Abstract
Graph theory (GT) is a mathematical field that involves the study of graphs or diagrams that contain points and lines to represent the representation of mathematical truth in a diagrammatic format. From simple graphs, complex network architectures can be built using graph operations. Topological indices (TI) are graph invariants that correlate the physicochemical and interesting properties of different graphs. TI deal with many properties of molecular structure as well. It is important to compute the TI of complex structures. The corona product (CP) of two graphs and gives us a new graph obtained by taking one copy of and copies of and joining the th vertex of to every vertex in the th copy of . In this paper, based on various CP graphs composed of paths, cycles, and complete graphs, the geometric index (GA) and atom bond connectivity (ABC) index are investigated. Particularly, we discussed the corona products , , , , and and GA and ABC index. Moreover, a few molecular graphs and physicochemical features may be predicted by considering relevant mathematical findings supported by proofs.
1. Introduction
Graph theory (GT) stands as a foundational mathematical discipline that explores the intricate interplay of graphs and diagrams as tools for visualizing and representing mathematical truths. Within this realm, the amalgamation of points and lines provides a canvas upon which complex relationships are elegantly portrayed in a diagrammatic format. The potency of graphs extends beyond mere visual representations, allowing for the construction of intricate network architectures through the application of various graph operations.
A central focus within graph theory is the notion of topological indices (TI), which serve as fundamental graph invariants connecting the realm of abstract graphs to the tangible world of physicochemical properties. These indices are powerful tools that encapsulate and correlate intriguing properties of graphs, reaching beyond the boundaries of pure mathematics into the realm of molecular structures. By providing insights into the structural characteristics of molecules, TI enable the deciphering of their physicochemical intricacies, offering a profound connection between mathematical abstraction and real-world phenomena.
In this context, the computation of TI, particularly within the framework of complex structures, emerges as a critical endeavor. This importance is magnified when considering the intricacies of intricate graphs generated through the corona product (CP) operation. The CP of two graphs, and , crafts a novel composite graph by combining a single instance of with copies of . This synthesis results in each vertex of intricately connecting to every vertex within its corresponding copy of , yielding a versatile platform for the exploration of structural intricacies within complex graph compositions.
The present research embarks on an exploratory journey through the landscape of CP graphs, placing particular emphasis on compositions involving paths, cycles, and complete graphs. A pivotal facet of this exploration resides in the investigation of two crucial indices: the geometric index (GA) and the atom bond connectivity (ABC) index. These indices bear the responsibility of encapsulating geometric patterns and atomic bonding structures within the domain of CP graph configurations. As the inquiry unravels, an intricate tapestry of relationships emerges, shedding light on the interplay between CP operations and the nuanced behaviour of these indices.
The trajectory of this investigation is guided by a constellation of foundational references, which collectively enrich the understanding of graph theory. Among these beacons are the works of Chartrand and Lesniak [1], Carlson [2], Afzal et al. [3], Alon and Lubetzky [4], Randic [5], Cash [6], and Lamprey and Barnes [7], among others. Each of these references contributes a unique thread to the complex fabric of graph theory, stitching together the intricate narrative that spans from mathematical abstraction to real-world application. In [8], the authors have calculated degree-based topological indices of generalized subdivision double-corona product. Moreover, readers may study some more literature in [9, 10].
As this journey unfolds, not only does it deepen our comprehension of mathematical relationships, but it also opens doors to the prediction of molecular graphs and the unraveling of physicochemical attributes. The nexus of mathematical rigor and tangible application establishes the groundwork for advancing both theoretical understanding and practical prediction, exemplifying the multifaceted impact of graph theory in diverse domains.
Through a symphony of mathematical insights and tangible applications, this research seeks not only to contribute to the ongoing discourse within graph theory but also to underscore the profound symbiosis between mathematical exploration and its real-world consequences. As we delve into the intricate worlds of CP graphs and their accompanying indices, we engage in a harmonious dance between abstraction and application, deepening our understanding of both mathematical beauty and physical reality.
Let and be 2 graphs, each with a group of vertex and and a group of edges and , respectively. We described the corơna product GoH as the prơduct of twơ graphs, and , achieved by combining each vertex of copies of .
Let be a nontrivial, simple, or undirected graph. An independent set of vertices in an adjacent graph is known as an independent set. A graph’s dominating set is a set of vertices in which every vertex in is not adjacent to a vertex in . A set that is both dominant and independent in a graph is called an independent set. To determine their properties, it is crucial to know these composite molecular graphs’ topological indices [3]. Topological indices of product graphs have been a fascinating area of study in recent years, and numerous articles offer formulas for various topological indices of various graph compositions [11]. Accordingly, researchers were taken by these results and were inspired to investigate the ABC index [10] and GA index [11] of the corona products of various graph architectures. The ABC index was introduced in 1998 [10]. As a result of this index, heat is used to characterize the way in which alkane production is affected by vertex degrees [10]. Detecting an independently dominating number of corona products of path, cycle, wheel, and ladder graphs will be investigated in this study. Consider a simple connected undirected graph with vertices; then, the Randic index is defined as follows: where is the degree of vertex .
Consider a simple connected undirected graph that has nodes, and then, the ABC index is defined as follows: where is the degree of vertex .
In addition, the GA index in 2009 [12] by considering the degrees of vertices in a graph.
The GA index is defined as follows:
The corona product of and is defined as the graph obtained by taking one copy of graph and copies of , where each vertex of the th copy of relates to the th vertex of and is denoted by .
Figure 1 shows the corona product of two graphs. We will discuss different families of corona product graphs and calculate their topological indices.
2. Main Results
Let , , and be a path, cycle, and complete graphs on vertices. In this section, we discuss the ABC index and the GA index of , , , , and .
Theorem 1. The ABC index and the GA index of the corona product of two path graphs and are given by the following equations:
Proof. Consider the corona product of two path graphs, denoted as . In the case where both and are greater than 1, it is evident that the vertices within this composite graph can be categorized into four distinct types based on their respective degrees. Specifically,
(1)the first type encompasses vertices with a degree of 2(2)the second type consists of vertices with a degree of 3(3)the third type comprises vertices with a degree of (4)the fourth type encompasses vertices with a degree of Notably, within this graph, the cardinality of the vertex set is given by , and the cardinality of the edge set is equal to , which simplifies to .
By carefully considering the degrees of these distinct vertex types, we discern a total of ten distinct types of edge partitions. These partitions, each characterized by specific vertex degree combinations, lend to a comprehensive understanding of the connectivity patterns within the composite graph. The specifics of these edge partitions can be observed in Table 1, encapsulating the diverse ways in which vertices of different degrees interact and contribute to the graph’s structure.
Now, substitute the value in Table 1 for each case.
Case 1. .
Case 2. .
Case 3. .
Case 4. .
Similarly, using Equation (6) and the values in Table 1, we obtain the required result for the , which completes the proof.
Theorem 2. The ABC index and the GA index and the corona product of the two cycles and are given by the following equations:
Proof. The theorem’s verification is straightforward: for and both exceeding 2, it becomes evident that the cardinality of the vertex set in the corona product of two cycle graphs, denoted as , equals . Additionally, the edge set’s cardinality, represented by , amounts to .
Furthermore, within this composite graph, a classification of vertices into two distinct types based on their degrees emerges. The first type encompasses vertices with a degree of 3, while the second type comprises vertices with a degree of . These differing degrees illuminate the diverse connectivity patterns within the graph, offering insights into the way vertices of distinct degrees interact and contribute to the overall structural makeup.
To concretize this insight, a comprehensive depiction of the edge partitions, discerned through a careful consideration of each vertex’s degree, is presented in Table 2. This table succinctly captures the distinct arrangements of edges based on the degrees of the respective vertices, shedding light on the intricate relationships and connectivity dynamics within the composite graph.
By substituting values in Table 2 in Equation (2) and simplifying the formula, we obtain
Similarly, using Equation (3) and the values in Table 2, we obtain the required result for .
Theorem 3. For the corona product, of two complete graphs and , ABC index and GA index are equal to the following equations, respectively:
Proof. By invoking the corona product definition, it becomes evident that when both and surpass 1, the cardinality of the vertex set in the corona product of two complete graphs, represented as , is . Additionally, the cardinality of the edge set, denoted as , equates to , where represents the binomial coefficient.
It is worth highlighting that, within this composite graph, a classification of vertices unfolds based on their degrees. Specifically, one classification pertains to vertices possessing a degree of , while the other involves vertices with a degree of . This duality of vertex degrees underscores the diverse interactions and contributions of vertices to the graph’s overall structure.
Consequently, this classification engenders the existence of three distinctive types of edge partitions, as eloquently displayed in Table 3. Each of these partitions corresponds to different configurations of edges, contingent upon the degrees of the participating vertices. This delineation illuminates the intricate interplay between vertex degrees and edge connections within the composite graph.
By substituting the values in Table 3 in Equation (13) and simplifying the formula, we obtain
Similarly, by substituting the values in Table 3 to Equation (14) and simplifying the formula, we have
This completes the proof.
Theorem 4. For the corona product of the complete graph and path graph and , ABC index and GA index are equal to the following equations, respectively:
Proof. Considering the corona product of , where both and exceed 1, a distinctive classification of vertices emerges based on their degrees. This categorization yields three primary vertex types:
(1)Vertices with a degree of 2 constitute the first type(2)The second type encompasses vertices possessing a degree of 3(3)The third type comprises vertices with a degree of In the context of this composite graph, the cardinality of the vertex set, , equates to . Correspondingly, the cardinality of the edge set, , is characterized by .
An insightful observation emerges upon evaluating the degrees of the vertices: the graph exhibits six distinct types of edge partitions, as eloquently depicted in Table 4. Each partition encapsulates a unique configuration of edges, shaped by the degrees of the connected vertices. This revelation accentuates the intricate interplay between vertex degrees and edge connections within the composite graph.
Now, substitute the values in Table 4 in Equation (17) for both cases.
Case 1. . By simplifying the formula, we obtain
Case 2. . Through the process of simplifying the formula, we achieve the desired outcome. Similarly, by plugging in the values from Table 4 into equation (18) and then simplifying the formula, we attain the necessary results for the GA index of the corona product graph ().
Theorem 5. The ABC index and the GA index of the corona product of the path graph and complete graph and are given by the following equations:
Proof. Let us delve into the corona product of the path graph and the complete graph, denoted as , where both and are greater than 1. Within this context, a classification of vertices emerges based on their respective degrees, leading to the identification of three distinct vertex types:
(1)The first type encompasses vertices with a degree of (2)The second type consists of vertices possessing a degree of (3)The third type comprises vertices with a degree of In relation to this composite graph, the cardinality of the vertex set, denoted as , corresponds to . Simultaneously, the edge set’s cardinality, represented by , can be expressed as .
A significant observation materializes as we assess the vertex degrees: the graph is characterized by six distinct types of edge partitions, as vividly portrayed in Table 5. Each partition embodies a unique arrangement of edges, intricately shaped by the degrees of the vertices they connect. This insight highlights the intricate interplay between vertex degrees and the network of edge connections within the composite graph.
Now, substitute the values in Table 5 in Equation (22) for both cases.
Case 1. . By simplifying the formula, we obtain
Case 2. . Through the process of simplifying the formula, we achieve the desired outcome. In a similar vein, by inserting the values from Table 5 into Equation (23) and subsequently simplifying the formula, we acquire the necessary results for the GA index of the corona product graph ().
Corollary 6. The Randic index, ABC index, and GA index of the corona product of graphs and are defined by the following equations (for particular cases, general result still can be worked out for such products): (i)(ii)(iii)
Proof. Let us examine the corona product of two graphs, , where and are both greater than 2. By analyzing the degrees of vertices, we can categorize them into four distinct types.
The first type consists of vertices with a degree of 3, while the second type comprises vertices with a degree of 4. The third type encompasses vertices with a degree of , and the fourth type includes vertices with a degree of . By delving into the degrees of these vertices, we can identify a total of six different partition types, as illustrated in Table 6.
Now, substitute the value in Table 6 for each case.
Case 1. .
Case 2. .
Case 3. .
Corollary 7. The Randic index, ABC index, and GA index of the corona product of graphs and are defined by the following equations: (i)(ii)(iii)
Proof. Let us explore the corona product of two graphs, , where both and exceed 3. By examining the vertex degrees, we can distinguish four distinct types of vertices. The initial type comprises vertices with a degree of 4, while the second type encompasses vertices with a degree of 5. The third type consists of vertices with a degree of , and the fourth type encompasses vertices with a degree of . By analyzing the vertex degrees in this manner, we can identify a total of seven distinct partition types, as illustrated in Table 7.
Now, substitute the value in Table 7 for each case.
Case 1. .
Case 2. .
Case 3. .
Corollary 8. The Randic index, ABC index, and GA index of the corona product of graphs and are defined by the following equations: (i)(ii)(iii)
Proof. Let us examine the corona product of two graphs, , where both and are greater than 4. By analyzing the degrees of vertices, we can classify them into four specific types.
The initial type encompasses vertices with a degree of 5, while the second type comprises vertices with a degree of 6. The third type consists of vertices with a degree of , and the fourth type includes vertices with a degree of . Through this analysis of vertex degrees, we can identify a total of nine distinct partition types, as depicted in Table 8.
Now, substitute the value in Table 8 for each case.
Case 1. .
Case 2. .
Case 3. .
3. Material and Methods
The authors studied the concepts of different operations for the graphs and found the corona product operation very interesting and useful. It has a waste application in chemistry, physics, and computer fields for constructing complex networks; some applications may be studied in [7, 13, 14]. Some recent developments and benefits of studying the degree-based topological indices for such graphs were greatly beneficial. Using the idea and developing special corona product graphs, we used the techniques to calculate some topological indices of these products by considering some special cases. Moreover, a combination of studying different graphs under different degrees, distances, or edge-based invariants can be seen in [4, 8–10, 13–17].
4. Conclusion
In this study, we generalized the concept of topological indices of different families of corona product graphs and described the corona graphs of the , , and . In the future, you will be able to find the topological indices based on the order and distance of these graph operations. The results are convenient for constructing and investigating topological indexes of complex network structures. In this study, we created corona products’ ABC and GA indexes when synthesizing paths, cycles, and complete graphs. In addition, a generalized corona product graph calculation example is given in different cases.
Open problems: find the GA and ABC index for the following families of corona products , , and .
Data Availability
The data is provided on the request to the authors.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All the authors equally contributed towards this work and agree to publish this paper under academic ethics.