Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 592 https://internationalpubls.com Prime Labeling of Bull Graph Dr. M. Ganeshan Assistant Professor, PG and Research Department of Mathematics, Agurchand Manmull jain college, University of Madras, Tamilnadu, chennai India. Email: sivananthan21oct@gmail.com Article History: Received: 27-09-2024 Revised: 02-11-2024 Accepted: 14-11-2024 Abstract: Let G be a graph. A bijection f:V → {1,2, … . . |V|} is called a prime labeling [3] if for each edge e = uv in E, we have GCD{ f(u), f(v)} = 1. A graph that admits a prime labeling is said to be a prime graph. In this paper we show that bull graph admits Prime labeling in the context of variety graph operations namely duplication of vertex, fusion of vertices and Switching in Bull graph. Keywords: Prime labeling, Bull graph, Duplication, Fusion and Switching. 1. INTRODUCTION Graph labeling is one of the stimulating areas with plentiful applications in various fields. In this paper we consider simple and finite graphs only. The notion of prime labeling was introduced by Roger Entringer and was discussed in a paper by A. Tout (1982 P 365-368). This paper is organized as follows. In section 2 we provide the preliminary definitions. In section 3, we prove the main results of the paper, where we prove the graph obtained by duplicating arbitrary vertex of bull graph is a Prime graph, The graph obtained by Switching of any vertex in a bull graph is a Prime graph and we also prove that in a bull graph fusion of any arbitrary vertex with 𝑣1 produces a Prime graph In section 4, we conclude the paper and also provide the insight for future work. For number theory concept refer [2]. 2.PRELIMINARY DEFINITIONS Definition [7]-2.1. Duplication of a vertex vi of a graph G produces a new graph G1 by adding a vertex vi ′ with N(vi ′) = N(vi) . In other words, a vertex vi ′ is said to be a duplication of vi if all the vertices adjacent to vi are now adjacent to vi ′ also. Definition [7]-2.2. Let u and v be two distinct vertices of a graph G. A new graph G1 is constructed by fusing two vertices u and v by a single vertex w such that every edge incident to u and v is now incident with w in G1. Definition [7] -2.3. A vertex switching Gu in a graph G is obtained by taking a vertex u of G, removing all the edges incident to u and adding edges joining u to every non-adjacent vertex of u in G. Definition [5], -2.4. The Bull graph is a graph with 5 vertices and 5 edges consisting of a triangle with two disjoint pendant edges. 3.MAIN RESULTS Theorem-3.1. The graph obtained by duplicating arbitrary vertex of bull graph is a Prime graph. