Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 11 https://internationalpubls.com An Application on Harmonic Mean Labeling of Variations in Triangular Snake Graphs T. Christy1, G. Palani2,*, E. Chandrasekaran3 1Assistant Professor, Department of Mathematics, Patrician College of Arts and Science, Affiliated to University of Madras, Tamil Nadu, India. 2Assistant Professor, Department of Mathematics, Dr. Ambedkar Government Arts College, Tamil Nadu, India. 3Professor of Mathematics, Veltech Rangarajan Dr Sagunthala R&D Institute of Science and Technology, Tamil Nadu, India. *Corresponding author: E-mail: gpalani32@yahoo.co.in Article History: Received: 15-01-2024 Revised: 26-03-2024 Accepted: 18-04-2024 Abstract: A graph 𝐺 with 𝑝 vertices and 𝑞 edges is called a harmonic mean(HM) labeling if it is possible to label the vertices 𝑥 ∈ 𝑣 with distinct labels 𝜌(𝑥) from {1,2, ⋯ , 𝑞 + 1} in such a way that each edge 𝑒 = 𝑎𝑏 is labeled with 𝜌(𝑎𝑏) = ⌈ 2𝜌(𝑎)𝜌(𝑏) 𝜌(𝑎)+𝜌(𝑏) ⌉ or ⌊ 2𝜌(𝑎)𝜌(𝑏) 𝜌(𝑎)+𝜌(𝑏) ⌋ then the edge labels are distinct. In this case 𝜌 is called Harmonic mean(HM) labeling of 𝐺. In this paper we introduce new graphs obtained from triangular snake graph 𝑇𝑆𝑛 such as 𝑇𝑆𝑛 ∘ 𝐾1, and prove that they are Harmonic Mean labeling graphs. Keywords: HM labeling, 𝑇𝑆𝑛 ∘ 𝐾1 graphs. 1. Introduction Let 𝐺 = (𝑉, 𝐸) be a (𝑝, 𝑞) graph with 𝑝 = |𝑉(𝐺)| vertices and 𝑞 = |𝐸(𝐺)| edges, where 𝑉(𝐺) and 𝐸(𝐺) respectively denote the vertex set and edge set of the graph G. In this paper, we consider the graphs which are simple, finite and undirected for graph theoretic terminology and notations we refer to Haray [2]. The Concept of graph labeling was introduced by Rosa in 1967.A detailed survey of graph labeling is available in Gallian [1]. The concept of Mean labeling of graph was introduced by S. Somasundaram, R. Ponraj and S.S. Sandhya [3]. Some of the harmonic mean graphs are investigated by S. Meena and M. Sivasakthi in [4]. The concept of Harmonic Mean labeling of graph was introduced by S. Somasundaram, R. Ponraj and S.S. Sandhya [5,6] and they investigated the existence of Harmonic mean labeling of several family of graphs such as this concept was then studied by several authors and studied their behavior in [7], [8], [9], and [10]. 