Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 139 https://internationalpubls.com A Reduction Analysis of the Weiner Index on Predicting the Boiling Point of Hydrocarbons Using Graph Technique R. Shankar*, Varadha Raj Manivannan1, M. Siva2 & E. Padmavathy3 *Assistant Professor, Department of Mathematics, School of Arts and Science, Vinayaka Mission’s Chennai Campus, Vinayaka Mission’s Research Foundation (Deemed to be University), Paiyanoor-603 104, Tamil Nadu, India. 1Assistant Professor, Department of Mathematics, Sri Chandrasekharendra Saraswathi Viswa Mahavidyalaya (Deemed to be University), Kanchipuram-631 561, Tamil Nadu, India. 2Assistant Professor, Department of Mathematics, St. Joseph’s Institute of Technology, OMR, Chennai-600 119, Tamil Nadu, India. 3Assistant Professor, Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Avadi, Chennai-600 062, Tamil Nadu, India. Email: rathinavelshankar@gmail.com, varadharaj.m219@gmail.com, sivam@stjosephstechnology.ac.in, padmavathyelu@gmail.com Article History: Received: 18-09-2024 Revised: 23-11-2024 Accepted: 31-11-2024 Abstract: Introduction: This study investigates the link between graph eccentricity and boiling points for nonanes and decanes, demonstrating graph theory's ability to predict the physical characteristics of organic compounds. Objectives: Inspired by Wiener’s work on topological indices, we create a novel power formula based on graph eccentricity to predict boiling points. This provides a more straightforward technique than typical wiener index computations. Methods: Our findings show a substantial association between graph eccentricity and boiling temperatures, demonstrating the importance of structural characteristics in predicting chemical responses. Results: This study advances chemical graph theory by proposing an alternate approach for estimating boiling points, underlining the importance of graph-based studies in understanding molecule structures and behaviours. Conclusions: This research examines the molecular graph’s eccentricity for 105 nonanes and decanes. Using a power formula, we discovered a substantial relationship between the boiling point and the eccentricity of the chemical structure. Keywords: Molecular graph, Nonanes and Decanes, Winer index, Eccentricity, R- Programming Language. 1. Introduction A graph 𝐺 consists of a set called vertices 𝑣 and edges 𝑒, such that each edge 𝑒𝑘 is identified with an unordered pair of vertices. The number of vertices is called the order of the graph |𝑉| = 𝑛 and the number of edges is called the size of the graph |𝐸| = 𝑚 in [2]. In 1875, Arthur Cayley (as instance of [1]) introduced a method for representing molecules using graph theory. In this model, vertices represent atoms, and edges represent chemical bonds between them. mailto:rathinavelshankar@gmail.com mailto:varadharaj.m219@gmail.com mailto:sivam@stjosephstechnology.ac.in mailto:padmavathyelu@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 140 https://internationalpubls.com This molecular graph approach provided a foundational tool for visualizing and analyzing molecular structures through graph theory. The Wiener index, introduced by Harold Wiener in [11], was the first graph metric applied in chemistry. The Wiener index (Wiener number) is a topological index of a molecule, defined as the sum of the lengths of the shortest paths between all pairs of vertices in the chemical graph representing the non-hydrogen atoms in the molecule (Harry Wiener, 1947). The Wiener index of G is the number 𝑊(𝐺) = 1 2 ∑ ∑ 𝑑𝐺(𝑖, 𝑗)𝑛 𝑗=1 𝑛 𝑖=1 is the entries of the distance matrix. In his 1947 study, Harry Wiener explored how the structure of paraffins influences their boiling points. He found that an organic compound’s boiling point and physical properties depend on the molecule's type, number, and arrangement