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 593 https://internationalpubls.com Proof: Figure - 1. Bull graph Case-1. Duplication of the vertex 𝑣1 Let 𝐺1 be the graph obtained by duplicating the vertex 𝑣1 Define ℬ: 𝑉( 𝐺1) → {1,2,3, … . . ,6} by ℬ(𝑣𝑖) = 𝑖 + 1 , 1 ≤ 𝑖 ≤ 5 and ℬ(𝑣1 ′ ) = 1 Evidently all the vertex labels are distinct For edges in 𝐺1 G.C.D ( ℬ(𝑣𝑖 ), ℬ(𝑣𝑖+1)) = 1, 1 ≤ 𝑖 ≤ 4 G.C.D ( ℬ(𝑣2), ℬ(𝑣4)) = 1 G.C.D ( ℬ(𝑣1 ′ ), ℬ(𝑣2)) = 1 Clearly ℬ is a prime labeling on 𝐺1.Hence 𝐺1 is a prime graph. Figure - 2. Prime labeling of duplication of vertex 𝑣1 in Bull graph Case-2. Duplication of the vertex 𝑣2 Let 𝐺2 be the graph obtained by duplicating the vertex 𝑣2 Define ℬ: 𝑉( 𝐺2) → {1,2,3, … . . ,6} by ℬ(𝑣𝑖) = 𝑖 + 1 , 1 ≤ 𝑖 ≤ 5 and ℬ(𝑣2 ′ ) = 1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 594 https://internationalpubls.com clearly all the vertex labels are distinct For edges in 𝐺2 G.C.D ( ℬ(𝑣𝑖 ), ℬ(𝑣𝑖+1)) = 1, 1 ≤ 𝑖 ≤ 4 G.C.D ( ℬ(𝑣2), ℬ(𝑣4)) = 1 G.C.D ( ℬ(𝑣2 ′ ), ℬ(𝑣1)) = 1 G.C.D ( ℬ(𝑣2 ′ ), ℬ(𝑣3)) = 1 G.C.D ( ℬ(𝑣2 ′ ), ℬ(𝑣4)) = 1 Therefore ℬ is a prime labeling on 𝐺2. Hence 𝐺2 is a prime graph. Figure - 3. Prime labeling of duplication of vertex 𝑣2 in Bull graph Case-3. Duplication of the vertex 𝑣3 Let 𝐺3 be the graph obtained by duplicating the vertex 𝑣3 Define ℬ: 𝑉( 𝐺3) → {1,2,3, … . . ,6} by ℬ(𝑣𝑖) = 𝑖 + 1, 1 ≤ 𝑖 ≤ 5 and ℬ(𝑣3 ′ ) = 1 clearly all the vertex labels are distinct For edges in 𝐺3 G.C.D ( ℬ(𝑣𝑖 ), ℬ(𝑣𝑖+1)) = 1, 1 ≤ 𝑖 ≤ 4 G.C.D ( ℬ(𝑣2), ℬ(𝑣4)) = 1 G.C.D ( ℬ(𝑣3 ′ ), ℬ(𝑣2)) = 1 G.C.D ( ℬ(𝑣3 ′ ), ℬ(𝑣4)) = 1 Thus ℬ is a prime labeling on 𝐺3. Hence 𝐺3 is a prime graph. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 595 https://internationalpubls.com Figure - 4. Prime labeling of duplication of vertex 𝑣3 in Bull graph Case-4. Duplication of the vertex 𝑣4 Let 𝐺4 be the graph obtained by duplicating the vertex 𝑣4 Define ℬ: 𝑉( 𝐺4) → {1,2,3, … . . ,6} by ℬ(𝑣𝑖) = 𝑖 + 1, 1 ≤ 𝑖 ≤ 5 and ℬ(𝑣4 ′ ) = 1 obviously all the vertex labels are distinct For edges in 𝐺4 G.C.D ( ℬ(𝑣𝑖 ), ℬ(𝑣𝑖+1)) = 1, 1 ≤ 𝑖 ≤ 4 G.C.D ( ℬ(𝑣2), ℬ(𝑣4)) = 1 G.C.D ( ℬ(𝑣4 ′ ), ℬ(𝑣2)) = 1 G.C.D ( ℬ(𝑣4 ′ ), ℬ(𝑣3)) = 1 G.C.D ( ℬ(𝑣4 ′ ), ℬ(𝑣5)) = 1 Clearly ℬ is a prime labeling on 𝐺4. Hence 𝐺4 is a prime graph. Figure - 5. Prime labeling of duplication of vertex 𝑣4 in Bull graph Case-5. Duplication of the vertex 𝑣5 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 596 https://internationalpubls.com Let 𝐺5 be the graph obtained by duplicating the vertex 𝑣5 Define ℬ: 𝑉( 𝐺5) → {1,2,3, … . . ,6} by ℬ(𝑣𝑖) = 𝑖 + 1 , 1 ≤ 𝑖 ≤ 5 and ℬ(𝑣5 ′ ) = 1 Evidently all the vertex labels are distinct For edges in 𝐺5 G.C.D ( ℬ(𝑣𝑖 ), ℬ(𝑣𝑖+1)) = 1, 1 ≤ 𝑖 ≤ 4 G.C.D ( ℬ(𝑣2), ℬ(𝑣4)) = 1 G.C.D ( ℬ(𝑣5 ′ ), ℬ(𝑣4)) = 1 Clearly ℬ is a prime labeling on 𝐺5. Hence 𝐺5 is a prime graph. Figure - 6. Prime labeling of duplication of vertex 𝑣5 of Bull graph Thus, in all the cases the graph obtained by duplication of any arbitrary vertex of bull graph is a Prime graph. Theorem-3.2. The graph obtained by Switching