2. Preliminaries Definition 2.1. Mean Labeling A function 𝜌 is called mean labeling for a graph 𝐺 = (𝑉, 𝐸) if 𝜌: 𝑉 → {0,1,2,3, ⋯ , 𝑞} is injective and the induce function 𝜌∗: 𝐸 → {1,2,3, ⋯ , 𝑞} defined as 𝜌 ∗= ⌈ 𝜌(𝑎)+𝜌(𝑏) 2 ⌉ or ⌊ 𝜌(𝑎)+𝜌(𝑏) 2 ⌋ is bijective for every edge. A graph 𝐺 is called mean labeling. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 12 https://internationalpubls.com Definition 2.2. Harmonic Mean (HM) Labeling A graph 𝐺 = (𝑉, 𝐸) with 𝑝 vertices and 𝑞 edges is called a harmonic mean (𝐻𝑀) graph if it is possible to label the vertices 𝑥 ∈ 𝑣 with distinct labels 𝜌(𝑥) from {1,2, ⋯ , 𝑞 + 1} in such 𝑎 way that when each edge 𝑒 = 𝑎𝑏 is labeled with 𝜌(𝑎𝑏) = ⌈ 2𝜌(𝑎)𝜌(𝑏) 𝜌(𝑎)+𝜌(𝑏) ⌉ or ⌊ 2𝜌(𝑎)𝜌(𝑏) 𝜌(𝑎)+𝜌(𝑏) ⌋ then the edge labels are distinct. Definition 2.3. 𝑇𝑆𝑛 ∘ 𝐾1 Graph The Triangular snake is obtained from the path 𝑃𝑛 by replacing each edge of the path by a cycle 𝐶3. Each vertex of triangular snake graph 𝑇𝑛 is attached with an edge and the graph obtained is called as 𝑇𝑆𝑛 ∘ 𝐾1 3. Main Results Theorem 3.1. For every 𝑛 ≥ 1, there exists a Triangular snake graph 𝑇𝑆𝑛 ∘ 𝐾1, which admits HM labeling. Proof. Consider the 𝑇𝑆𝑛 ∘ 𝐾1 graph with Vertex set 𝑉 = {𝑎𝑖, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖; 1 ≤ 𝑖 ≤ 𝑛} and Edge set 𝐸 = {(𝑎𝑖𝑏𝑖) ∪ (𝑐𝑖𝑏𝑖) ∪ (𝑐𝑖𝑐𝑖+1) ∪ (𝑐𝑖𝑑𝑖); 1 ≤ 𝑖 ≤ 𝑛} Now let us define a function 𝜌: 𝑣 → {1,2,3, ⋯ 𝑞 + 1} Let us label the vertices as follows. 𝜌(𝑎1) = 6; 𝜌(𝑎𝑖+1) = 5𝑖 + 3; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌(𝑏𝑖) = 5𝑖 − 1; 𝜌(𝑐𝑖) = 5𝑖 − 3; 1 ≤ 𝑖 ≤ 𝑛 𝜌(𝑑1) = 1; 𝜌(𝑑𝑖+1) = 5𝑖; 1 ≤ 𝑖 ≤ 𝑛 − 1 The induced edge labeling is as follows 𝜌∗(𝑎1𝑏1) = 4; 𝜌∗(𝑎𝑖+1𝑏𝑖+1) = 5𝑖 + 3; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌∗(𝑏𝑖𝑐𝑖) = 5𝑖 − 3; 𝜌∗(𝑐𝑖+1𝑏𝑖) = 5𝑖; 1 ≤ 𝑖 ≤ 𝑛 𝜌∗(𝑐1𝑐2) = 3; 𝜌∗(𝑐𝑖+1𝑐𝑖+2) = 5𝑖 + 4; 1 ≤ 𝑖 ≤ 𝑛 𝜌∗(𝑑𝑖𝑐𝑖) = 5𝑖 − 4; 1 ≤ 𝑖 ≤ 𝑛 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 13 https://internationalpubls.com Thus, 𝑇𝑆𝑛 ∘ 𝐾1 is a HM labeling. Fig 1. 