of atoms. Wiener noted that differences in physical properties are due to changes in structural relationships, even though the number and type of atoms within an isomeric group remain constant. He also linked the Wiener polarity index to the Wiener index through a specific equation. ∆𝑡 = 98 𝑛2 ∆𝑤 + 5.5∆𝑝𝑝 (1) where ∆𝑡 is the boiling point of a group of isomers, ∆𝑤 = 𝑤0 − 𝑤, where 𝑤0 = ( 1 6 ) (𝑛 − 1)(𝑛)(𝑛 + 1) distances between any two carbon atoms in a molecule, in terms of carbon-carbon bonds. We denote ∆𝑝 = 𝑝0 − 𝑝, where 𝑝0 = 𝑛 − 3 and 𝑝, the polarity number is defined as the number of pairs of carbon atoms separated by three carbon-carbon bonds. A list of the detailed results obtained by applying this equation to the 37 paraffins from 𝐶4𝐻10 to 𝐶8𝐻18 and further extended this method for the boiling point data available for the nonanes and decanes. After Wiener, numerous researchers developed various techniques to determine physical and chemical properties from topological indices. The polarity number 𝑝 is the number of pairs of carbon atoms separated by three carbon-carbon bonds. He further defined the path number as the sum of the distances between two carbon atoms in a molecule in terms of carbon-carbon bonds. Later, these topological indices were named the Wiener index and the Wiener polarity index. Following its introduction, researchers have developed extensions of the Wiener index and created other indices, such as the Hosoya index in [7], the Gutman and Schultz indices in [6], and many other distance-based topological indices in [4], to predict various molecular properties. Figure 1: The Chemical Compound and Graph Structure Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 141 https://internationalpubls.com Shi, Kosari, Ahmad, Hameed, and Akhter’s [9], examined topological indices based on vertex degree, particularly within flabellum graphs. Their study visualized these indices, offering insights into their behaviour within this graph class. Additionally, they applied the fuzzy first Zagreb index to analyze branching patterns in cyber flabellum graphs, contributing to understanding cybercrime. A chemical graph is a labeled graph whose vertices correspond to the atoms of the compound and whose edges correspond to chemical bonds. Figure 1 shows a chemical compound's molecular graph and graph structure. The eccentricity 𝑒(𝑣) of a vertex 𝑣 in a connected graph 𝐺 is 𝑚𝑎𝑥 𝑑𝐺(𝑢, 𝑣) for all 𝑢 in 𝐺. The minimum eccentricity is the radius and it is denoted by 𝑟(𝐺). The maximum eccentricity is the diameter and it is denoted by 𝑑𝑖𝑎𝑚(𝐺). 𝑝𝐾𝑎-value is defined as a negative base-10 logarithm of the acid dissociation constant 𝐾𝑎 of a solution is 𝑝𝐾𝑎 = 𝑙𝑜𝑔10 𝐾𝑎. Divya and Yamuna determined the pKa value of local anaesthesia with similar structures and properties, such as boiling point, melting point, and vapour pressure, which are compressed by compressing the drug graph and reducing the calculation in [5]. Some more researchers investigated and determined the problem using various types of topological indices in [8], [10], and [12]. This paper analyses the molecular graph and graph eccentricity for the hydrocarbons. Using a power formula, our observations reveal a strong correlation between the boiling point and the eccentricity of the chemical structure. This formula can be applied to estimate the boiling point of a chemical based on its molecular structure. We also calculate the correlation coefficients between boiling point and Wiener index and between boiling point and eccentricity. 