of any vertex in a bull graph is a Prime graph. Proof. Figure - 7. Bull graph Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 597 https://internationalpubls.com Case-1. switching the vertex 𝑣1 Let 𝐺1 be the graph obtained by switching the vertex 𝑣1 Define ℘: 𝑉( 𝐺1) → {1,2,3, … . . ,5} by ℘(𝑣1) = 1, ℘(𝑣2) = 5, ℘(𝑣3) = 4, ℘(𝑣4) = 3, ℘(𝑣5) = 2 Evidently all the vertex labels are distinct For edges in 𝐺1 G.C.D ( ℘(𝑣𝑖 ), ℘(𝑣𝑖+1)) = 1, 2 ≤ 𝑖 ≤ 4 G.C.D ( ℘(𝑣2), ℘(𝑣4)) = 1 G.C.D ( ℘(𝑣1), ℘(𝑣3)) = 1 G.C.D ( ℘(𝑣1), ℘(𝑣4)) = 1 G.C.D ( ℘(𝑣1), ℘(𝑣5)) = 1 Thus ℘ is a prime labeling on 𝐺1. Hence 𝐺1 is a prime graph. Figure - 8. Prime labeling of switching of vertex 𝑣1 in Bull graph Case-2. switching the vertex 𝑣2 Let 𝐺2 be the graph obtained by switching the vertex 𝑣2 Define ℘: 𝑉( 𝐺2) → {1,2,3, … . . ,5} by ℘(𝑣1) = 1, ℘(𝑣2) = 5, ℘(𝑣3) = 4, ℘(𝑣4) = 3, ℘(𝑣5) = 2 clearly all the vertex labels are distinct For edges in 𝐺2 G.C.D ( ℘(𝑣𝑖 ), ℘(𝑣𝑖+1)) = 1, 3 ≤ 𝑖 ≤ 4 G.C.D ( ℘(𝑣2), ℘(𝑣5)) = 1 Hence ℘ is a prime labeling on 𝐺2. Thus 𝐺2 is a prime graph. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 598 https://internationalpubls.com Figure - 9. Prime labeling of switching of vertex 𝑣2 in Bull graph Case-3. switching the vertex 𝑣3 Let 𝐺3 be the graph obtained by switching the vertex 𝑣3 Define ℘: 𝑉( 𝐺3) → {1,2,3, … . . ,5} by ℘(𝑣1) = 5, ℘(𝑣2) = 4, ℘(𝑣3) = 1, ℘(𝑣4) = 3, ℘(𝑣5) = 2 Visibly all the vertex labels are distinct For edges in 𝐺3 G.C.D ( ℘(𝑣1), ℘(𝑣2)) = 1 G.C.D ( ℘(𝑣1), ℘(𝑣3)) = 1 G.C.D ( ℘(𝑣2), ℘(𝑣4)) = 1 G.C.D ( ℘(𝑣4), ℘(𝑣5)) = 1 G.C.D ( ℘(𝑣3), ℘(𝑣5)) = 1 Therefore ℘ is a prime labeling on 𝐺3. Hence 𝐺3 is a prime graph. Figure - 10. Prime labeling of switching of vertex 𝑣3 in Bull graph Case-4. switching the vertex 𝑣4 Let 𝐺4 be the graph obtained by switching the vertex 𝑣4 Define ℘: 𝑉( 𝐺4) → {1,2,3, … . . ,5} by ℘(𝑣1) = 1, ℘(𝑣2) = 5, ℘(𝑣3) = 4, ℘(𝑣4) = 3, ℘(𝑣5) = 2 Clearly all the vertex labels are distinct Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 599 https://internationalpubls.com For edges in 𝐺4 G.C.D ( ℘(𝑣𝑖 ), ℘(𝑣𝑖+1)) = 1, 1 ≤ 𝑖 ≤ 2 G.C.D ( ℘(𝑣1), ℘(𝑣4)) = 1 Hence ℘ is a prime labeling on 𝐺4. Therefore 𝐺4 is a prime graph Figure - 11. Prime labeling of switching of vertex 𝑣4 in Bull graph Case-5. switching the vertex 𝑣5 Let 𝐺5 be the graph obtained by switching the vertex 𝑣5 Define ℘: 𝑉( 𝐺5) → {1,2,3, … . . ,5} by ℘(𝑣1) = 2, ℘(𝑣2) = 3, ℘(𝑣3) = 4, ℘(𝑣4) = 5, ℘(𝑣5) = 1 Visibly all the vertex labels are distinct For edges in 𝐺5 G.C.D ( ℘(𝑣𝑖 ), ℘(𝑣𝑖+1)) = 1, 1 ≤ 𝑖 ≤ 3 G.C.D ( ℘(𝑣2), ℘(𝑣4)) = 1 G.C.D ( ℘(𝑣1), ℘(𝑣5)) = 1 G.C.D ( ℘(𝑣2), ℘(𝑣5)) = 1 G.C.D ( ℘(𝑣3), ℘(𝑣5)) = 1 Hence ℘ is a prime labeling on 𝐺5. So 𝐺5 is a prime graph Figure - 12. Prime labeling of switching of vertex 𝑣5 in Bull graph Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 600 https://internationalpubls.com