Abstract
Graph operations are utilized for developing complicated graph structures from basic graphs, and these basic graphs can help to understand the properties of complex networks. While on the other side, the topological descriptor is known as a numeric value that is associated with the graph of a network. It has enormous practical applications in chemistry and other fields of science. This particular work in this draft is the extended work and investigated the first, second, first multiplicative, first reformulated Zagreb indices, and the forgotten index of subdivision double corona and subdivision double neighborhood corona products.
1. Introduction
A topological index is a number associated with a graph of some network. With the help of this number, we can describe some properties of the network. Especially in organic chemistry, topological indices are used to predict some physical, chemical, or biological properties of organic compounds. Topological indices are key topics in the study of quantitative structural properties of a chemical network [1–5].
For a graph , the vertex and edge sets are denoted by and . The number of elements in and is called the order and the size , respectively, of . A graph of order and size is denoted by . The set of vertices adjacent to the vertex is called the neighborhood set of and the number of elements in the neighborhood set the degree of in is denoted by [6–10].
The study of the topological index started in 1947, and after that, hundreds of topological indices have been presented depending on the nature of applications for different chemical compounds. In 1972, the researchers in [11] introduced the first and second Zagreb indices. For a graph , these topological indices are defined as
The researchers of [12, 13], introduced multiplicative variants of ordinary Zagreb indices. These topological indices are used to study molecular chirality, complexity, heterosystems, and Ze-isomerism. The first and second multiplicative Zagreb indices are defined as
In 2015, researchers in [14, 15] suggested a forgotten topological index that is comparable to the first Zagreb index in its applications. The forgotten topological index is also known as -index, and it is defined as
In 2004, Milicevic et al. [16] proposed reformulated Zagreb indices using edge-degrees rather than vertex-degrees. Mathematically, it is expressed as
For , the above expression is known as the first hyper Zagreb index .
Complex network structures or large molecular structures can be constructed by applying some graph operations on simple graphs. Furthermore, these simple graphs can help to describe some properties of these structures. For example, the Cartesian product provides a significant model for connecting computers [17, 18].
For graphs and , the corona product is obtained by taking one copy of , copies of , and joining vertex of to every vertex copy of [19]. A special graph obtained by attaching a vertex in the each edge of is called the subdivision graph of and is symbolized by [20].
Let , , and be three graphs. The operation known as subdivision double corona product of , , and and symbolized by , and is attained by making single copy of copies of copies of , and after that, by attaching the old vertex, , of to each vertex of the copy of and new vertex of to each vertex of the copy of [21]. An illustration of subdivision double corona product is shown in Figure 1.
For the above three graphs, the subdivision double neighborhood corona product, , is the graph attained by making single copy of copies of copies of and after that, by attaching the neighborhood vertices of the old vertex of to every vertex of the duplicate of and joining the neighborhood vertices of the new vertex of to every vertex of the duplicate of [21]. Figure 2 explains the notation of .
In [22], the authors investigated the first and second Zagreb indices of the Cartesian, composition, join, disjunction, and symmetric difference graph operations. The author in [23] computed the forgotten topological index of different corona products of graphs and the author in [24] gave the exact expressions of Zagreb indices of the generalized hierarchical product of graphs. For more discussion and results, we refer to [25, 26]. There are some new and recent topics related to this study is found, one can see [27–31].
The Laplacian spectrum of double neighborhood corona graphs are found in the literature of [21], and the main results are presented.
Theorem 1. Let be a t-regular graph on n vertices, m edges, and be any two graphs on and vertices, respectively. Then, the Laplacian spectrum of comprises(i), for ;(ii) repeated times each;(iii) repeated times, for ;(iv) repeated times, for .
Theorem 2 (see [21]). Let be a t-regular graph on n vertices, m edges, and be any two graphs on and vertices, respectively. Then, the Laplacian spectrum of comprises(i)all the roots of the equation(ii) repeated times;(iii)1 repeated times;(iv) + 1 repeated n times, for ;(v) + 1 repeated n times, for .