2. Results In chemical graph theory, the Wiener index (also Wiener number) is a topological index of a molecule, defined as the sum of the lengths of the shortest paths between all pairs of vertices in the chemical graph representing the non-hydrogen atoms in the molecule. Let 𝐺 be a graph with 𝑛 vertices. For each pair 𝑖, 𝑗 of vertices, let 𝑑𝑖𝑗 denote the distance between i and j. The Wiener Index of G is the number 𝑊(𝐺) = 1 2 ∑ ∑ 𝑑𝐺(𝑖, 𝑗)𝑛 𝑗=1 𝑛 𝑖=1 . An example of calculating the Wiener index for 2,3-dimethyl hexane is shown in Figure 2. Figure 2: 2,3-dimethyl hexane Structure and Graphical Representation We observe that while calculating the Wiener index for a graph 𝐺 with n vertices, we need to determine the distance between every pair of vertices. 𝑊(𝐺) = 1 2 (20 + 14 + 12 + 14 + 18 + 24 + 20 + 18) = 70. Moreover, we have to create 𝑛 ∗ 𝑛 matrix and determine the row sums. If a boiling point can be determined by using any other graph property that involves less calculation, then it becomes simple to Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 142 https://internationalpubls.com determine the physical properties, which is attempted in this article. We recollect that the structural arrangement of atoms and variations in physical properties are related. To satisfy this, Wiener defined a topological index based on distance between vertices. Using this property, we try to simplify the calculation process involved in determining the topological index. Moreover, while defining this index care has been taken to include all the vertices of 𝐺. Figure 3. 3-ethyl pentane Structure and Graphical Representation Graph eccentricity is a graph property that involves distance and can be defined on every vertex of 𝐺. Graph eccentricity given for any molecular structure. Determine the molecular graph. Find the eccentricity of all the vertices of the molecular graph. We define the sum of the eccentricity of all the vertices has the eccentricity of the graph. The example for calculation of eccentricity of 3- Ethyl pentane as shown in Figure 4. Figure 4. The Eccentricity value is 23 for 3-ethyl pentane 2.1 Determination of Power Formula An equation of the form 𝐵 = 𝛼𝑊𝛽 relating the boiling point 𝐵 and the Wiener index 𝑊This is an approximated power formula determined by Colin Adams and Robert Franzosa in [3], shown in Equation (2). We try to develop a similar kind of formula using the eccentricity of any given molecular graph. 𝐵 = 181 × 𝑊0.1775 (2) Figure 5 displays a graph showing the correlation between the original boiling point and eccentricity. We observe that a strong correlation between boiling point and eccentricity of the chemical structure can be approximated by an increasing curve. Fitting a power equation 𝐵 = 𝛼𝐸𝛽 and find that the relation between the boiling point 𝐵 and eccentricity 𝐸 for the data in Table 1 is approximated by 𝐵 = 82 × 𝐸0.15 (3) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 143 https://internationalpubls.com Figure 5. A scatter diagram of the original boiling point and eccentricity index Table 1: The Boiling Points for Nonanes and Decanes using Power Formula SI.No Name Wiener Index Eccentricity Original Boiling point (°𝑪) Calculated Boiling Point (°𝑪) from power formula 𝑩𝒑 = 𝟏𝟖𝟏𝑾𝟎.𝟏𝟕𝟕𝟓 Calculated Boiling Point (°𝑪) from power formula 𝑩𝑬 = 𝟖𝟐𝑬𝟎.𝟏𝟓 1 n-Nonane 120 56 150.8 150.25 149.9 2 2-methyoctane 114 51 143.3 146.85 147.2 3 3-methyoctane 120 50 144.2 150.25 147.4 4 4-methyoctane 108 49 142.5 142.85 147 5 3-Ethylheptane 104 44 143 139.85 144.6 6 4-Ethylheptane 102 42 141.2 138.15 143.6 7 2,2- Dimethylheptane 104 45 130.5 139.85 145.1 8 2,3- Dimethylheptane 102 44 140.5 138.15 144.6 9 2,4- Dimethylheptane 102 43 133 138.15 144.1 10 2,5- Dimethylheptane 104 44 136 139.85 144.6 11 2,6- Dimethylheptane 108 45 135.2 142.85 145.1 12 3,3- Dimethylheptane 98 43 137.3 135.25 144.1 13 3,4- Dimethylheptane 98 42 140.5 135.25 143.6 14 2-Methyl-3- ethylhexane 96 38 139 133.85 141 15 2,2,3- Trimethylhexane 92 38 133.4 130.85 141 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 144 https://internationalpubls.com 16 2,2,4- Trimethylhexane 94 38 126.5 131.85 141 