𝑇𝑆4 ∘ 𝐾1 Theorem 3.2. For every 𝑛 ≥ 1, there exists a Alternate Triangular snake graph 𝐴(𝑇𝑆)𝑛 ∘ 𝐾1, which admits 𝐻𝑀 labeling. Proof. Consider the 𝐴(𝑇𝑆)𝑛 ∘ 𝐾1 graph with Vertex set 𝑉 = {𝑎𝑖, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖; 1 ≤ 𝑖 ≤ 𝑛} and Edge set 𝐸 = {(𝑎𝑖𝑏𝑖) ∪ (𝑐𝑖𝑏𝑖) ∪ (𝑐𝑖𝑐𝑖+1) ∪ (𝑐𝑖𝑑𝑖); 1 ≤ 𝑖 ≤ 𝑛} Now let us define a function 𝜌: 𝑣 → {1,2,3, ⋯ 𝑞 + 1} Let us label the vertices as follows 𝜌(𝑎𝑖) = 7𝑖 − 6; 𝜌(𝑏𝑖) = 7𝑖 − 5; 1 ≤ 𝑖 ≤ 𝑛. 𝜌(𝑐1) = 4; 𝜌(𝑐2𝑖) = 7𝑖 − 2; 𝑖 ≡ 0(mod2) 𝜌(𝑐2𝑖+1) = 7𝑖 + 3; 𝑖 ≡ 1(mod2) 𝜌(𝑑2𝑖) = 7𝑖; 𝑖 ≡ 0(mod2) 𝜌(𝑑2𝑖−1) = 7𝑖 − 1; 𝑖 ≡ 1(mod2) The induced edge labeling is as follows 𝜌∗(𝑎𝑖𝑏𝑖) = 7𝑖 − 6; 1 ≤ 𝑖 ≤ 𝑛 𝜌∗(𝑏2𝑖−1𝑐2𝑖−1) = 7𝑖 − 5; 𝑖 ≡ 1(mod2) 𝜌∗(𝑏2𝑖𝑐2𝑖) = 7𝑖 − 4; 𝑖 ≡ 0(mod2) 𝜌∗(𝑐2𝑖−1𝑐2𝑖) = 7𝑖 − 3; 𝜌∗(𝑐2𝑖+1𝑐2𝑖) = 7𝑖; 1 ≤ 𝑖 ≤ 𝑛 𝜌∗(𝑑2𝑖−1𝑐2𝑖−1) = 7𝑖 − 2; 𝑖 ≡ 1(mod2) 𝜌∗(𝑑2𝑖𝑐2𝑖) = 7𝑖 − 1; 𝑖 ≡ 0(mod2) Thus, 𝐴(𝑇𝑆)𝑛 ∘ 𝐾1 is a HM labeling. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 14 https://internationalpubls.com Fig. 2. 𝐴(𝑇𝑆)3 ∘ 𝐾1 Theorem 3.3. For every 𝑛 ≥ 1, there exists a Double Triangular snake graph DTS 𝑆𝑛 。 𝐾1, which admits HM labeling. Proof. Consider the 𝐷𝑇𝑆𝑛 ∘ 𝐾1 graph with Vertex set 𝑉 = {𝑎𝑖, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖 , 𝑎𝑖 ′, 𝑏𝑖 ′; 1 ≤ 𝑖 ≤ 𝑛} and Edge set 𝐸 = {(𝑎𝑖𝑏𝑖) ∪ (𝑑𝑖𝑏𝑖) ∪ (𝑐𝑖𝑑𝑖) ∪ (𝑑𝑖𝑑𝑖+1) ∪ (𝑏𝑖 ′𝑑𝑖 ′),∪ (𝑎𝑖 ′𝑏𝑖 ′); 1 ≤ 𝑖 ≤ 𝑛}. Now let us define a function 𝜌: 𝑣 → {1,2,3, ⋯ 𝑞 + 1} Let us label the vertices as follows. 𝜌(𝑎1) = 1; 𝜌(𝑎𝑖+1) = 8𝑖 + 3; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌(𝑏1) = 2; 𝜌(𝑏𝑖+1) = 8𝑖 + 4; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌(𝑐1) = 4; 𝜌(𝑐𝑖+1) = 8𝑖 − 2; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌(𝑑1) = 5; 𝜌(𝑑𝑖+1) = 8𝑖 + 2; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌(𝑏𝑖 ′) = 8𝑖 − 1; 𝜌(𝑎𝑖 ′) = 8𝑖 + 1; 1 ≤ 𝑖 ≤ 𝑛 The induced edge labeling is as follows 𝜌∗(𝑎1𝑏1) = 1; 𝜌∗(𝑎𝑖+1𝑏𝑖+1) = 8𝑖 + 3; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌∗(𝑑1𝑐1) = 4; 𝜌∗(𝑑𝑖+1𝑐𝑖+1) = 8𝑖 − 1; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌∗(𝑎𝑖 ′𝑏𝑖 ′) = 8𝑖; 𝜌∗(𝑑𝑖𝑏𝑖) = 8𝑖 − 6; 1 ≤ 𝑖 ≤ 𝑛 𝜌∗(𝑑2𝑏1) = 3; 𝜌∗(𝑑𝑖+1𝑏𝑖) = 8𝑖 + 6; 2 ≤ 𝑖 ≤ 𝑛 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 15 https://internationalpubls.com 𝜌∗(𝑏1 ′ 𝑑1) = 5; 𝜌∗(𝑏𝑖+1 ′ 𝑑𝑖+1) = 8𝑖 + 4; 𝜌∗(𝑏𝑖 ′𝑑𝑖+1) = 8𝑖 + 1; 1 ≤ 𝑖 ≤ 𝑛 𝜌∗(𝑑1𝑑2) = 6; 𝜌∗(𝑑𝑖+1𝑑𝑖+2) = 8𝑖 + 5; 1 ≤ 𝑖 ≤ 𝑛 − 1 Thus, 𝐷𝑇𝑆𝑛 ∘ 𝐾1 is a HM labeling. Fig. 3. 