The Degree-Based Topological Indices for Two Special Families of Graphs of Diameter Three

Department of Mathematics, Government Graduate College for Women Samanabad, Lahore, Pakistan Yildiz Technical University, Faculty of Arts and Science, Department of Mathematics, Esenler, 34210 Istanbul, Turkey Department of Mathematics, Government Postgraduate College, Abbottabad, Pakistan Université de Sousse, Institut Supérieur d’Informatique et Des Techniques de Communication, H. Sousse 4000, Tunisia China Medical University Hospital, China Medical University, Taichung 40402, Taiwan Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa Department of Mathematics and Statistics, International Islamic University Islamabad, Pakistan Quantum Leap Africa (QLA) AIMS Rwanda Center, Remera Sector KN 3, Kigali, Rwanda Institut de Mathématiques et de Sciences Physiques (IMSP/UAC). Laboratoire de Topologie Fondamentale, Computationnelle et Leurs Applications (Lab-ToFoCApp), BP 613 Porto-Novo, Benin African Center for Advanced Studies, P.O. Box 4477, Yaounde, Cameroon


Introduction
Around the center of a century ago, theoretical experts found some interesting relationships between different properties of organic substances and those of molecular structure by analyzing a few invariants of the underlying molecular graph. These graph invariants are useful for molecular objects and are named as topological indices or topological descriptors. These are numerical parameters of graphs which can be used to describe the physiochemical properties like boiling point, molecular weight, density and refractive index of a molecule [1,2]. The three main categories of topological indices are distance-based topological indices, degree-based indices and spectrum-based topological indices. Among these, degree-based topological indices are of great significance [1, 3,4]. Some vertex degreebased topological indices are Randic or connectivity index, first and second Zagreb indices, Narumi Katayama and multiplicative Zagreb indices, atom bond connectivity index, augmented Zagreb index, geometric arithmetic index, harmonic index, and sum connectivity index [5]. The topological descriptors are beneficial tools to chemists that are provided by graph theory. The use of these theoretical invariants in QSPR/QSPR modeling has been a popular work in the past years [4,6].
The first most widely used topological index in chemistry is the Wiener index. Initially, it was presented by the American Chemist Harold Wiener in 1947 and is denoted by WðGÞ. After its success, many other topological indices have been introduced that are based on Wiener's work [3,7].
Milan Randic introduced the Randic index in 1975. It was the most popular index among all topological indices [2,8]. The mathematical form of this descriptor is with a summation going with overall pairs of adjacent vertices of the graph G. Later, mathematicians did not pay any attention to this index for almost two decades. However, Erdos together with Bollobas worked on this index and discovered the mathematical hidden in it. In 1998, they published their first paper on this index. When the mathematical community began to understand the importance of this descriptor, they started to do researches on it. There have been hundreds of articles and several books dedicated to this structural descriptor [9]. The chemists [1] found out some expressions during the study of molecular structure dependency of total pielectron energy which includes the terms of the form Immediately, it was identified that these two terms provide quantitative measures of the branching of the molecular carbon atom skeleton. In chemical theory, M 1 and M 2 are called first and second Zegreb indices. These two indices have been introduced by Gutman and Trinajestic in 1972. Initially, the first Zagreb index was also known as the Gutman index. But, Balaban et al. did not want to introduce these indices by the names of the discovers. After 1982, Balaban et al. included these two terms in the topological indices and named them as Zagreb indices. In 2012, multiplicative versions of multiplicative Zagreb indices have been proposed by Todeshine et al. and are known as Narum Katayama indices. They were the first topological index defined by the product of some graph theoretical quantities. In 1984, Narumi and Katayama determined the term Q μ d μ ðGÞ 2 and named it as the first multiplicative Zagreb index. It is denoted by Q 1 ðGÞ. The second multiplicative Zagreb index and modified first multiplicative Zagreb index are defined, respectively, by Y with a product going with overall adjacent vertices of a graph G [10,11]. The first, second, and modified multiplicative Zagreb indices are known as Narumi Katayama multiplicative Zagreb indices. The geometric arithmetic index is the modified form of the Randic index. This topological descriptor was introduced by Vukicevic and Furtula in 2009. The mathematical form [12] of this index is Another vertex degree-based topological index that keeps the spirit of Randic index is the atom bond connectivity index. This topological descriptor was introduced by Estrada et al. in 1998. This index has turned out to be an important index for the stability of Alkanes. The mathematical form [8,13] of this index is Furtula et al. introduced the modified form of atom bond connectivity index in 2009 and named it as the augmented Zagreb index. The mathematical form [8,14] of this descriptor is In the investigation of heat creation in heptanes and octanes, the prediction capability of the augmented Zagreb index outperforms that of the atom bond connectivity index. Ali et al. investigated the correlation abilities of 20 vertex degree-based topological descriptors for the case of normal heats of formation and ordinary boiling factor of octane isomers, finding that the AZI produces the best results [15].
In 1980, Fajtlowiez introduced another topological index. In 2012, Zhang worked on this index and named it as the harmonic index. This index is defined as No chemical applications of this index have been found so far, but in the last few years, this index has attracted the great attention of theoreticians [16,17]. Another vertex degree-based topological index was proposed by Zhou and Trinajstic and was named as the sum connectivity index. They observed in the definition of Randic index that the term d μ × d ν can be replaced by d μ + d ν which is defined by ([2, 18,19]).
Many topological indices are bond-additive, i.e., they can be presented as a sum of edge contributions and have the following form: where gðμÞ are usually degrees or the sum of distances from μ to all other vertices of G. Inspired by the most successful indices of this type, such as the Randic index, the second Zagreb index and others, a whole family of Adriatic indices [20] was defined. A particularly intriguing subclass of these descriptors is made up of 148 discrete Adriatic indices. They were examined on the testing sets provided by the International Academy of Mathematical Chemistry, and it had been shown that they have good predictive properties in many cases and it was found that they have good predictive properties in many cases. One of these useful discrete Adriatic indices is the symmetric division deg (SDD) index, which is defined as where EðGÞ is the edge set of a molecular graph G (a graph in which vertices correspond to atoms of a (hydrogen-suppressed) molecule and edges correspond to bonds between the atoms) and d μ and d ν denote the degrees of the vertices μν ∈ VðGÞ, respectively [21].
In this paper, we compute the degree-based topological indices, namely, Randic or connectivity 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 for two special families of graphs of diameter three. These graphs are undirected having no loops and multiple edges. In Section 2, the topological indices of families of graphs C 4 ðK n Þ are determined, while in Section 3, the topological indices of families of graphs C 4 ðK n Þ + v 1 v 3 are computed.

