Abstract

Topological indices are graph invariants that are used to correlate the physicochemical properties of a chemical compound with its (molecular) graph. In this study, we study certain degree-based topological indices such as Randić index, Zagreb indices, multiplicative Zagreb indices, Narumi–Katayama index, atom-bond connectivity index, augmented Zagreb index, geometric-arithmetic index, harmonic index, and sum-connectivity index for the bistar graphs and the corona product , where represents the complement of complete graph .

1. Introduction

Graph theory techniques [1–4] have applications in chemistry, biology, physics, and computer science. Topological indices are graph invariants that are used to study graph topology. Along with computer networks, graph theory is a powerful tool in other research areas such as coding theory, database management systems, circuit design, secret-sharing schemes, and theoretical chemistry. Cheminformatics is the combination of technology, graph theory, and chemistry. It links organic substances structure and physiochemical properties by using some useful graph invariants and their associated molecular graph. A molecular graph contains sets of points and lines which shows the atoms and covalent bond in the molecule. These points and lines are known as vertices and edges in graph theory, respectively [5]. Theoretical study of underlying chemical structure using useful graph invariants is an appealing area of research in mathematical chemistry due to its practical applications in QSAR/QSPR investigation [6]. Topological indices are used to estimate the physicochemical properties of the chemical compounds. A topological index can be considered as a function that maps a graph to a nonnegative real number [7].

Some degree-based topological indices are Randić index, Zagreb indices, Narumi–Katayama and multiplicative Zagreb indices, atom-bond connectivity index, augmented Zagreb index, geometric-arithmetic index, harmonic index, and sum-connectivity index [8].

Throughout this work, we use standard notations for graph, set of vertices, set of edges, and degree of vertex in .

For any simple graph , denotes the adjacent vertices and in graph .

Definition 1. Randić index is the most popular topological index among all topological indices and is defined as [9]

Definition 2. The first Zagreb index of a graph is defined asThe second Zagreb index of a graph is defined asThe first and second Zagreb index are known as Zagreb indices [10].

Definition 3. The first multiplicative Zagreb index of a graph is defined aswith product going overall vertices of a graph .
The second multiplicative Zagreb index of a graph is defined asThe modified first multiplicative Zagreb index of a graph is defined asThe first, second, and modified first multiplicative Zagreb indices are known as Narumi–Katayama indices [8, 11].

Definition 4. The atom-bond connectivity index of a graph is defined as [12]

Definition 5. The augmented Zagreb index of a graph is defined as [13]

Definition 6. The geometric-arithmetic index of a graph is defined as [14]

Definition 7. The harmonic index of a graph is defined as [15, 16]

Definition 8. The sum-connectivity index of a graph is defined as [17, 18]Randić index was introduced by Milan Randić in 1975 [19]. After that, mathematicians did not pay attention to this index for nearly two decades. However, Pal, Erdos, and Bela Bollobas soon worked on this index and discovered the useful mathematics hidden within it. They published their first paper on this index in 1998. When the mathematical communities realized the value of the Randić index, they started to do research studies and soon a flood of publications on this descriptor were started. Randić also wrote two articles on this descriptor [8].
The work on Randić index encouraged theoretical chemists and mathematicians to discover other topological indices that depend on vertex degree.
During the study of total pi electron energy on molecular structure, some expressions which included the terms of the formwere occurred [8]. In the chemical theory, and are called first Zagreb index and second Zagreb index. These two indices were discovered by Trinajestic and Gutman in 1972 [8, 10]. At first, the first Zagreb index was also known as the Gutman index, but Balaban et al. did not want to introduce these descriptors by the names of the discoverers. So, they included and in the topological indices after 1982 and named them Zagreb group indices. However, soon, the name Zagreb group indices was changed to Zagreb indices. The second Zagreb index did not get any attention of mathematicians or mathematical chemists and that is why not a single property of was introduced in [8]. Narumi–Katayama [8] introduced the first product descriptor in 1984 and named it simple topological index. This index is defined asThis product index was later renamed the Narumi–Katayama index by Tomovic and Gutman. In 2010, Todeshine et al. proposed multiplicative versions of multiplicative Zagreb indices. These two graph invariants are known as the first and second multiplicative Zagreb indices, after Gutman. Eliasi et al. and Iranmanesh and Gutman proposed a multiplicative version of the first Zagreb index. The authors termed it as modified multiplicative Zagreb index, respectively [20].
The first geometric-arithmetic index also known as geometric-arithmetic index is the modified version of the Randić index. This index was proposed by Vukicevic and Furtula in 2009 [6, 8].
The atom-bond connectivity index is another topological index that is similar to the Randić index. This topological descriptor was introduced by Estrada et al. in 1998 [6, 13]. A recent study of this index has caught the attention of some researchers. Furtula et al. [8] introduced the augmented Zagreb index in 2009, a modified version of the atom-bond connectivity index. This descriptor has higher predictive power than the atom-bond connectivity index. Tosovic and Gutman investigated in 2013 that AZI produced nice effects in the formation of heat and also yielded the best results in boiling factors of octane isomers. Furtula et al. investigated the structure sensitivity of twelve topological indices using trees and found that the AZI has the best structure sensitivity [13].
In 1980, Siemion Fajtlowiez introduced another topological index. In 2012, Zhang worked on this index and named it as harmonic index [8]. In sharp construct to other topological indices, no chemical applications of this index were found, but in mathematical chemistry, such research studies are expected very much [8]. In the last few years, this index has attracted the great attention of theoreticians [15, 16].
Bo Zhou and Nenad Trinajstic discovered that the term in the Randić index can be replaced by and named it sum-connectivity index [8]. This index got the attention of both applied and pure researchers. Some recent work on this index can be found in [21].
There are a number of studies of various topological indices of graphs establishing formulas for computing the indices and also providing upper and lower bounds on the values of such indices. In this study, certain vertex degree-based topological indices are studied. We have determined and computed the closed formulas of these indices for two special families of graphs of diameter three. A graph formed by joining the centers of two-star graphs of order and , i.e., and by an edge, is called the bistar graph and is denoted by . The corona product of two graphs is defined as the graph obtained by taking one copy of of order and copies of and then joining each vertex of the copy of to the vertex of . These graphs are undirected having no loops and multiple edges. These results are novel and significant contributions in graph theory and network science, and they provide a good basis to understand the topology of these graphs and networks (Figures 1 and 2).