𝐷𝑇𝑆4 ∘ 𝐾1 Theorem 3.4. For every 𝑛 ≥ 1, there exists a Alternate Double Triangular snake graph 𝐴(𝐷𝑇𝑆)𝑛 ∘ 𝐾1, which admits 𝐻𝑀 labeling. Proof. Consider the 𝐴(𝐷𝑇𝑆)𝑛 ∘ 𝐾1 graph with Vertex set 𝑉 = {𝑎𝑖, 𝑏𝑖 , 𝑐𝑖, 𝑑𝑖, 𝑎𝑖 ′, 𝑏𝑖 ′; 1 ≤ 𝑖 ≤ 𝑛} and Edge set 𝐸 = {(𝑎𝑖𝑏𝑖) ∪ (𝑑𝑖𝑏𝑖) ∪ (𝑐𝑖𝑑𝑖) ∪ (𝑑𝑖𝑑𝑖+1) ∪ (𝑏𝑖 ′𝑑𝑖) ∪ (𝑎𝑖 ′𝑏𝑖 ′); 1 ≤ 𝑖 ≤ 𝑛} Now let us define a function 𝜌: 𝑣 → {1,2,3, ⋯ 𝑞 + 1} Let us label the vertices as follows. 𝜌(𝑎1) = 1; 𝜌(𝑎𝑖+1) = 10𝑖 + 3; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌(𝑏𝑖) = 10𝑖 − 8; 1 ≤ 𝑖 ≤ 𝑛 𝜌(𝑑1) = 5; 𝜌(𝑑2𝑖) = 10𝑖; 𝑖 ≡ 0(mod2) 𝜌(𝑑2𝑖−1) = 10𝑖 + 1; 𝑖 ≡ 1(mod2) 𝜌(𝑐2𝑖) = 10𝑖 − 4; 𝑖 ≡ 0(mod2) 𝜌(𝑐2𝑖−1) = 10𝑖 − 6; 𝑖 ≡ 1(mod2) 𝜌(𝑏𝑖 ′) = 10𝑖 − 3; 𝜌(𝑎𝑖 ′) = 10𝑖 − 1; 1 ≤ 𝑖 ≤ 𝑛 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 16 https://internationalpubls.com The induced edge labeling is as follows 𝜌∗(𝑎1𝑏1) = 1; 𝜌∗(𝑎𝑖+1𝑏𝑖+1) = 10𝑖 + 2; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌∗(𝑑1𝑐1) = 4; 𝜌∗(𝑑2𝑖𝑐2𝑖) = 10𝑖 − 3; 𝑖 ≡ 0(mod2) 𝜌(𝑑2𝑖+1𝑐2𝑖+1) = 10𝑖 + 3; 𝑖 ≡ 1(mod2) 𝜌∗(𝑑1𝑏1) = 2; 𝜌∗(𝑑2𝑖+1𝑏𝑖+1) = 10𝑖 + 1; 1 ≤ 𝑖 ≤ 𝑛 𝜌∗(𝑑2𝑏1) = 3; 𝜌∗(𝑑2𝑖+2𝑏𝑖+1) = 10𝑖 + 6; 1 ≤ 𝑖 ≤ 𝑛 𝜌∗(𝑑2𝑑1) = 3; 𝜌∗(𝑑2𝑖𝑑2𝑖+1) = 10𝑖; 𝜌∗(𝑑2𝑖+1𝑑2𝑖+2) = 10𝑖 + 5; 1 ≤ 𝑖 ≤ 𝑛 𝜌∗(𝑏𝑖+1 ′ 𝑑2𝑖+1) = 10𝑖 + 4; 𝜌∗(𝑏𝑖 ′𝑑𝑖+1) = 10𝑖 − 1; 1 ≤ 𝑖 ≤ 𝑛 𝜌∗(𝑎1 ′ 𝑏𝑖 ′) = 10𝑖 − 2; 1 ≤ 𝑖 ≤ 𝑛 Thus, 𝐴(𝐷𝑇𝑆)𝑛 ∘ 𝐾1 is a HM labeling. Fig. 4. 𝐴(𝐷𝑇𝑆)3 ∘ 𝐾1 Theorem 3.5. For every 𝑛 ≥ 1 there exists a Triple Triangular snake graph 𝑇3𝑆𝑛 ∘ 𝐾1, which admits HM labeling. Proof. Consider the 𝑇3𝑆𝑛 ∘ 𝐾1 graph with Vertex set 𝑉 = {𝑎𝑖, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖 , 𝑎𝑖 ′, 𝑏𝑖 ′, 𝑑𝑖 ′; 1 ≤ 𝑖 ≤ 𝑛} and Edge set 𝐸 = {(𝑎𝑖𝑏𝑖) ∪ (𝑑𝑖𝑏𝑖) ∪ (𝑐𝑖𝑑𝑖) ∪ (𝑑𝑖𝑑𝑖+1) ∪ (𝑏𝑖 ′𝑑𝑖),∪ (𝑎𝑖 ′𝑏𝑖 ′) ∪ (𝑑𝑖 ′𝑑𝑖); 1 ≤ 𝑖 ≤ 𝑛} Now let us define a function 𝜌: 𝑣 → {1,2,3, ⋯ 𝑞 + 1} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 17 https://internationalpubls.com Let us label the vertices as follows. 