First General Zagreb Co-Index of Graphs under Operations

Topological indices are graph-theoretic parameters which are widely used in the subject of chemistry and computer science to predict the various chemical and structural properties of the graphs respectively. Let G be a graph; then, by performing sub-division-related operations S , Q , R , and T on G , the four new graphs S ( G ) (subdivision graph), Q ( G ) (edge-semitotal), R ( G ) (vertex-semitotal), and T ( G ) (total graph) are obtained, respectively. Furthermore, for two simple connected graphs G and H , we define F -sum graphs (denoted by G + F H ) which are obtained by Cartesian product of F ( G ) and H , where F ∈ S,R, Q,T { } . In this study, we determine first general Zagreb co-index of graphs under operations in the form of Zagreb indices and co-indices of their basic graphs.


Introduction
Graph theory has given di erent valuable tools in which likely the best tool is known as topological index (TI) that is used to predict structural and chemical properties of graphs such as connectivity, solubility, freezing point, boiling point, critical temperature, and molecular mass, see [1]. e medical behaviors and drugs' particles of the di erent compounds are discussed with the help of various TIs in the pharmaceutical industries, see [2]. In addition, for the study of molecules, the quantitative structures' activity relationships (QSAR) and quantitative structures' property relationships (QSPR) are very useful techniques which are mostly performed with the help of TIs [3].
ere are three basic types of TIs depending on the parameters of degree, distance, and polynomial. According to recent review [4], the degree-based TIs are mostly studied. First of all, Wiener calculated the boiling point of para n with the help of a degree-based TI, see [5]. Gutman and Trinajsti introduced Zagreb indices and used them to compute the di erent structure-based characteristics of the molecular graphs [6].
Later on, Shenggui and Huiling characterized the graphs for the rst general Zagreb index [7]. Bedratyuk and Savenko calculated the ordinary generating function and linear recurrence relation for the sequence of the general rst Zagreb index [8].
Recently, Ashra et al. de ned Zagreb co-index and computed it for graphs which are formed using various operations, see [9,10]. Kinkar et al. computed the rst Zagreb co-indices of trees under di erent conditions, see [11]. Mansour and Song established relationship between Zagreb indices and co-indices of graphs [12]. Huaa and Zhang computed sharp bounds on the rst Zagreb co-index in terms of Wiener, eccentric distance sum, eccentric connectivity, and degree distance indices [13]. Gutman et al. calculated relations between the Zagreb indices and co-indices of a graph G and of its complement G [14,15].
In graph theory, the operations (union, intersection, complement, product, and subdivision) play an important role to develop new structure of graphs. Yan et al. computed the Wiener index for new graphs using five different operations L, S, Q, R, and T on a graph G such as line graph L(G), subdivided graph S(G), line superposition graph Q(G), triangle parallel graph R(G), and total graph T(G), respectively, see [16]. After that, Eliasi and Taeri computed Wiener indices of newly defined F-sum graphs represented as (G 1+F G 2 ), where F ∈ S, R, Q, T { } [17]. Furthermore, Deng et al. calculated Zagreb indices [18], Ibraheem et al. [19] forgot co-index, Liu et al. obtained first general Zagreb (FGZ) index [20], and Javaid et al. [21] computed the bounds of first Zagreb co-index; furthermore, they also studied the connection-based Zagreb index and co-index [22] of these graphs.
In this study, we computed FGZ co-index of graphs under operations such as M 1 (G +S H), M 1 (G +R H), M 1 (G +Q H), and M 1 (G +T H). e reset of the study is settled as follows. Section 2 contains preliminaries. In Section 3, the main results of our work are discussed, and Section 4 has the conclusion of work.

Preliminaries
Let G be a simple and connected graph with vertex and edge set denoted by V(G) and E(G), respectively. e degree of vertex any vertex v in G is the number of edges incident on it and denoted by d(v). Let G be a graph; then, its complement is defined as |V(G)| � |V(G)| and uv ∉ G iff uv ∈ G denoted as G. Gutman and Trinajsti introduced the first and second Zagreb indices as [6] (1) Ashrafi ȇt al. defined first Zagreb co-index M 1 (G) as follows, see [10]: (2) Let G be a graph; then, S(G) is obtained by adding one vertex in every edge of G.
(i) R(G) is obtained from S(G) by inserting an edge between the vertices that are adjacent in G (ii) Q(G) is obtained from S(G) by inserting an edge between new vertices that adjacent edges of G (iii) Apply both R(G) and Q(G) on S(G); then, T(G) is obtained Suppose two connected graphs G and H; then, their F − sum graph is represented by

Main Results
is section contains results about FGZ co-index of graphs under operations.

Theorem 1. Let G +S H be S-sum graph; then, its first general Zagreb co-index is given as
2 Journal of Mathematics Proof. Using equation (2), we have The graph P 4 The graph C 3 The graph P 4 + S C 3 The graph P 4 + Q C 3 The graph P 4 + R C 3 The graph P 4 + T C 3 Figure 2: F-sum graphs for P 5 and P 6 .
The graph S (C 6 ) The graph Q (C 6 ) The graph T (C 6 ) The graph R (C 6 ) The graph C 6 Figure 1: We arrived at desired result by putting the values in equation (4).
Proof. Using equation (2), we have Using equation 4, we directly have Journal of Mathematics We arrived at desired result by putting the values in equation (6). □ Theorem 3. Let G +Q H be Q-sum graph; then, its first general Zagreb co-index is given as