Predictive Model among Graph Configurations

Introduction
In recent times, graphs have emerged as a crucial mathematical tool across diverse fields, encompassing theory, operational research, chemistry, genetics, linguistics, electrical engineering, geography, sociology, and architecture. They constitute a significant mathematical discipline, with varied and extensive application areas actively utilized across numerous scientific branches. Understanding the fundamental structure of graphs is imperative for leveraging graph theory across different disciplines and applications. Presenting academic research and its findings in the scientific realm is greatly enhanced through the use of figures, pictures, graphics, and tables.
\par Graph theory, the study of graphs, aids in manipulating graph structures to illustrate relationships between graphs and their configurations. Originating from Leonard Euler's work on the Seven Bridges problem in 1735, it led to the discovery of Eulerian graphs. Graph theory is a significant and multifaceted field widely applied in professional domains such as engineering, architecture, programming, and design. Particularly, it offers valuable tools for modeling complex problems and serves as a guide in problem-solving. It enables tackling intricate issues like low-cost routing, mapping, and program analysis. The utility of graph theory extends to optimizing routing in traffic flows within network systems' routers and switches. Additionally, it finds effective application in science across various domains, including electronic circuitry, urban planning, transportation networks, and computer networks.
\par Topological indices, which are scalar numerical values linked to molecular compositions, are essential for establishing the relationships between chemical structure and physical, biological, and reactive characteristics. The origins of topological indices can be found in Wiener's investigation of paraffin's boiling point \cite{WH}, which launched the field of graphical molecular descriptor research. A graphical index serves as a structurally invariant number, generating a mathematical real number from a graph structure. There are hundreds of topological descriptors. The origin of their definition parameters can easily classify them. Degree, distance, and eigenvalue characterize topological molecular descriptors.
\par A fundamental understanding of its basic structure is imperative to utilize graph theory effectively across diverse disciplines. Graphs comprise two sets, typically denoted as $G=(V, E)$. In this representation, the set of nodes is depicted as $V={v_{1}, v_{2},…, v_{n}}$, while the set of edges is represented as $E={e_{1}, e_{2},…, e_{n}}$, with $(E \subset V\times V)$. The number of edges within a graph is referred to as the order of the graph. Additionally, each edge can be assigned a "weight" signifying the degree of connectivity between the edge and its attached vertex. The degree of a vertex denotes the number of edges connected to it, offering insights into the relationship between nodes and edges within a graph. Two elements in a graph, a vertex and an edge, are deemed incident to each other if they are directly connected. \par 
Graphs are categorized into various types based on factors such as the presence or absence of directed edges, the number of nodes, or the nature of the edges. Examples of basic graph types include simple, weighted, connected, directed, undirected, and regular graphs. Understanding these distinctions lays the groundwork for applying graph theory effectively in diverse contexts. Graph configurations serve as foundational structures in graph theory, providing fundamental building blocks for understanding and analyzing more complex graph structures. These configurations, including complete graphs, cycles, paths, and trees, form the basis of numerous theoretical and practical applications across various disciplines. Predictive modeling is a process used in data science and machine learning to develop models that predict future outcomes based on historical data. It involves building mathematical algorithms that learn from existing data to make predictions or decisions about unseen or future events. Jose M Rodriguez et al. studied multiplicative degree-based topological indices to perform a detailed analytical and statistical study of random networks \cite{AS}. They find analytical inequalities involving these multiplicative indices. Also, they applied multiplicative indices to three models of random networks: Erdos Renyi networks, random geometric graphs, and bipartite random networks. Recently, within a statistical approach to degree-based topological indices (TIs) on random graphs, it has been shown that the average values of TIs like the average Randic index, the average harmonic index, and the average of some Sombor indices are highly correlated with the average Shannon entropy of the eigenvectors of the corresponding adjacency matrix. \cite{RA, CT, CT2}. They studied the predicted comparison of topological and spectral properties of different graph configurations involving random geometric graphs.

Applications of Graphs and Topological Indices in Scientific Studies
Below are the most common applications of Graph Theory and topological indices.

Engineering Sciences
The ongoing need to analyze physical networks remains a crucial aspect for planners. Graphs play a vital role in the planning and analysis of electricity grids within mega cities. Moreover, they find applications in designing circuit connections in electrical engineering, known as topologies. Various topologies, such as series, bridge, star, and parallel topologies, are utilized in this context. Circuit boards employ planar graphs, enabling the study of their topologies through different topological indices.

chemistry
Graph theory, an integral branch of mathematics, holds significant importance in chemistry, providing a powerful tool for modeling and comprehending complex systems. Graph theory finds diverse applications within chemistry, underscoring its relevance and impact. Specifically, chemical graph theory operates on organic structures by representing compounds as chemical graphs, establishing a one-to-one correspondence between atoms and vertices, and bonds to edges. These topological descriptors are valuable for generating regression models to predict physicochemical characteristics such as sublimation enthalpies, formation, combustion, and vaporization \cite{AA}, normal boiling points \cite{GI}, and various

Social Networks
Graph theory has found numerous applications in various domains, particularly in complex networks where social networks exhibit diverse topological characteristics such as dynamics, heterogeneity, high transitivity, relatively short average distances between vertices, and degree associativity. These characteristics vary significantly across individual vertices within a network \cite{OE}. Recently, social networks have emerged as prominent platforms for information-sharing and communication. Information analytics firms leverage social trends and user sentiments towards brands and employ techniques like web and data mining, sentiment analysis, and personality analysis. Social networks enable billions of Internet users worldwide to create and consume digital content, facilitating rapidly disseminating information to a large audience. Graphs are essential tools for analyzing and visualizing this propagation process, allowing representation of the influence and trust relationships among users within a social network. Utilizing graph theory and topological indices, complex models involving vast numbers of users can be effectively represented, providing insights into the dynamics and structure of these networks.

Data Science
Data science plays a pivotal role in extracting valuable insights from extensive datasets, employing diverse principles, processes, and techniques to uncover patterns and generate knowledge. Grounded in mathematics and statistics, the data science journey starts with real-world data and leads to the development of innovative products or insights applied in practical contexts. Visual representation through data science graphs proves indispensable, clarifying complex relationships and structures within data. These graphs and graph theory provide researchers with a robust framework to explore, analyze, and convey insights effectively.