17 2,2,5- Trimethylhexane 98 39 124.1 135.35 142 18 2,3,3- Trimethylhexane 90 37 138 129.15 140 19 2,3,5- Trimethylhexane 96 38 131.4 133.85 141 20 2,4,4- Trimethylhexane 92 37 131 130.85 140 21 3,3,4- Trimethylhexane 88 36 139 127.85 140.3 22 3,3-Diethylpentane 88 30 146.5 127.85 137 23 2,2-Dimethyl-3- ethylpentane 88 31 133.8 127.85 137.2 24 2,3-Dimethyl-3- ethylpentane 86 30 142 125.85 136.5 25 2,4-Dimethyl-3- ethylpentane 90 31 136.7 129.15 137.2 26 2,2,3,3- Tetramethylpentane 82 30 140.2 122.85 136.5 27 2,2,3,4- Tetramethylpentane 86 31 133 125.85 137.2 28 2,2,4,4- Tetramethylpentane 88 32 122.3 127.85 137 29 2,3,3,4- Tetramethylpentane 84 30 141.5 123.85 136.5 30 n-Decane 165 70 174 174.85 155 31 2-Methylnonane 158 64 166.8 170.85 153 32 3-Methylnonane 153 63 167.8 168.85 152.7 33 4-Methylnonane 150 62 165.7 167.35 152.3 34 5-Methylnonane 149 61 165.1 166.85 152 35 2,4-Dimethyloctane 142 54 153.2 163 149 36 2,5-Dimethyloctane 143 56 159 163.6 150 37 2,6-Dimethyloctane 146 57 160 165.2 150.4 38 2,7-Dimethyloctane 151 58 160.2 167.8 151 39 3,3-Dimethyloctane 138 56 161.2 160.85 150 40 3,6-Dimethyloctane 141 54 160.8 162.85 149.2 41 4,5-Dimethyloctane 135 54 161 158.85 149.2 42 4,n-Propylheptane 138 48 161.7 160.85 147 43 4-Isopropylheptane 131 44 158.6 156.85 145 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 145 https://internationalpubls.com 44 2-Methyl-5- ethylheptane 138 50 158.4 160.85 147.5 45 2,2,4- Trimethylheptane 131 44 147 156.25 145 46 2,2,6- Trimethylheptane 145 42 148.9 164.85 144 47 2,3,3- Trimethylheptane 127 49 160 154.85 147 48 2,3,6- Trimethylheptane 136 51 155.3 159.85 148 49 2,4,4- Trimethylheptane 127 47 151 154.85 146.1 50 2,4,6- Trimethylheptane 135 49 147.6 158.85 147 51 2,3,5- Trimethylheptane 131 50 152.8 156.85 147.5 52 3,4-Diethylhexane 125 42 160.7 153.35 144 53 2,2-Diehtyl-4- ethylhexane 126 43 148 153.85 144.2 54 2,2,3,4- Tetramethyhexane 118 38 156.5 148.85 142 55 2,2,4,5- Tetramethylhexane 124 43 145.8 152.85 144.2 56 2,2,5,5- Ttramethylhexane 127 44 136.8 154.85 145 57 3,5- Dimethylheptane 100 40 136 136.85 142.6 58 2-methyl-4- ethylhexane 98 38 134 135.25 141 59 3-Ethyloctane 145 57 168 160.85 150.4 60 4-Ethyloctane 141 55 163.6 162.85 150 61 2,2-Dimethyloctane 146 58 155 165.25 150.8 62 2,3-Dimethyloctane 143 57 164 163.85 150.4 63 3,4-Dimethyloctane 137 55 163 159.85 150 64 3,5-Dimethyloctane 138 55 158.6 160.85 150 65 4,4-Dimethyloctane 134 54 161 158.85 149.2 66 2-Methyl-3- ethylheptane 134 50 161.9 158.85 147.5 67 2-Methyl-4- ethylheptane 134 45 156 158.85 145.1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 146 https://internationalpubls.com 68 3-Methyl-4- ethylheptane 129 47 154.17 155.85 146.1 69 3-Methyl-5- ethylheptane 88 43 155 127.85 144.2 70 4-Methyl-3- ethylheptane 130 48 163.1 155.85 146.6 71 4-Methyl-4- ethylheptane 126 46 151.8 153.85 145.6 72 2,2,3- Trimethylheptane 130 50 157.6 155.85 147.5 73 2,2,5- Trimethylheptane 134 50 150 158.85 147.5 74 2,3,4- Trimethylheptane 128 48 161.7 154.85 146.6 75 2,3,5- Trimethylheptane 131 49 157.9 156.85 147 76 2,4,5- Trimethylheptane 130 48 155.7 155.25 146.6 77 3,3,4- Trimethylheptane 123 47 162.8 151.85 146.1 78 3,3,5- Trimethylheptane 126 48 155.8 153.85 146.6 79 3,4,4- Trimethylheptane 122 46 161.4 151.85 145.6 80 3,4,5- Trimethylheptane 125 47 162.2 153.35 146.1 81 3-Methyl-3- isopropylhexane 124 43 166.7 152.85 144.2 82 3,3-Diethylhexane 121 42 167.6 150.85 144 83 2,2-Dimethyl-3- ethylhexane 122 43 155.2 151.85 144.2 84 2,3-Dimethyl-3- ethylhexane 119 42 164.1 149.85 144 85 2,3-Dimethyl-4- 161.1ethylhexane 123 42 161.2 152.03 144 86 2,4-Dimethyl-4- ethylhexane 122 42 161.1 151.85 144 87 2,4-Dimethyl-3- ethylhexane 122 38 159 151.85 142 88 2,5-Dimethyl-3- ethylhexane 127 43 153.5 154.85 144.2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 147 https://internationalpubls.com 