Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 414 https://internationalpubls.com Binomial Labeling of Intersection Mirror Lagoon Step Graph and Intersection Opposite Lagoon Step Graph with Python Implementation 1J. Vasanthi, 2N. Ramya 1Research Scholar, Department of Mathematics, Bharath Institute of Higher Education & Research, Chennai, Tamil Nadu, India. 2Professor, Department of Mathematics, Bharath Institute of Higher Education & Research, Chennai,Tamil Nadu ,India. Article History: Received: 27-10-2024 Revised:01-12-2024 Accepted:28-12-2024 Abstract: Assume G = {V, E} be the graph with n vertices and e edges. The binomial labeling classification has been used for Intersection Mirror Lagoon Step Graph IM LnSm and Intersection Opposite Lagoon Step Graph IO Ln Sm in this paper. The concepts stated below in the paper have been implemented using Python coding. Keywords: Intersection Mirror Lagoon Step Graph, Intersection Opposite Lagoon Step Graph, Binomial labeling. 1. Introduction Rosa [1] first proposed the idea of graph labeling in 1967. Gallian [2] updates a dynamic overview of graph labeling techniques on a regular basis, and the Electronic Journal of Combinatory publishes it. We adhere to Bondy and Murthy's [3] notation and nomenclature for several aspects of graph theory. Binomial labeling introduced by Chandrakala and Sekar [4] et.al. The Binomial labeling for Intersection Mirror Lagoon Step Graph IM Ln Sm and the Intersection Opposite Lagoon Step Graph IO Ln Sm are constructed here using Python code [5][6]. 2. Preliminaries Definition 1 :A Lagoon Step graph Ln Sm is obtained from Lagoon Ln (L Shape) graph with vertices n (n≥3, must be an odd) joining with step graph Sm (m≥1,m is number of steps with m =(n-1)/2 , the graph consists of n+2m-1 vertices and n+2m-1 edges and is described below Figure 1: L5S2 & L7S3 Definition 2 :Intersection Mirror Lagoon Step graph IM Ln Sm obtained by attaching Lagoon step graph with Mirror image Lagoon step graph consists of 3n+m-4 vertices and 3n+m-3 edges (with n ≥ 3 must be an odd and m = (n-1)/2, m ≥ 1) described below Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 415 https://internationalpubls.com Figure 2: IM L7S3 Definition 3: Intersection Opposite Lagoon Step graph IO LnS m obtained by attaching Lagoon step graph with Opposite direction of Lagoon step graph consists of vertices 2n+4m-4 vertices and edges 2n+4m-3(with n ≥ 3 must be an odd and m = (n-1)/2, m ≥ 1) described below Figure 3: IO L7S3 Definition 4 : For positive integers n and r such that 0 ≤ r ≤ n, then Binomial coefficient nCr = n!/(r!(n-r)!) is always an integer. Definition 5: Consider G = (p, q) be a graph. If the graph is an injective map f: V(G) → {1, 2, 3, … q+1} where the induced f*: E(G) → N is provided by, then G has a Binomial Labeling by f* (u v) = M Cm = M!/(m!(M-m)!) where M = max {f(u), f(v)}, m=min {f(u), f(v)}assigns distinct labels for the edges. Definition 6: The term "binomial graph" refers to a graph that has the Binomial Labeling. 3. Main Results 3. 1 Theorem: Intersection Mirror Lagoon step graph IM Ln Sm is a Binomial graph. Proof: Consider the graph G be the intersection Mirror Lagoon step graph IM Ln Sm with 3n+m-4 vertices and 3n+m-3 edges when n = 3, 5, 7, 9… then m = 1, 2, 3… respectively. Case i : When n = 5, 7, 9… then m = 2, 3,4… respectively. Let vi be the outer vertices starts from left step corner to right step corner, w i be the vertical inner vertices and xi be the base vertices. Classify f: V (G) → {1, 2, 3… 3n+m-2} is given below f(vi) = i, i = 1, 2 … n+2m-2, when n = 5, 7, 9…, then m = 2, 3, 4… respectively, f(wi) = 2n+2m+j-1, which can be expressed as i n m j 1 5 2 0 1, 2 7 3 0, 1 1, 2, 3 9 4 0, 1, 2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 416 https://internationalpubls.com . . . . . . . . . . . . respectively. f(xi) = n+2m+i-2 which can be expressed as n m i 5 2 1, 2, 3, 4, 5 7 3 1, 2, 3, 4, 5, 6, 7 9 4 1, 2, 3, 4, 5, 6, 7, 8, 9 . . . . . . . . . respectively. The equation f* (e=uv) = R Cr = R!