𝜌(𝑎1) = 1; 𝜌(𝑎𝑖+1) = 10𝑖 + 4; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌(𝑏1) = 2; 𝜌(𝑏𝑖+1) = 10𝑖 + 5; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌(𝑑1) = 5; 𝜌(𝑑𝑖+1 ′ ) = 10𝑖 + 2; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌(𝑑1) = 6; 𝜌(𝑑𝑖+1) = 10𝑖 + 1; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌(𝑐𝑖) = 10𝑖 + 3; 𝜌(𝑏𝑖 ′) = 10𝑖 − 2; 𝜌(𝑐𝑖 ′) = 10𝑖 − 1; 1 ≤ 𝑖 ≤ 𝑛 The induced edge labeling is as follows 𝜌∗(𝑎1𝑏1) = 1; 𝜌∗(𝑎𝑖+1𝑏𝑖+1) = 10𝑖 + 4; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌∗(𝑑1 ′ 𝑑1) = 6; 𝜌∗(𝑑𝑖+1 ′ 𝑑𝑖+1) = 10𝑖 + 1; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌∗(𝑑1 ′ 𝑏1) = 2; 𝜌∗(𝑑𝑖+2 ′ 𝑏𝑖+1) = 10𝑖 + 3; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌∗(𝑑2 ′ 𝑏1) = 3; 𝜌∗(𝑑2𝑖 ′ 𝑏𝑖+1) = 10𝑖 + 8; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌∗(𝑑1 ′ 𝑐1) = 4; 𝜌∗(𝑑2𝑖+1 ′ 𝑐𝑖+1) = 10𝑖 + 2; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌∗(𝑑2 ′ 𝑐1) = 5; 𝜌∗(𝑑2𝑖 ′ 𝑐𝑖+1) = 10𝑖 + 7; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌∗(𝑑2 ′ 𝑑2) = 8; 𝜌∗(𝑑2𝑖+1 ′ 𝑐2𝑖+2) = 10𝑖 + 6; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌∗(𝑏1 ′ 𝑑1 ′ ) = 7; 𝜌∗(𝑑𝑖+2 ′ 𝑏𝑖+1 ′ ) = 10𝑖 + 5; 1 ≤ 𝑖 ≤ 𝑛 − 1 𝜌∗(𝑏𝑖 ′𝑑2𝑖 ′ ) = 10𝑖; 𝜌∗(𝑎𝑖 ′𝑏𝑖 ′) = 10𝑖 − 1; 1 ≤ 𝑖 ≤ 𝑛 Thus, 𝑇3𝑆𝑛 ∘ 𝐾1 is a HM labeling. Fig. 5. 𝑇3𝑆3 ∘ 𝐾1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 18 https://internationalpubls.com 4. Application Definition 4.1. Plain text An original intelligible message is called as Plain text. Definition 4.2. Cipher text The Transformed message after coding is called as Cipher text. In this section, We used HM Labeling of 𝑇𝑆𝑛 ∘ 𝐾1 Graph to encode a message and created novel encoding and decoding techniques that increase the secrecy of the coded message. 4.3. Algorithm for Encoding • First assign the values of the alphabet ranges over 1-26. • It is the original position of the alphabet. Then shift the alphabet using the formula 𝑦 = (𝑥 + 𝑛)(mod26),where x denotes the original position of the alphabet and n denotes the length of the word. In this way, encoding table was created. • Convert the