Ratio Mathematica Volume 44, 2022 On Forgotten Index of Stolarsky-3 Mean Graphs Sree Vidya.M 1 Sandhya. S. S 2 Abstract The Forgotten index of a graph G is defined as F(G) = over all edges of ,where , are the degrees of the vertices u and v in , respectively. In this paper, we introduced Forgotten index of some standard Stolarsky-3 Mean Graphs. Keywords: Forgotten index, Stolarsky-3 Mean Graphs. AMS Subject Classification: 05C12 3 1 Research Scholar, Sree Ayyappa College for Women, Chunkankadai 2 Research Supervisor, Department of Mathematics, Sree Ayyappa College for Women, Chunkankadai. [Affiliated to Manonmaniam Sundaranar University, Abishekapatti – Tirunelveli - 627012, Tamilnadu, India] Email: witvidya@gmail.com & sssandhya2009@gmail.com 3 Received on June 9 th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.915. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published under the CC-BY licence agreement. 272 mailto:witvidya@gmail.com%20& Sree Vidya.M & Sandhya. S. S 1. Introduction Let G be a simple graph corresponding to a drug structure with vertex (atom) set V(G) and edge (bond) set E(G). The edge joining the vertices u and v is denoted by uv. Thus, if then u and v are adjacent in G. The degree of a vertex u, denoted by d(u), is the number of edge incident to u. Several topological indices such as Estrada index, Zagreb index, PI index, eccentric index, and wiener index have been introduced in the literature to study the chemical and pharmacological properties of molecules. The forgotten topological index of a graph G is defined as the sum of weights over all edges of ,where and are the degrees of the vertices u and v in , respectively. In this paper, we characterize the external properties of F-index (forgotten topological index). We first introduce some graph transformation which increase or decrease this index. Recently in 2015 Furtula and Gutman was introduced another topological index called index or F-index as F(G) = . On the basis of this work, we introduce a new concept Forgotten index ofStolarsky3Meangraphs.Inthispaperweinvestigate Forgotten index of some standard graphs which admit Forgotten Mean graphs. We will provide a brief summary of definitions and other information which are necessary for our present investigation. Definition:1.1 A graph with vertices and edges is called a Stolarsky-3 Mean graph, if each vertex with distinct labels from and eachedge is assigned the distinct labels then the resulting edge labels are distinct. In this case is called Stolarsky-3 Mean labeling of . Definition:1.2 Let G be a Stolarsky-3 Mean graph. The Forgotten index of a graph F(G) is defined by F(G) = , where d(u) is the degree of vertex u in G. Theorem 1.3: Any Path is a Stolarsky-3 mean graph. Theorem 1.4: Any Cycle C is a Stolarsky-3 mean graph. Theorem 1.5: Any Comb ⨀ 1is a Stolarsky 3 mean graph. Theorem 1.6: The ladder graph is a Stolarsky-3 mean graph. Theorem 1.7: ATriangular Snake graph is a Stolarsky-3 mean graph. Theorem 1.8: AQuadrilateral Snake graph is a Stolarsky-3 mean graph. 273 On Forgotten Index of Stolarsky-3 Mean Graphs Remark 1.9: If is a Stolarsky 3 mean graph, then ‘1’ must be a label of one of the vertices of , Since, an edge should get label ‘1’. Remark1.10: If u gets label ‘1’, then any edge incident with must get label 1 (or) 2 (or) 3. Hence this vertex must have a degree ≤ 3. 2. Main results Theorem 2.1: Let G = Pn be a Stolarsky-3 mean graph. Then the Forgotten index of a path Pn is F(Pn) =5n+4. Proof. Let G = Pn be a Stolarsky-3 mean graph. Figure: 1 Path Pn We have and . Therefore, by the definition of forgotten topological index, we obtain F(G) = = = = = F(G) = Example 2.2. Forgotten index of P6 is given below Figure: 2 Path P6 F(P6) = = = = = Theorem 2.3. The Forgotten index of cycle is . Proof. Let be a Stolarsky-3 mean graph u1 u2 u3 un-1 un 274 Sree Vidya.M & Sandhya. S. S Figure: 3 We have and . Therefore, by the definition of forgotten topological index, we obtain F(G) = = = = F(G) = Example 2.4. Forgotten index of is given below Figure: 4 F( ) = = = = 48 Theorem 2.5. The Forgotten index of Comb graph ⨀ . Proof. Let G = Pn ʘ K1 be a Stolarsky – 3 Mean graph. 275 On Forgotten Index of Stolarsky-3 Mean Graphs Figure: 5 Comb PnʘK1 We have and . Therefore, by the definition of forgotten topological index, we obtain F(G) = = = = F(G) = Example 2.6: Forgotten index of P6ʘK1 is given below. Figure: 6 Comb P6ʘK1 F(P6ʘK1) = [ = = Theorem 2.7: Forgotten index of ladder graph is Proof. Let G = Pn be a Stolarsky- 3 Mean graph Figure: 7 G = Pn Case (i) if n = 2 F ( ) = 276 Sree Vidya.M & Sandhya. S. S = = = Case (ii) if n 2 F( ) = = = = = F ( ) = Example 2.8. Forgotten index of L4 is given below. Figure: 8 L4 F( ) = = = = = 140 Theorem 2.9. Forgotten index of Triangular Snake graph is 58n-6. Proof. Let us consider a Stolarsky-3 Mean graph be a Stolarsky-3 mean graph. Figure: 9 F(G) = 277 On Forgotten Index of Stolarsky-3 Mean Graphs = = = = = F(G) = Example 2.10. Forgotten index of T3 is given below. Figure: 10 F(T3) = = = = 120+16+32 = 168 Theorem 2.11: Forgotten index of Quadrilateral Snake graph is 80n-48. Proof. Consider be a Stolarsky-3 mean graph. Figure:11 F ( = = = = 278 Sree Vidya.M & Sandhya. S. S = F ( = Example 2.12. Forgotten index of is given below. Figure: 12 F( ) = = = = = Theorem 2.13: Forgotten index of Crown graph ⨀ is 28n. Proof. Consider ⨀ be a Stolarsky-3 mean graph. F ( = = = = = F ( = Example 2.14. Forgotten index of ⨀ is given below. Figure: 13 ⨀ 279 On Forgotten Index of Stolarsky-3 Mean Graphs F ( ⨀ ) = = = = References [1] F. Harary. Graph Theory, Narosa Publishing House: New Delhi; 2001. [2] B. Furtula and I. Gutman, “A forgotten topological index “, Journal of mathematical chemistry, vol. 53, no. 4, pp. 1184-1190,2015. [3] Toufik MANSOUR, Mohammad Ali ROSTAMI, On the bounds of the forgotten topological index Turkish Journal of Mathematics, (2017) 41:1687-1702. [4] S. S. Sandhya, S. Somasundaram, and S. Kavitha “Stolarsky 3 Mean Labeling of Graphs” Journal of Applied Science and Computaions, Vol.5, Issue 9, pp. 59 – 66. [5] Sree Vidya. M and Sandhya. S. S. “Degree Splitting of Stolarsky 3 Mean Labeling of Graphs” International Journal of Computer Science, ISSN 2348-6600, Volume 8, Issue 1, No 2, 2020, Page No: 2413 – 2420. [6] Sree Vidya. M and Sandhya. S. S. “Decomposition of Stolarsky 3 Mean Labeling of Graphs” International Journal for Innovative Engineering Research, Volume 1, Issue 1, March, 2022, Page No: 08-12. 280