Abstract
Numbering a graph is a very practical and effective technique in science and engineering. Numerous graph assignment techniques, including distance-based labeling, topological indices, and spectral graph theory, can be used to investigate molecule structures. Consider the graph , with the injection from the node set to , where is the sum of the number of nodes and links. Assume that the induced link assignment is the ceiling function of the average of root square, harmonic, geometric, and arithmetic means of the vertex labels of the end vertices of each edge. If the union of range of of the node set and the range of of the link set is the set , then is called a super classical average assignment (SCAA). This is known as the SCAA criterion. In this study, the graphical structures corresponding to chemical structures based on the SCAA criterion are demonstrated. The graphical depiction of chemical substances was first defined and second, the union of any number of cycles , the tadpole graph, the graph extracted by identifying a line of any two cycles and , and the graph extracted by joining any two cycles by a path are all examined in this work.
1. Introduction
The graph theory is also used to study molecules in chemistry [1]. The molecular graph theory is a branch of the graph theory that focuses on the study of the properties of molecules using graphs. In this context, a molecular graph is a graph where the vertices represent atoms and the edges represent chemical bonds between the atoms. The molecular graph theory uses various graph-theoretic concepts and techniques to analyze the properties of molecules. One of the main applications of the molecular graph theory is to study the topological properties of molecules, such as their connectivity, degree sequence, and distance matrix. For example, the degree sequence of a molecular graph represents the number of bonds that each atom has, which can be used to identify functional groups and other structural features of the molecule. Another application of the molecular graph theory is to study the chemical properties of molecules, such as their reactivity, stability, and biological activity. Graph-theoretic measures such as the Wiener index, the Randi index, and the Zagreb indices have been used to predict the biological activity of molecules based on their molecular graph [2, 3]. It is also used in computational chemistry, where it is used to model and simulate the behavior of molecules. For instance, molecular graphs are used in molecular dynamics simulations to depict the structure of molecules and to mimic their motion and interactions through time. It is an important tool for understanding the properties of molecules and their behavior in chemical reactions. It has applications in drug discovery, material science, and many other fields of chemistry. Graph labeling is an important tool in the molecular graph theory, which studies the properties of molecules using the graph theory. In this context, graph labeling is used to represent atoms and bonds in a molecule and to label these atoms and bonds with various attributes, such as atomic number, bond order, and hybridization. Quantitative structure-activity relationship (QSAR) studies use mathematical models to predict the biological activity of compounds based on their chemical structure. Graph labeling is used to represent the chemical structure of the compounds, and the labels can be used as input features for the QSAR model. Graph labeling is used to represent chemical reaction networks, which are networks of chemical reactions that occur in a system [4]. The labels on the nodes and edges of the graph represent the chemical species and the reactions, respectively. It can also be used to measure the similarity between two molecules. The labels on the atoms and bonds of the molecules are used to calculate a similarity score between the two molecules, which can be used to predict their biological activity or other properties.
A graph is described as a finite, undirected, and simple graph throughout this work. Considering the graph , which has vertices and edges, researchers are using the notations and contexts [5–8] and proposing [9–16] for a comprehensive review of the graph theory and its applications. represents a path containing vertices, while represents a cycle containing vertices. The graph of graphs and with disjoint vertex sets and and edge sets and are the graphs with and . The union of number of cycles of equal length is indicated by .
2. Literature Survey
In [17], Barrientos examined at graceful labelings of chain and corona graphs. The mean labeling [18] of graphs was defined by Somosundaram and Ponraj, and super mean labeling [19] was discussed by Vasuki and Arockiaraj. Baskar and Arockiaraj proposed the principles of -geometric mean labeling [20] and super geometric mean labeling [21] and highlighted them for several standard graphs. In [22], the authors introduced -exponential mean graphs. In [23], Durai Baskar and Saratha Devi created the idea of super C-logarithmic mean labeling. The super fibonacci graceful labelings of several graphs were described by Vaidya and Prajapati [24]. Muhiuddin et al. defined classical mean graph labeling [25], whereas Alanazi et al. expanded it to specific ladder graphs [26]. Recently, the authors explained that the classical mean labeling of various graphs has been developed by duplicating operations [27]. Here, the researchers extend and investigate a super classical mean-labeling techniques based on graph operation strategies in a significant number of investigators in the area of graph labeling.
3. Methodology
Consider the graph , with the injection from the node set to , where is the sum of the number of nodes and links. Assume that the induced link assignment is the ceiling function of the average of root square, harmonic, geometric, and arithmetic means of the vertex labels of the end vertices of each edge. If the union of range of of the node set and the range of of the link set is the set , then is called a super classical average assignment (SCAA). This is known as the SCAA criterion. Here, the researchers have demonstrated the SCAA by using the ceiling function and established the SCAA of some standard structures.
