Abstract

The first general Zagreb (FGZ) index (also known as the general zeroth-order Randić index) of a graph can be defined as , where is a real number. As is equal to the order and size of when and , respectively, is usually assumed to be different from 0 to 1. In this paper, for every integer , the FGZ index is computed for the generalized F-sums graphs which are obtained by applying the different operations of subdivision and Cartesian product. The obtained results can be considered as the generalizations of the results appeared in (IEEE Access; 7 (2019) 47494–47502) and (IEEE Access 7 (2019) 105479–105488).

1. Introduction

Graph theory concepts are being utilized to model and study the several problems in different fields of science, including chemistry and computer science. A topological index (TI) of a (molecular) graph is a numeric quantity that remained unchanged under graph isomorphism [1,2]. Many topological indices have found applications in chemistry, especially in the quantitative structure-activity/property relationships studies; for detail, see [3–13].

Wiener index is the first TI introduced by Harry Wiener in 1947, when he was working on the boiling point of paraffin [14]. In 1972, Trinajstić and Gutman [15] obtained a formula concerning the total energy of electrons of molecules where the sum of square of valences of the vertices of a molecular structure was appeared. This sum is nowadays known as the first Zagreb index. In this paper, we are concerned with a generalized version of the first Zagreb index, known as the general first Zagreb index as well as the general zeroth-order Randić index.

There are several operations in graph theory such as product, complement, addition, switching, subdivision, and deletion. In many cases, graph operations may be helpful in finding graph quantities of more complicated graphs by considering the less complicated ones. In chemical graph theory, by using different graph operations, one can develop large molecular structures from the simple and basic structures. Recently, many classes of molecular structures are studied with the assistance of graph operations.

In 2007, Yan et al. [6] listed the five subdivision operations with the help of their vertices and edges. They also discussed the different features of Wiener index of graphs under these operations. After that, Eliasi and Taeri [16] introduced the -sum graphs with the assistance of Cartesian product on graphs and , where is obtained by applying the subdivision operations , and . They also defined the Wiener indices of these resulting graphs , , , and . Later on, Deng et al. [17] calculated the 1st and 2nd Zagreb topological indices, and Imran and Akhtar [18] calculated the forgotten topological index of the -sums graph. In 2019, Liu et al. [19] computed the first general Zagreb index of -sums graphs.

Recently, Liu et al. [20] introduced the generalized version of the aforesaid subdivided operations of graphs denoted by , where is counting number. They also defined the generalized F-sums graphs using these generalized operations and calculated their 1st and 2nd Zagreb indices. In the present work, we compute the 1st general Zagreb index of the generalized -sums graphs for . The remaining work is arranged as follows: Section 2 contains some basic definitions, Section 3 contains the key outcomes, and Section 4 contains the some particular applications. Conclusions of the obtained results are presented in Section 5.

2. Preliminaries

Let be a simple graph having the order the size of a graph, where is considered as node set and is a bond set. Every vertex is considered as an atom in a graph, and bonding within the two atoms is known as edge. The valency or degree of any node is the number of total edges which are incident to the node. Now, few useful TI’s are explained given below:

Definition 1. If be a connected graph, then the 1st and 2nd Zagreb topological indices asThese two descriptors of the graph were introduced by Trinajsti and Gutman [15]. Such type of TI’s have been utilized to discuss the QSAR/QSPR of the different chemical structures such as chirality, complexity, hetero-system, ZE-isomers, electron energy, and branching [9, 10].

Definition 2. If is the real number, , and be a connected graph, so the 1st general Zagreb topological index is given as

Definition 3. If is the real number, , and be a connected graph, so the general Randi is given aswhere is considered as the classical Randi connectivity topological index.
The generalized F-sums graph is defined in [20] as follows:(i) graph is obtained by inserting vertices in each edge of .(ii) is obtained from by joining the old vertices which are adjacent .(iii) is obtained from by joining the new vertices lying on edge to the corresponding new vertices of other edge, if these edges have some common vertex in .(iv) is union of and graphs. For further details, see Figure 1.

(a)

(b)

(c)

(d)

(e)

(a)
(b)
(c)
(d)
(e)

Figure 1 

(a) , (b). , (c) , (d) , and (e) .

Definition 4. If be two connected molecular structures, and be a structure obtained after using on with bonds (edges) and nodes (vertices) . So, the generalized F-sums graph () is a structure with nodes:in such a way two nodes of are adjacent if [] or []. For more details, see Figures 2 and 3.

(a)

(b)

(c)

(d)

(a)
(b)
(c)
(d)

Figure 2 

(a) . (b) . (c) . (d) .

(a)

(b)

(a)
(b)

Figure 3 

(a) . (b) .

Lemma 1. For and , the degree of in is

3. Main Results

The main results of FGZ index of the generalized F-sum graphs are presented in this section.

Theorem 1. Let and be two simple graphs and . The FGZ index of the generalized -sum graph iswhere is the set of natural numbers and .

Proof. LetFor , then the above equation is considered asFor every vertex and edge , then 1st term of (8) will beSince . So, for every and with , and ; then the 2nd term of (8) isand the 3rd term of equation (8) will beSince in this case , we haveBy using (9), (10), (12) in (8), we get

Theorem 2. Let and be two simple graphs and . The FGZ index of the generalized -sum graph iswhere is the set of natural numbers and .

Proof. Then by definition, we have,For , the above equation is consider asFor every vertex edge , then the 1st term of (16) isFor every vertex edge , then the 2nd term of (16) will beFor every vertex edge . Since we have also . So the 3rd term of (16) will beHere and :and the 4th term of (5) isSince in this case , we haveUsing (17), (18), (20), and (22) in (16), then we have