Figure 1 

A representation of bistar graph of order and .

Figure 2 

A representation of corona product of two graphs.

2. Topological Indices of Families of Bistar Graphs, i.e.,

By looking at the earlier results for computing the topological indices for families of bistar graphs, here we introduce new degree-based topological indices to compute their values for these families of graphs.

Theorem 1. The sum-connectivity index of families of bistar graphs is

Proof. To find the sum-connectivity index of , firstly, we select a vertex on of degree . There are vertices of degree 1 which are adjacent to . For the vertices and , where , the sum is obtained asSince the degree of is , the other vertex which is adjacent to is of degree . For the vertices and , we haveNow, we select a vertex of degree on ; there are vertices of degree 1 which are adjacent to . For the vertices and , the sum is obtained asAdding above equations, we have  = sum-connectivity index of :

Theorem 2. The Zagreb indices of the families of bistar graphs are

Proof. To find the first Zagreb index of , we select a vertex on of degree . For the vertex , we haveSince there are vertices of of degree 1 which are adjacent to , thus, we haveAs the degree of is , the other vertex which is adjacent to is of degree . For the vertex , we haveSince the degree of is , the other vertices which are adjacent to are in numbers of degree 1:Adding equations (20) to (23), we have  = first Zagreb index of :Similarly,  = second Zagreb index of is

Theorem 3. The atom-bond connectivity index and augmented Zagreb index of families of bistar graphs are

The proof of this Theorem is the same as Theorem 1.

Theorem 4. The geometric-arithmetic index and harmonic index of families of bistar graphs are

The proof of this theorem is the same as Theorem 1.

Theorem 5. The Narumi–Katayama and multiplicative Zagreb indices of families of bistar graphs are

Proof. To find the first multiplicative Zagreb index of , we select a vertex on of degree . For the vertex , we haveSince there are vertices of of degree 1, thus, we haveSimilarly, for the vertex , we haveAlso, for the vertices of of degree 1, we haveMultiplying equations (29) to (32), we have  = first multiplicative Zagreb index of families of bistar graphs :Similarly, second multiplicative Zagreb index and modified first multiplicative Zagreb index of families of bistar graphs are

Example 1. Topological indices of bistar graph are shown in Table 1.

3. Topological Indices of Families of Corona Product of Graphs, i.e.,

Based on previous results for computing topological indices for families of corona product of graphs, we present new degree-based topological indices to compute their values for these families of graphs.

Theorem 6. The Randić index and sum-connectivity index of families of corona product of graphs are

Proof. To find the Randić index of the family of graphs obtained by taking the corona product of complete graph of order and the complement of of order , firstly, we select a vertex on as is adjacent to vertices and vertices . Therefore, the degree of is .
For the vertices and , where , the sum is obtained asAs the graph is symmetric, the same result is obtained for the remaining vertices . So, combining all the vertices equation (36) becomesSince is also adjacent to vertices of degree , for the vertices and , where , the sum is obtained asSince is symmetric, the same result is obtained for remaining vertices . Combining the result for all vertices , we haveAdding equations (37) and (39), we have  = Randić index of :Similarly,  = sum-connectivity index of is

Theorem 7. The atom-bond connectivity index and augmented Zagreb index of families of corona product of graphs are

The proof of this theorem is the same as Theorem 6.

Theorem 8. The geometric-arithmetic index and harmonic index of families of corona product of graphs are

The proof of this theorem is the same as Theorem 6.

Theorem 9. The Zagreb indices of families of corona product of graphs are

Proof. For the families of corona product of graphs there are vertices of degree one and vertices of degree , respectively. For the vertices , we haveSince there are vertices of degree 1, combining the above result for all vertices, thus, the above equation becomesAlso, for the vertex , we haveSince the graph is symmetric, the same result is obtained for remaining vertices :Adding equations (46) and (48), we have  = first Zagreb index of families of corona product of graphs :Similarly,  = second Zagreb index of families of corona product of graphs :

Theorem 10. The Narumi–Katayma and multiplicative Zagreb indices of families of corona product of graphs are