Topological Indices of Families of
Graphs C 4 ðK n Þ In this section, we have determined and computed the closed formulas of degree-based topological indices for families of graphs C 4 ðK n Þ. The results are analyzed, and the general formulas are derived for these families of graphs.
Theorem 1. The Randic and sum connectivity indices of families of graphs C 4 ðK n Þ are Proof. To find the Randic or connectivity index of C 4 ðK n Þ as shown in Figure 1, we first select a vertex u 1 on C 4 ðK n Þ of degree n + 1. There are ðn − 1Þ vertices u 2 , u 3 , ⋯, u n of degree n − 1, which are adjacent to u 1 . For the vertices u 1 and u i , where i = 2, 3, 4, ⋯, n, the sum is obtained as Since the degree of u 1 is n + 1, the other two vertices which are adjacent to u 1 , are v 1 and v 3 of degree two. For the vertices u 1 and v 1 , we have Similarly, for the vertices u 1 are v 3 , we have Now, we select a vertex u 2 in C 4 ðK n Þ. Since the degree of u 2 is n − 1, the other ðn − 2Þ vertices which are adjacent to u 2 , are u 3 , u 4 , ⋯, u n . For the vertices u 2 and u j where j = 3, 4, 5, ⋯, n, the sum is obtained as Since K n is symmetric, the same result is obtained for the remaining ðn − 2Þ vertices u 3 , u 4 , ⋯, u n . So combining the result for all ðn − 1Þ vertices u 2 , u 3 , ⋯, u n , we have Now, we select v 1 on C 4 ðK n Þ. The degree of v 1 is two, the other vertex which is adjacent to v 1 is v 2 of degree two. For the vertices v 1 and v 2 , we have

Advances in Mathematical Physics
Similarly, we select v 2 on C 4 ðK n Þ. The degree of v 2 is two, the other vertex which is adjacent to v 2 is v 3 of degree two. For vertices v 2 and v 3 , we have Adding equations (12) to (18), we have RðGÞ = Randic or connectivity index of C 4 ðK n Þ is Similarly, SCIðGÞ = sum connectivity index of C 4 ðK n Þ is Theorem 2. The atom bond connectivity index and augmented Zagreb index of families of graphs C 4 ðK n Þ are defined, respectively, by The proof is the same as that of Theorem 1.

Theorem 3.
The geometric arithmetic index and harmonic index of families of graphs C 4 ðK n Þ are defined, respectively, by The proof is similar to the proof of Theorem 1.

