Ratio Mathematica Volume 44, 2022 Odd Fibonacci Stolarsky-3 Mean Labeling of Some Special Graphs Sree Vidya. M 1 , Sandhya S. S 2 Abstract Let G be a graph with p vertices and q edges and an injective function where each is a odd Fibonacci number and the induced edge labeling are defined by and all these edge labeling are distinct is called Odd Fibonacci Stolarsky-3 Mean Labeling. A graph which admits a Odd Fibonacci Stolarsky-3 Mean Labeling is called a Odd Fibonacci Stolarsky-3 mean graph. Keywords: Stolarsky-3 Mean Labeling of Graphs, Odd Fibonacci Stolarsky-3 Mean Labeling of Graphs, Bull graph, Wheel graph, (m, n)-tadpole graph, Fire Cracker graph, Pan graph, Gear graph, Star graph. AMS Subject Classification: 05C78 3 1 Research Scholar, Sree Ayyappa College for Women, Chunkankadai. 2 Assistant Professor, Department of Mathematics, Sree Ayyappa College for Women, Chunkankadai. [Affiliated to Manonmaniam Sundaranar University, Abishekapatti – Tirunelveli - 627012, Tamilnadu, India] Email: witvidya@gmail.com 1 sssandhya2009@gmail.com 2 3 Received on June 10th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.916. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published under the CC-BY licence agreement. 281 mailto:witvidya@gmail.com1 mailto:sssandhya2009@gmail.com2 Sree Vidya. M, Sandhya. S. S 1. Introduction The graph considered here will be finite, undirected and simple graph with p vertices and q edges. For all detailed survey of graph labeling, we refer to Galian [1]. For all other standard terminology and notations, we follow Harary [2]. S. S. Sandhya, S. Somasundaram and S. Kavitha introduced the concept of Stolarsky 3 Mean labeling of graphs in [3]. In this paper, we introduced a new concept namely Odd Fibonacci Mean Labeling of graphs. Definition: 1.1 The Fibonacci numbers can be defined by linear recurrence . This generates the infinite sequence of integer beginning 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144, 233, …. Definition: 1.2 Let G be a graph with p vertices q edges. An injective function where each isa odd Fibonacci number and the induced edge labeling defined by edge e =uv is labeled with f=(e=uv)= (or) , then all the edge labels are distinct and are from Odd Fibonacci number where . A graph that admits Odd Fibonacci Stolarsky-3 Mean labeling is called Odd Fibonacci Stolarsky-3 Mean graph. Definition: 1.3 The Bull graph is a planar undirected graph with 5 vertices and 5 edges in the form of a triangle with two disjoint pendent edges. Definition: 1.4 The Wheel graph is join of the graphs , . Definition: 1.5 The (m, n)-tadpole graph is a graph is a special consisting of a graph on m (at least 3) vertices and a path graph on n vertices connected with a bridge. Definition: 1.6 An Firecracker is a graph obtained by the concatenation of stars by linking one leaf. Definition: 1.7 Gear graph is obtained from the wheel by adding a vertex between every pair of adjacent vertices of the cycle. The gear graph has 2n+1 vertices and 3n edges. Definition:1.8 The pan graph is the graph obtained by joining a cycle graph to a singleton graph with a bridge. Definition: 1.9A Star graph with n vertices is a tree with one vertex having degree n-1 282 Odd Fibonacci Stolarsky-3 Mean Labeling of Some Special Graphs and other n-1 vertices having degree 1. A star graph with n+1 vertices . 2. Main Results Theorem 2.1: The Bull graph is a Odd Fibonacci Stolarsky-3 mean labeling. Proof. Let be a Bull graph. Here , Define a function where each is a odd Fibonacci number. Then the induced edge labeling is all distinct. Hence, we proved a Bull graph admits Odd Fibonacci Stolarsky-3 mean labeling. Example 2.2: Odd Fibonacci Stolarsky-3 mean labeling of Bull graph is shown below. Figure 2.1: Bull graph Theorem 2.3: The Wheel graph admits Odd Fibonacci Stolarsky-3 mean labeling. Proof. Let be a Wheel graph. Here , Define a function where each is a odd Fibonacci number. Then the induced edge labeling are all distinct. Hence, we proved a wheel graph admits Odd Fibonacci Stolarsky-3 mean labeling. Example 2.4: Odd Fibonacci Stolarsky-3 mean labeling of Wheel graph is shown below. 