89 3,3-Dimethyl-4- ethylhexane 118 41 162.1 148.85 143.1 90 3,4-Dimethyl-3- ethylhexane 117 41 162.1 148.35 143.1 91 2,2,3,3- Tetramethylhexane 115 42 161.3 147.05 144 92 2,2,3,5- Tetramethylhexane 123 43 149.4 152.05 144.2 93 2,2,4,4- Tetramethylhexane 119 42 152.4 149.85 144 94 2,3,3,4- Tetramethylhexane 115 41 163.9 147.05 143.1 95 2,3,3,5- Tetramethylhexane 120 46 154.5 150.25 146 96 2,3,4,4- Tetramethylhexane 116 41 162.2 147.65 143.1 97 2,3,4,5- Tetramethylhexane 121 42 156.2 150.85 144 98 3,3,4,4- Tetramethylhexane 111 40 168.4 144.85 143 99 2,4-Dimethyl-3- isopropylpentane 117 35 154.9 148.25 140 100 2-Methyl-3,3- diethylpentane 114 34 169.5 146.35 139.2 101 2,2,3-Trimethyl-3- ethylpentane 110 34 166.6 143.65 139.2 102 2,2,4-Trimethyl-3- ethylpentane 115 35 154.5 147.05 140 103 2,3,4-Trimethyl-3- ethylpentane 112 34 169.4 145.05 139.2 104 2,2,3,3,4- Pentamethylpentane 108 34 166.1 142.35 139.2 105 2,2,3,4,4- Pentamethylpentane 111 35 159.6 144.85 139.8 3. Results and Discussion The molecular graph’s eccentricity for 105 nonanes and decanes. Using a power formula, we discovered a substantial relationship between the boiling point and the eccentricity of the chemical structure in Table 1. The molecular graph comparison Figure 6 between the original boiling point Vs boiling point calculated from determined power formula in Equation 3. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 148 https://internationalpubls.com Figure 6. Comparision on Original Boiling Point and Determinate Power Formula We observe that the power formula developed can be used to determine the approximate boiling point of any chemical whose molecular structure is known. Let 𝐵𝑊, 𝐵𝑃 and 𝐵𝐸 denote the boiling points of nonanes and decanes using formulas (1), (2), (3) respectively. The following Figure 7 and Figure 8 provides an R program for determining the correlation coefficient between 𝐵𝑊, 𝐵𝐸 and 𝐵𝑃, 𝐵𝐸 respectively. Figure 7. The Correlation Coefficient Between 𝐵𝑊, 𝐵𝐸 and 𝐵𝑃, 𝐵𝐸 Figures 9 and 10 show a good correlation between the formulas. From this, we understand that the graph eccentricity can be used instead of the Wiener index for determining the boiling point of nonanes and decanes. Figure 9. The Correlation Coefficient Between 𝐵𝑊, 𝐵𝐸 Figure 10. The Correlation Coefficient Between 𝐵𝑊, 𝐵𝑃, 𝐵𝐸 4. Conclusion This research examines the molecular graph’s eccentricity for 105 nonanes and decanes. Using a power formula, we discovered a substantial relationship between the boiling point and the eccentricity of the chemical structure. This formula may be used to predict the boiling point of a chemical based on its molecular structure, where (𝑛2 − 𝑛) calculations are reduced. We also calculate the correlation coefficients between the boiling point and the Wiener index and the boiling point and eccentricity. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 149 https://internationalpubls.com 5. Acknowledgement The authors would like to express their sincere gratitude to the management and faculty members of the School of Arts and Science, Vinayaka Mission’s Chennai Campus, Vinayaka Mission’s Research Foundation (Deemed to be University), Sri Chandrasekharendra Saraswathi Viswa Mahavidyalaya (Deemed to be University), St. Joseph’s Institute of Technology, and Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology for their encouragement and support throughout this research. We especially thank our colleagues for their valuable suggestions, and we extend our heartfelt thanks to the reviewers for their constructive feedback, which helped refine this paper. Conflict of interest statement: The authors declared no conflict of interest in the manuscript. References [1] Burch KJ, Wakefield DK, Whitehead EG. Boiling point models of alkanes. 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