Theorem 4. The Zagreb indices of families of graphs C 4 ðK n Þ are
Proof. For the family of graphs C 4 ðK n Þ, there are ðn − 1Þ vertices u 2 , u 3 , ⋯, u n of degree n − 1, one vertex u 1 of degree n + 1, and three vertices v 1 , v 2 , and v 3 of degree two, respectively. For the vertex u 1 , we have For the vertices v 1 , v 2 and v 3 , we have For vertices u 2 , u 3 , ⋯, u n , we have Adding equations (24) to (26), we have M 1 ðGÞ = first Zagreb index of C 4 ðK n Þ is Similarly, M 2 ðGÞ = second Zagreb index of C 4 ðK n Þ is Theorem 5. The Katayma and multiplicative Zagreb indices of families of graphs C 4 ðK n Þ are Proof. Since there are ðn − 1Þ vertices u 2 , u 3 , ⋯, u n of degree n − 1, we have one vertex of degree n + 1 and three vertices of degree two in C 4 ðK n Þ graph. Thus, we have

Advances in Mathematical Physics
For the vertex u 1 , we have Similarly, for the vertices v 1 , v 2 , and v 3 , we have Multiplying equations (30) to (32), Similarly, second multiplicative and modified first multiplicative Zagreb indices of C 4 ðK n Þ are defined, respectively, by Example 6. Topological indices of graph C 4 ðK 6 Þ are shown in Table 1.

Topological Indices of Families of
By looking at the earlier results for computing the topological indices for different families of graphs, here, we introduce new degree-based topological indices to compute their values for families of graphs C 4 ðK n Þ + v 1 v 3 .
Theorem 7. The Randic and sum connectivity indices of families of graphs C 4 ðK n Þ + v 1 v 3 are defined, respectively, by Proof. To find the Randic or connectivity index of C 4 ðK n Þ + v 1 v 3 as presented in Figure 2, we first select a vertex u 1 on C 4 ðK n Þ + v 1 v 3 of degree n + 1. There are ðn − 1Þ vertices u 2 , u 3 , ⋯, u n of degree n − 1, which are adjacent to u 1 . For the vertices u 1 and u i where i = 2, 3, 4, ⋯, n, the sum is obtained as Since the degree of u 1 is n + 1, the other two vertices which are adjacent to u 1 , are v 1 and v 3 of degree 3. For the vertices u 1 and v 1 , we have Similarly, for the vertices u 1 are v 3 , we have Now, we select a vertex u 2 on C 4 ðK n Þ. Since the degree of u 2 is n − 1, the other ðn − 2Þ vertices which are adjacent to u 2 , are u 3 , u 4 , cdots, u n . For the vertices u 2 and u j where j = 3, 4, 5, ⋯, n, the sum is obtained as Since K n is symmetric, the same result is obtained for the remaining n − 2 vertices u 3 , u 4 , ⋯, u n . By using equation (16), the above equation can be written as  Figure 2: G = C 4 ðK 5 Þ + v 1 v 3 (graph obtained by identifying one of the vertices of K 5 with one vertex of C 4 + v 1 v 3 ).

Advances in Mathematical Physics
Now, we select v 1 on C 4 ðK n Þ. Since the degree of v 1 is three, the other two vertices which are adjacent to v 1 , are v 2 and v 3 . For vertices v 1 and v 2 , we have Similarly, for the vertices v 1 and v 3 , we have Now, we select v 2 on C 4 ðK n Þ. The degree of v 2 is two, the other vertex which is adjacent to v 2 , is v 3 of degree three. For vertices v 2 and v 3 , we have Adding equations (37) to (44) RðGÞ = Randic or connectivity index of C 4 ðK n Þ + v 1 v 3 is RðGÞ = ðn − 1Þ/ ffiffiffiffiffiffiffiffiffiffiffi ffi n 2 − 1 p + ð1/ ðn − 1ÞÞððn 2 − 3n + 2Þ/2Þ + ð2/ ffiffiffiffiffiffiffiffiffiffiffiffi 2n + 2 p Þ + 1. Similarly, SCIðGÞ = sum connectivity index of C 4 ðK n Þ + v 1 v 3 is SCI G ð Þ = n − 1 ffiffiffiffiffi 2n p + 2 ffiffiffiffiffiffiffiffiffiffi n + 4 p + 1 ffiffiffiffiffiffiffiffiffiffiffiffi 2n − 2 p n 2 − 3n + 2 2 + 2 ffiffi ffi 5 p + 1 ffiffi ffi 6 p : Using the same arguments as in Theorem 6, we determine some other topological indices of C 4 ðK n Þ + v 1 v 3 . Theorem 8. The atom bond connectivity index and augmented Zagreb index of families of graphs C 4 ðK n Þ + v 1 v 3 are defined, respectively, by