plaintext message into a sequence of integers using the encoding table. • Construct the 𝑇𝑆𝑛 ∘ 𝐾1 graph corresponding to the length ' 𝑛 ' of the plaintext message 𝑇𝑆𝑛 ∘ 𝐾1 graph, Let 𝑏𝑖, 𝑐𝑖 be the vertices of a Triangular snake . Let 𝑎𝑖, 𝑑𝑖 be the pendent vertices join 𝑎𝑖, 𝑏𝑖 and 𝑐𝑖, 𝑑𝑖 • 𝑣𝑖 denotes the vertices in the graph. 𝑒𝑖 denotes the edges in the graph. 𝑤𝑖(𝑒𝑖) denotes the weights of each edges. • Next,the weights 𝑤1, 𝑤2, … 𝑤𝑛 to each to the edges 𝑒1, 𝑒2, … 𝑒𝑛 in such a way that 𝑊1(𝑒1) < 𝑊2(𝑒2) … 𝑊𝑛(𝑒𝑛). 4.3.1. Method for calculating weight of edges • find the weight of each edge 𝑤𝑖(𝑒𝑖) by the formulation. 𝑤𝑖(𝑒𝑖) = (Numerical value of vertex 𝑣𝑖 corresponding to the edge 𝑒𝑖 )- 10𝑖, where 𝑖 = 1,2,3 … 𝑛. • The resulting values are the weight of each edge. Now assign the weight of each edge in the 𝑇𝑆𝑛 ∘ 𝐾1 graph. • Finally send this graph to the receiver by hiding the values of the vertex which is known as encrypted message along with the public key 10. 4.4. Algorithm for Decoding • Arrange the weights of edge in ascending order of mod values and add the multiples of 10 respectively. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 19 https://internationalpubls.com • Decode the cipher text into plain text message from the encoding table to get the original plain text message. • Plaintext: Bishop 4.5. Encoding • Length of the message = 6, 𝑦 = (𝑥 + 𝑛)(mod26) • Construct the 𝑇𝑆𝑛 ∘ 𝐾1 graph by assigning the above sequence of integers to the vertices of the 𝑇𝑆𝑛 ∘ 𝐾1 graph. Table 1. Encoding Table A B C D E F G H I J K L M 7 8 9 10 11 12 13 14 15 16 17 18 19 N O P Q R S T U V W X Y Z 20 21 22 22 23 24 25 26 1 2 3 5 6 Each letter from the table above represented as follows B I S H O P 8 15 25 14 21 22 Fig. 6. Encoding Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 20 https://internationalpubls.com • Applying weights to the 𝑤𝑖, 𝑖 = 1,2,3,4,5,6 to the corresponding edges of the vertices 𝑤1(8) < 𝑤2(15) < 𝑊3(25) < 𝑤4(14) < 𝑤5(21) < 𝑤6(22). • Weights are obtained by subtracting the multiples of 10 from each adjacent numeric value of the vertices. • Weight of edge 𝑒1 = 𝑤1 = 8 − 10 = −2 𝑒2 = 𝑤2 = 15 − 20 = −5 𝑒3 = 𝑤3 = 25 − 30 = −5 𝑒4 = 𝑤4 = 14 − 40 = −26 𝑒5 = 𝑤5 = 21 − 50 = −29 𝑒6 = 𝑤6 = 22 − 60 = −38 • Cipher graph is created by assigning the weight of each edge and hiding the values of the each vertices. 