Thus, in all the cases the graph obtained by Switching of any arbitrary vertex of bull graph is a Prime graph. Theorem-3.3. In a bull graph fusion of any arbitrary vertex with 𝑣1 produces a Prime graph. Proof. Figure - 13. Bull graph Case-1. Fusion of 𝑣2 with 𝑣1 Let 𝐺1 be the graph obtained by fusion of 𝑣2 with 𝑣1 Define 𝒰: 𝑉( 𝐺1) → {1,2,3,4} by 𝒰(𝑣1 = 𝑣2) = 1, 𝒰(𝑣3) = 2, 𝒰(𝑣4) = 3, 𝒰(𝑣5) = 4 Evidently all the vertex labels are distinct For edges in 𝐺1 G.C.D ( 𝒰(𝑣𝑖 ), 𝒰(𝑣𝑖+1)) = 1, 3 ≤ 𝑖 ≤ 4 G.C.D ( 𝒰(𝑣1 = 𝑣2 ), 𝒰(𝑣3)) = 1 G.C.D ( 𝒰(𝑣1 = 𝑣2 ), 𝒰(𝑣4)) = 1 Hence 𝒰 is a prime labeling on 𝐺1. So 𝐺1 is a prime graph Figure - 14 . Prime labeling of fusion of vertices 𝑣2 with 𝑣1 in Bull graph Case-2. Fusion of 𝑣3 with 𝑣1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 601 https://internationalpubls.com Let 𝐺2 be the graph obtained by fusion of 𝑣3 with 𝑣1 Define 𝒰: 𝑉( 𝐺1) → {1,2,3,4} by 𝒰(𝑣1 = 𝑣3) = 1, 𝒰(𝑣2) = 2, 𝒰(𝑣4) = 3, 𝒰(𝑣5) = 4 Clearly all the vertex labels are distinct For edges in 𝐺2 G.C.D ( 𝒰(𝑣1 = 𝑣3 ), 𝒰(𝑣2)) = 1 G.C.D ( 𝒰(𝑣1 = 𝑣3 ), 𝒰(𝑣4)) = 1 G.C.D ( 𝒰(𝑣2 ), 𝒰(𝑣4)) = 1 G.C.D ( 𝒰(𝑣4 ), 𝒰(𝑣5)) = 1 Hence 𝒰 is a prime labeling on 𝐺2. Therefore 𝐺2 is a prime graph Figure - 15 . Prime labeling of fusion of vertices 𝑣3 with 𝑣1 in Bull graph Case-3. Fusion of 𝑣4 with 𝑣1 Let 𝐺3 be the graph obtained by fusion of 𝑣4 with 𝑣1 Define 𝒰: 𝑉( 𝐺1) → {1,2,3,4} by 𝒰(𝑣1 = 𝑣4 ) = 1, 𝒰(𝑣2) = 2, 𝒰(𝑣3) = 3, 𝒰(𝑣5) = 4 Evidently all the vertex labels are distinct For edges in 𝐺3 G.C.D ( 𝒰(𝑣1 = 𝑣4 ), 𝒰(𝑣2)) = 1 G.C.D ( 𝒰(𝑣1 = 𝑣4 ), 𝒰(𝑣3)) = 1 G.C.D ( 𝒰(𝑣1 = 𝑣4 ), 𝒰(𝑣5)) = 1 G.C.D ( 𝒰(𝑣2), 𝒰(𝑣3)) = 1 Hence 𝒰 is a prime labeling on 𝐺3. So 𝐺3 is a prime graph Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 602 https://internationalpubls.com Figure - 16 . Prime labeling of fusion of vertices 𝑣4 with 𝑣1 in Bull graph Case-4. Fusion of 𝑣5 with 𝑣1 Let 𝐺4 be the graph obtained by fusion of 𝑣5 with 𝑣1 Define 𝒰: 𝑉( 𝐺1) → {1,2,3,4} by 𝒰(𝑣1 = 𝑣5) = 4, 𝒰(𝑣2) = 3, 𝒰(𝑣3) = 2, 𝒰(𝑣4) = 1 Obviously all the vertex labels are distinct For edges in 𝐺4 G.C.D ( 𝒰(𝑣2), 𝒰(𝑣3)) = 1 G.C.D ( 𝒰(𝑣2), 𝒰(𝑣4)) = 1 G.C.D ( 𝒰(𝑣3), 𝒰(𝑣4)) = 1 G.C.D ( 𝒰(𝑣1 = 𝑣5 ), 𝒰(𝑣2)) = 1 G.C.D ( 𝒰(𝑣1 = 𝑣5 ), 𝒰(𝑣4)) = 1 Thus 𝒰 is a prime labeling on 𝐺4. Hence 𝐺4 is a prime graph. Figure - 17. Prime labeling of fusion of vertices 𝑣5 with 𝑣1 in Bull graph Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 603 https://internationalpubls.com Thus, in all the cases the graph obtained by fusion of any arbitrary vertex to 𝑣1 of bull graph is a Prime graph. 4.CONCLUSION AND FUTURE WORK In this paper we have proved that bull graph admits Prime labeling in the context of graph operations namely duplication, fusion and switching. There exist many such graphs that admit Prime labeling. 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