1-ethyl-2-methylcyclobutane is an organic compound with the molecular formula [1]. Let us break down the name to apprehend its composition as follows: “1-ethyl” designates that there is an ethyl group attached to the first carbon atom of the cyclobutane ring. “2-methyl” designates that there is a methyl group committed to the second carbon atom of the cyclobutane ring. Now, let us visualize the molecule. It has a cyclobutane ring, which is a four-membered carbon ring, where each carbon atom is attached to two hydrogen atoms. The first carbon atom in the ring has an ethyl group attached to it, which comprises two carbon atoms and five hydrogen atoms. The second carbon atom in the ring has a methyl group attached to it, which consists of one carbon atom and three hydrogen atoms. The chemical formulation of 1-ethyl-2-methylcyclobutane can be written as , representing it has 8 carbon atoms and 16 hydrogen atoms in its structure. This molecule is an illustration of a cycloalkane with substituent groups, making it an attention-grabbing compound in organic chemistry. It is a cyclobutane derivative with one ethyl group and one methyl group attached to the cyclobutane ring. Figure 1 depicts 1-ethyl-2-methylcyclobutane and its corresponding SCAA in the hydrogen-depleted molecular graph. Here, the number of points and links are 7. Based on the SCAA, the vertex and edge sets are and , respectively.
4. Results and Discussion
The assignment of various simple graphs and some chain graphs based on super classical averages are discussed. By implementing these super classical averages, it provides more accurate assignment values. Also, the collection of unions of node and link assignment values are , where is the sum of nodes and links.
4.1. Super Classical Meanness of Some Simple Graphs
Theorem 1. For , the union of any number of cycles meets its SCAA criterion.
Proof. Let the graph be the union of cycles. Let be the vertices of the cycle with and .
Assign the following values to .
When is odd,The values associated with the influenced line assignment will be provided byWhen is even,The values associated with the influenced line assignment will be provided byLet and , thenThe values associated with the influenced line assignment will be provided byHence, satisfies the required numbering and also the resultant graph admits required labeling. Figure 2 depicts an SCAA of .
Theorem 2. The graphical structure meets its SCAA criterion.
Proof. Let and . is constructed by identifying the vertex and while and and while is even. Case (i). . Sub case (i). is odd. Assign the following values to . The values associated with the influenced line assignment will be provided by Sub case (ii). is even. Assign the following values to . The values associated with the influenced line assignment will be provided by Case (ii). n = 4. Assign the following values to . The values associated with the influenced line assignment will be provided by Hence, satisfies the required numbering and also the resultant graph admits required labeling. Figure 3 depicts an SCAA of .
Theorem 3. For any and , the graphical structure obtained by identifying an edge of any two cycles and meets its SCAA criterion.
Proof. Let and . Let be the resultant graph obtained by identifying the edge of the cycle with the edge of the cycle when is odd and of with the edge of when is even. Case (i). and are odd. Assign the following values to . The values associated with the influenced line assignment will be provided by Case (ii). is odd and Assign the following values to . The values associated with the influenced line assignment will be provided by Case (iii). is even and . Assign the following values to . The values associated with the influenced line assignment will be provided by Case (iv). and are even. Assign the following values to . The values associated with the influenced line assignment will be provided by Hence, satisfies the required numbering and also the resultant graph admits required labeling.
Theorem 4. For any and , the graphical structure obtained by joining any two cycles and by a path meets its SCAA criterion.
Proof. Let and . Let be the vertices of the path with when is odd, when is even, and . Case (i). is odd. Assign the following values to . The values associated with the influenced line assignment will be provided by Case (ii). is even. Assign the following values to . The values associated with the influenced line assignment will be provided by Hence, satisfies the required numbering and also the resultant graph admits required labeling.
5. Conclusion
The assignment of various simple graphs and some chain graphs based on super classical averages were discussed. By the implementation of the SCAA criterion on the various chain-related graphs, namely, the union of any number of cycles , the tadpole graph, the graph extracted by identifying a line of any two cycles and , and the graph extracted by joining any two cycles by a path were discussed in detail. Also, the collection of unions of vertex and edge assignment values are , where is the sum of nodes and links. The following are the primary benefits of the suggested method: For getting the assignment values of chemical compounds, they will be constructed as a corresponding graph. So, this interpretation provides a simple graph structure. This method can be applied in the chemical sciences, and so many complicated chemical structures are easily analyzed by its characterization. The following are some restrictions on this work: This work mainly focuses on some standard graphs. This method is tedious when the number of vertices and edges is huge. To find out whether the graph admits that the assignment itself is a difficult problem, it is better to study for the class of graphs. For example, a complete graph has no such assignment. The future potential studies determined on the SCAA of the lines and linkages in the structures involve (i) preservation of line graph operation, (ii) Cartesian product operation of two graphs, (iii) tensor products of two graphs, etc. Using alternative graph operations, similar results can be found for a variety of other graphs. Furthermore, the results can be used to develop and investigate unknown electrochemical systems using graph-labeling techniques.
Data Availability
The data used to support the findings of this study are available from the corresponding author without undue reservation.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to this work. All the authors have read and approved the final version manuscript.
Acknowledgments
This research was funded by Princess Nourah Bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R59), Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia.