Mapping Connectivity Patterns: Degree-Based Topological Indices of Corona Product Graphs

. Graph theory (GT) is a mathematical ﬁ eld 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 di ﬀ erent 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 G and H gives us a new graph obtained by taking one copy of G and V G copies of H and joining the i th vertex of G to every vertex in the i th copy of H . 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 P s ⨀ P t , C t ⨀ C s , K t ⊙ K s , K t ⊙ P s , and P s ⊙ K t and GA and ABC index. Moreover, a few molecular graphs and physicochemical features may be predicted by considering relevant mathematical ﬁ ndings supported by proofs


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, G and H, crafts a novel composite graph by combining a single instance of G with V G copies of H.This synthesis results in each vertex of G intricately connecting to every vertex within its corresponding copy of H, 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 G and H be 2 graphs, each with a group of vertex V G and V H and a group of edges E G and E H , respectively.We described the corơna product GoH as the prơduct of twơ graphs, G and H, achieved by combining each vertex of V G copies of H.
Let G = V, E 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 D of vertices in which every vertex in S is not adjacent to a vertex in D. A set that is both dominant and indepen-dent 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 n vertices; then, the Randic index is defined as follows: where d u is the degree of vertex u.Consider a simple connected undirected graph G V, E that has n nodes, and then, the ABC index is defined as follows: where d u is the degree of vertex u.
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 G 1 and G 2 is defined as the graph obtained by taking one copy of graph G 1 and V G 1 copies of G 2 , where each vertex of the ith copy of G 2 relates to the ith vertex of G 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.

Main Results
Let P t , C t , and K t be a path, cycle, and complete graphs on n vertices.In this section, we discuss the ABC index and the GA index of 1: Corona product of two graphs.

Journal of Applied Mathematics
Theorem 1.The ABC index and the GA index of the corona product of two path graphs P t and P s are given by the following equations: Proof.Consider the corona product of two path graphs, denoted as P t ⊙ P s .In the case where both t and s 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 n + 1 (4) the fourth type encompasses vertices with a degree of n + 2 Notably, within this graph, the cardinality of the vertex set V P t ⊙ P s is given by t + ts, and the cardinality of the edge set E P t ⊙ P s is equal to ts + t − 1 + t s − 1 , which simplifies to 2ts − 1.
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.
Similarly, using Equation ( 6) and the values in Table 1, we obtain the required result for the G P t ⨀P s , which completes the proof.
Theorem 2. The ABC index and the GA index and the corona product of the two cycles C t and C s are given by the following equations: Proof.The theorem's verification is straightforward: for tand s both exceeding 2, it becomes evident that the cardinality of the vertex set in the corona product of two cycle graphs, denoted as V C t ⨀C s , equals n + nm.Additionally, the edge set's cardinality, represented by E C t ⨀C s , amounts to t + 2ts.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 s + 2. 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 Journal of Applied Mathematics 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 G C t ⨀C s .Theorem 3.For the corona product, of two complete graphs K t and K s , 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 t and s surpass 1, the cardinality the product of two complete graphs, represented as V K t ⊙ K s , is t + ts.Additionally, the cardinality of the edge set, denoted as E K t ⊙ K s , equates to ts + C t, 2 + tC s, 2 , where C n, k 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 s, while the other involves vertices with a degree of t + s − 1.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 K t and P s , ABC index and GA index are equal to the following equations, respectively:

Journal of Applied Mathematics
Proof.Considering the corona product of K t ⊙ P s , where both t and s 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 n + m − 1 In the context of this composite graph, the cardinality of the vertex set, V K t ⊙ P s , equates to n + nm.Correspondingly, the cardinality of the edge set, E K t ⊙ P s , is characterized by ts 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. 19 By simplifying the formula, we obtain 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 (K t ⊙ P s ).
Theorem 5.The ABC index and the GA index of the corona product of the path graph and complete graph P s and K t are given by the following equations: Proof.Let us delve into the corona product of the path graph and the complete graph, denoted as P s ⊙ K t , where both t and s 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 t (2) The second type consists of vertices possessing a degree of t + 1 (3) The third type comprises vertices with a degree of t + 2 In relation to this composite graph, the cardinality of the vertex set, denoted as V P s ⊙ K t , corresponds to s + ts.Simultaneously, the edge set's cardinality, represented by E P s ⊙ K t , can be expressed as ts + s − 1 + sC t, 2 .
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.
By simplifying the formula, we obtain 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 (P s ⊙ K t ).Corollary 6.The Randic index, ABC index, and GA index of the corona product of graphs W n and P m are defined by the following equations (for particular cases, general result still can be worked out for such products): Proof.Let us examine the corona product of two graphs, W n ⨀P m , where n and m 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 n + 3, and the fourth type includes vertices with a degree of m + 2. 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.
Proof.Let us explore the corona product of two graphs, K n ⨀L m , where both n and m 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 n, and the fourth type encompasses vertices with a degree of n + m − 1.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   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 n + 1, and the fourth type includes vertices with a degree of m + 2. 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.

( i )
Randic index L n ⨀K m = n = 4, m = 6 = 34 872 (ii) ABC index L n ⨀K m = n = 4, m = 6 = 90 728 (iii) GA index L n ⨀K m = n = 4, m = 6 = 117 195Proof.Let us examine the corona product of two graphs, L n ⨀K m , where both n and m are greater than 4. By analyzing the degrees of vertices, we can classify them into four specific types.

Table 1 :
Number of the edges in each partition of P t ⨀P s based on the degree of the end vertices of each edge.

Table 2 :
Number of edges in each the partition of C t ⨀C s based on the degree of end vertices of each edge.

Table 3 :
Number of edges in each partition of K t ⨀K s based on the degree of the end vertices of each edge.

Table 4 :
Number of edges in each partition of K t ⨀P s based on the degree of end vertices of each edge.

Table 5 :
Number of edges in each partition of P s ⨀K t based on the degree of end vertices of each edge.The Randic index, ABC index, and GA index of the corona product of graphs k n and L m are defined by the following equations: for each case.The Randic index, ABC index, and GA index of the corona product of graphs L n and K m are defined by the following equations:

Table 6 :
Number of edges in each partition of W n ⨀P m based on the degree of end vertices of each edge.

Table 7 :
Number of edges in each partition of K n ⨀L m on the basis of the degree of end vertices of each edge.