphs obtained by attaching *K*_{1,2 }at each external vertex of the circular ladder graph *CL*_{n} and proved that the constructed graphs are prime graphs when *n *≥ 3 and *n **≠* 1 (*mod3*) . Finally, focus on another particular type of simple undirected finite graph called a scorpion graph, denoted by *S*(2*p*,2*q*,*r*) . The Scorpion graph gets its name from shape, which resembles a scorpion, having 2*p* + 2*q* + *r* vertices *p* ≥ 1, *q *≥ 2, *r* ≥ 2)are placed in the head, body, and tail respectively. To prove that the scorpion graph has prime labeling, we used two results that have already been proved for ladder graphs.
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How to Cite:
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Weerarathna, M.D.M.C.P., Thennakoon, T.R.D.S.M., Dhananjaya, K.D.E. and Perera, A.A.I., 2021. Prime labeling of special graphs. *Journal of Science*, 12(1), pp.1–12. DOI: http://doi.org/10.4038/jsc.v12i1.30
Published on
05 Aug 2021.

Peer Reviewed

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