In this paper, we extend the work and investigated some degree-based topological indices of these graph operations.
2. Main Results
The current section contained the main results which include the formulation of some degree-based topological indices such as first and second Zagreb, first multiplicative Zagreb, first reformulated Zagreb, the forgotten indices of subdivision double corona product, and subdivision double neighborhood corona product of graphs.
Following is the famous relationship between arithmetic and geometric means.
Lemma 1. (AM-GM Inequality). Let be non-negative numbers. Then,and the equality holds if and only if are equal.
Next to lemmas are the direct results from the definitions of the subdivision double corona product and subdivision neighborhood corona product of graphs.
Lemma 2. Let , , and are three graphs having order , , and , respectively. Then, the degree behavior of the vertices in subdivision double corona product is given as
Lemma 3. Let , , and are graphs having order , , and , respectively, then the degrees of the vertices in subdivision double neighborhood corona product is
Following is our first main result, which gives the first Zagreb index of the subdivision double corona product in terms of the first Zagreb indices of basic graphs, their orders, and sizes.
Theorem 3. Let , , and be the simple connected graphs. Then, the first Zagreb index of the subdivision double corona product, , is given as
Proof. From the concept of topological descriptor named the first Zagreb index, we have gotNow, we apply Lemma 2,Hence, the required expression.
The next results put a bound on the first multiplicative Zagreb index for the subdivision double corona product.
Theorem 4. Let , , and be the simple connected graphs. Then, the first multiplicative Zagreb index of the subdivision double corona product is given as
Proof. From using the concept of the first multiplicative Zagreb index and Lemma 2, we have gotThe inequality is due to the Lemma 1. Equality in the last expression holds if and only if , , and are regular graphs.
Theorem 5. Let , , and are three simple graphs and be the subdivided graph of . Then, the second Zagreb index of subdivision double corona product is given as
Proof. From using the concept of the second Zagreb index and Lemma 2, we have gotAfter some simplification, we can get the required result.
The next result is about the first reformulated Zagreb index of the subdivision double corona product of graphs.
Theorem 6. Let , , and be the simple graphs and is the subdivision of the graph . Then, the first reformulated Zagreb index of the subdivision double corona product is given as
Proof. Using the Lemma 2 and the concept of the first reformulated Zagreb index, we have gotHence, the proof is done.
Theorem 7. For graphs , , and , then, the first Zagreb index of the subdivision double neighborhood corona product is given as
Proof. From the concept of the first Zagreb index we haveNow, we apply Lemma 3,Hence, the proof is done.
Theorem 8. Let , , , and be the simple connected graphs. Then, the second Zagreb index of subdivision double neighborhood corona product is given as
Proof. From the concept of the second Zagreb index, we haveNow, we apply Lemma 3,Hence, the proof is done.
Theorem 9. Let , and be the simple connected graphs. Then, the forgotten topological index of subdivision double neighborhood corona product is given as
Proof. From the concept of the forgotten index, we haveNow, we apply Lemma 3,Hence, the proof is done.
Theorem 10. Let , , and be the simple connected graphs. Then, the first multiplicative Zagreb index of subdivision double neighborhood corona product is given as
Proof. From the concept of the first multiplicative Zagreb index, we haveNow, we apply Lemma 3,By Lemma 1,Hence, equality holds in 13 iff , , and are regular graphs.
Hence, the proof is done.
Theorem 11. Let , , , and be the simple connected graphs. Then, the first reformulated Zagreb index of subdivision double neighborhood corona product is given as
Proof. From the concept of the first reformulated Zagreb, we haveNow, we apply Lemma 3,Hence, the proof is done.
3. Conclusion
The first and second Zagreb indices, the first multiplicative Zagreb index, the first reformed Zagreb index, and the forgotten topological index were explored in this work, and their exact expressions were investigated. Other degree and distance-based topological indices of these complicated network operations could be calculated in the future work.
Data Availability
All the data supporting the results are included in the manuscript.
Conflicts of Interest
The authors declare that they have no conflicts of interest.