4.6. Encrypted Message Fig. 7. Encryption 4.7. Decryption • The receiver receives the encrypted message. • Arrange the weights of edges in ascending order of mod values | − 2| < | − 5| < | − 5| < | − 26| < | − 29| < | − 38| • Adding the multiples of 10 to each adjacent value: | − 2 + 1(10)| < | − 5 + 2(10)| < | − 5 + 3(10)| < | − 26 + 4(10)| < | − 29 + 5(10)| < | − 38 + 6(10)| • Sequence of integers is 8,15,25,14,21,22 • 8,15,25,14,21,22 = B, I, S, H, O, P Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 21 https://internationalpubls.com Therefore, we deduce from the encoding table that the plain text is BISHOP. 5. Conclusion In this paper, we investigated some families Triangular Snake graphs satisfy the condition of Harmonic mean labeling. We presented secret coding by using a revised GMJ technique with new labeling and numbering of alphabets based on vowels. In the future, we intend to introduce new labeling technique and prove coding, utilising various graphs in conjunction with various methods of alphabet numbering. It is very interesting and challenging as well as to investigate graph families which admit Harmonic Mean Labeling graph. We have presented new results on the Harmonic mean labeling of certain classes of graphs like 𝑇𝑆𝑛 ∘ 𝐾1.Analogous work can be carried out for other families and in the context of different types of graph labeling techniques. References [1] Gallian, J. A. (2018). A dynamic survey of graph labeling. Electronic Journal of combinatorics, 1(DynamicSurveys), DS6. [2] Harary, Graph theory, Addison Wesley, Massachusetts, 1969. [3] Somasundaram, S., & Ponraj, R. (2003). Mean labelings of graphs. National academy Science letters, 26(7-8), 210- 213. [4] David Raj, C., Jayasekaran, C., & Sandhya, S. S. (2016). Few families of harmonic mean graphs. 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