283 Sree Vidya. M, Sandhya. S. S Figure 2.2 Wheel graph Theorem 2.5: The (m, n)- tadpole graph admits Odd Fibonacci Stolarsky-3 mean labeling. Proof. Let G be a (m, n)- tadpole graph. Here , Define a function where each is a odd Fibonacci number. Then the induced edge labeling is all distinct. Hence, we proved a (m, n)- tadpole graph admits Odd Fibonacci Stolarsky-3 mean labeling. Example 2.6: Odd Fibonacci Stolarsky-3 mean labeling of (5,1)- tadpole graph is shown below. Figure 2.3 (5, 1)- tadpole graph Theorem 2.7: A Fire Cracker graph admits Odd Fibonacci Stolarsky-3 mean labeling. Proof. Let G be a Fire Cracker graph. Here , Define a function where each is a odd Fibonacci number and assignment of vertex labeling are 284 Odd Fibonacci Stolarsky-3 Mean Labeling of Some Special Graphs Then the induced edge labeling is all distinct. Hence, we proved a Fire Cracker admits Odd Fibonacci Stolarsky-3 mean labeling. Example 2.8: Odd Fibonacci Stolarsky-3 mean labeling of Fire Cracker graph is shown below. Figure 2.4: Fire Cracker graph Theorem 2.9: The Pan graph admits an Odd Fibonacci Stolarsky-3 mean labeling. Proof. Let G be a Pan graph. Here , Define a function where each is a odd Fibonacci number and assignment of vertex labeling are Then the induced edge labeling are all distinct. Hence, we proved a Pan admits Odd Fibonacci Stolarsky-3 mean labeling. Example 2.10: Odd Fibonacci Stolarsky-3 mean labeling of Pan graph is shown below. Figure 2.5: Pan graph Theorem 2.11: The Gear graph admits Odd Fibonacci Stolarsky-3 mean labeling. Proof. Let be Gear graph. 285 Sree Vidya. M, Sandhya. S. S Here , Define a function where each is a odd Fibonacci number. Then the induced edge labeling is all distinct. Hence, we proved a Gear graph admits Odd Fibonacci Stolarsky-3 mean labeling. Example 2.12: Odd Fibonacci Stolarsky-3 mean labeling of Gear graph is shown below. Figure 2.6 Gear graph Theorem 2.13: The Star graph admits an Odd Fibonacci Stolarsky-3 mean labeling. Proof. Let G be a Star graph. Here , Define a function where each is a odd Fibonacci number and assignment of vertex labeling are Then the induced edge labeling is all distinct. Hence, we proved a Star graph admits Odd Fibonacci Stolarsky-3 mean labeling. Example 2.14: Odd Fibonacci Stolarsky-3 mean labeling of Star graph is shown below. 286 Odd Fibonacci Stolarsky-3 Mean Labeling of Some Special Graphs Figure 2.7 Star graph 3. Conclusion we have introduced a new labeling namely Odd Fibonacci Stolarsky-3 Mean Labeling of graphs. We prove that Bull graph, Wheel graph, (m,n)-tadpole graph, Fire Cracker graph, Pan graph, Gear graph, Star graph. Extending the study to other families of graphs is an open area of research. References [1] Galian, J. A. (2019) A Dynamic Survey of Graph Labeling. The Electronic Journal of combinatories. [2] Harary, F. (1988) Graph Theory. Narosa Publishing House Reading, New Delhi. [3] S. Somasundaram and R. Ponraj, “Mean Labeling of graphs”, National Academy of Science Letters vol.26, p.210-213. [4] S. S. Sandhya, S. Somasundaram, S. Kavitha, Stolarsky 3 Mean Labeling of Graphs Global Journal of pure and Applied Mathematics, Vol.14, 2018, no.14, pp.39 - 47. [5] K. Thirugnanasambandam, G. Chitra, Y. Vishnupriya, Prime Odd Mean Labeling of Some Special Graphs, Journal of Emerging Technologies and Innovative Research (JETIR), volume 5, Issue 6, 2018 JETIR June 2018. [6] Sree Vidya. M & Sandhya, S.S 2020, Degree Splitting of Stolarsky-3 Mean Labeling of graphs, International Journal of Computer Science, ISSN 2348-6600, vol.8, no.2, pp. 2413 – 2420. [7] Sree Vidya. M & Sandhya, S.S 2022, Decomposition of Stolarsky-3 Mean Labeling of Graphs, International Journal for Innovative Engineering Research, volume, Issue 1, pp. 08-12. 287