/(r!(R-r)!) yields f*: E(G) → N, wherein R = max{f(u), f(v)} and r = min{ f(u), f(v) } for every uv ϵ E(G) .Every edge in this case has a unique label. Figure 4: Binomial graph IML5S2 Figure 5: Binomial graph IM L7S3 Special case IM L3S1: Consider v1, v2, v3, v4, v5 and v6 be the six vertices of IM L3S1. Vertex labeling are f(vi) = i, i = 1, 2, 3, 4, 5 and f(vi) = i+1, i = 6. Figure 6: Binomial graph IML3S1 Every edge in this case has a unique label. IMLnSm is a binomial graph as a result. 3. 2 Theorem : Intersection Opposite Lagoon step graph IO LnSm is a Binomial graph. Proof :Consider the graph G be the Intersection Opposite Lagoon Step graph with 2n+4m-4 vertices and 2n+4m-3 edges when n = 3, 5, 7…, then m = 1, 2, 3, 4… respectively. Case i: When n = 5, 7, 9… then m = 2, 3, 4… respectively. Let vi be the outer vertices except last vertex vi1 where vi1 be the last vertex. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 417 https://internationalpubls.com Classify f: V(G) → {1, 2,3… 2n+4m-2} as follows f(vi) = i, i = 1, 2,3…2n+4m-5 when n = 5, 7, 9… then m = 2, 3, 4… respectively, f(vi1)) = i + 1, where i = 2n+4m-4 when n = 5, 7, 9… then m = 2, 3, 4… respectively. f*: E(G) → N is given by f* (e=uv) = R Cr = R!/(r!(R-r)!) for all uv ϵ E(G) where R = max {f(u), f(v)} and r = min {f(u), f(v)}.Here all the edges have distinct labels. Figure 7: Binomial graph IOL5S2 Figure 8: Binomial graph IOL7S3 Special case IO L3S1: Let v1, v2, v3, v4, v5 and v6 be the six vertices of IO L3S1. Vertex labelings are f(vi) = i, i = 1, 2, 3, 4, 5 and f(vi) = i+1, i = 6. Figure 9: Binomial graph IOL3S1 Every edge in this case has a unique label. IOLnSm is a binomial graph as a result. 4. Verification Using Python Coding 4. 1 Verification of the Binomial of IM L9 S4 and IM L11 S5 using Python Implementation Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 418 https://internationalpubls.com 4. 2 Verification of the Binomial of IO L9 S4 and IO L11 S5 using Python Implementation Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 419 https://internationalpubls.com 5. Conclusion The demonstration of graphs like Intersection Mirror Lagoon step graph and Intersection Opposite Lagoon step graph exist when it comes to Binomial labeling. This paper also discusses the limitations and possible uses of this type of labeling. Python coding for Binomial labeling has been implemented for the above graphs. In the near future, more research under the Lagoon Step graph using various labeling strategies is needed. References [1] A. Rosa, On Certain Valuations of the Vertices of a Graph, In Theory of Graphs (Internat. Sympos. Rome. (1966) (1967), 349–359, Gordan and Breach. Newyork. Dunod. Paris. [2] J.A. Gallian, A Dynamic Survey of Graph Labeling, Electronic Journal of Combinatorics, 17 (DS6) (2016). [3] Bondy .J.A and Murthy U.S.R, “Graph Theory and Application” (North Holland). New York (1976). [4].S. Chandrakala and C.Sekar, “Binomial Labeling of Some Graphs”,International Journal for Research in Engineering Application & Management ISSN: 2454-9150 ,Vol.04,Issue. 08, Nov 2018. [5] D. Amuthavalli, O. V. Shanmuga Sundaram, Super Fibonacci graceful anti – magic labeling for flower graphs and python coding,Tuijin Jishu/Journal of Propulsion Technology, Vol. 44 No. 3,2023. [6] J.A.Gadhiya and R.Solanki, “ Labeling of Some Graphs Using Python Programming,” vol.20,no.17,pp.2288 - 2294,2022.