# First General Zagreb Co-Index of Graphs under Operations

Authors: Muhammad Javaid; Muhammad Ibraheem; Ebenezer Bonyah; Uzma Ahmad; Shaohui Wang
Journal: Journal of Mathematics
 (2022)

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## Abstract

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. LetG be a graph; then, by performing subdivision-related operations S, Q, R, and T on G, the four new graphs SG (subdivision graph), QG (edge-semitotal), RG (vertex-semitotal), and TG (total graph) are obtained, respectively. Furthermore, for two simple connected graphs G and H, we define F-sum graphs (denoted by G+FH) which are obtained by Cartesian product of FG 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.


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## Body

## 1. Introduction

Graph theory has given different 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]. The medical behaviors and drugs’ particles of the different 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].There 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 paraffin with the help of a degree-based TI, see [5]. Gutman and Trinajsti introduced Zagreb indices and used them to compute the different structure-based characteristics of the molecular graphs [6].Later on, Shenggui and Huiling characterized the graphs for the first general Zagreb index [7]. Bedratyuk and Savenko calculated the ordinary generating function and linear recurrence relation for the sequence of the general first Zagreb index [8].Recently, Ashrafi et al. defined Zagreb co-index and computed it for graphs which are formed using various operations, see [9, 10]. Kinkar et al. computed the first Zagreb co-indices of trees under different 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 first 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 operationsL, S, Q, R, and T on a graph G such as line graph LG, subdivided graph SG, line superposition graph QG, triangle parallel graph RG, and total graph TG, respectively, see [16]. After that, Eliasi and Taeri computed Wiener indices of newly defined F-sum graphs represented as G1+FG2, 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 asM¯1G+SH, M¯1G+RH, M¯1G+QH, and M¯1G+TH. The 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.


## 2. Preliminaries

LetG be a simple and connected graph with vertex and edge set denoted by VG and EG, respectively. The 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 VG¯=VG and uv∉G¯ iff uv∈G denoted as G¯. Gutman and Trinajsti introduced the first and second Zagreb indices as [6](1)M1G=∑p1p2∈EGdGp1+dGp2,M2G=∑p1p2∈EGdGp1dGp2.Ashrafi ȇt al. defined first Zagreb co-indexM¯1G as follows, see [10]:(2)M¯1G=∑y1y2∉EGdGy1+dGy2.LetG be a graph; then, SG is obtained by adding one vertex in every edge of G.(i)
RG is obtained from SG by inserting an edge between the vertices that are adjacent in G(ii)
QG is obtained from SG by inserting an edge between new vertices that adjacent edges of G(iii)
Apply bothRG and QG on SG; then, TG is obtainedSuppose two connected graphsG and H; then, their F−sum graph is represented by G+FH having vertex set VG+FH=VG∪EG×VH and y1,y2z1,z2∈EG+FH iff y1=z1∈VG and y2∽z2∈Hy2=z2∈VH and y1∽z1∈FG, where F∈S,R,Q,T.For details, see Figure1 and 2.Figure 1
Subdivision ofC5.Figure 2
F-sum graphs for P5 and P6.


## 3. Main Results

This section contains results about FGZ co-index of graphs under operations.Theorem 1.
LetG+SH be S-sum graph; then, its first general Zagreb co-index is given as(3)M¯1γG+SH=2θn22e12−n2e1++2θe12n1e2+e2¯+n2n1−2+∑i=0θθiM1θ−iGM1¯i+1H+M1γ−iGM1iH+M1i+1HM1γ−iG+M1¯γ−iG+M1¯i+1HM1γ−iG+M1¯γ−iG+M1iHM1¯θ−iG+M1γ−iG+M1γ−iGM1¯iH+M1¯γ−iGM1iH+M1¯iH,where θ=γ−1.Proof.
Using equation (2), we have(4)M¯G+SH=∑y1,y2z1,z2∉EG+SHdy1,z1+dy2,z2,M¯1γG+SH=∑z1,z2∈H∑y1,y2∈VSG−GdG+SHθy1,z1+dG+SHθy2,z2+∑y1,y2∈VGdG+SHθy1,z1+dG+SHθy2,z2+∑y1,y2∈VSGy1∈VGy2∈VSG−GdG+SHθy1,z1+dG+SHθy2,z2=∑A+∑B+∑C,∑A=∑y1,y2∈VSG−G∑z1,z2∈VG2dG+SHθy1,z1+dG+SHθy2,z2,=∑y1,y2∈VSG−G∑z1,z2∈VG22θ+2θ,∑A=2θn22e12−n2e1,∑B=∑B1+∑B2+∑B3+∑B4+∑B5+∑B6,∑B1=∑y∈VG∑z1z2∉EHdG+SHθy,z1+dG+SHθy,z2=∑y∈VG∑z1z2∉EH∑i=0θθidGθ−iydHiz1+dGθ−iydHiz2=∑y∈VG∑z1z2∉EH∑i=0θθidGθ−iydHiz1+dHiz2,=∑i=0θθiM1θ−iGM1¯i+1H,∑B2=∑y1y2∈EG∑z∈VHdG+SHθy1,z+dG+SHθy2,z=∑y1y2∈EG∑z∈VH∑i=0θθidGθ−iy1dHiz+dGθ−iy2dHiz=∑y1,y2∈VG∑z∈H∑i=0θθidHizdGθ−iy1dGθ−iy2,=∑i=0θθiM1γ−iGM1iH,∑B3=∑y1y2∈EG∑z1z2∈EHdG+SHθy1,z1+dG+SHθy2,z2=∑y1y2∈EG∑z1z2∈EH∑i=0θθidGθ−iy1dHiz1+dGθ−iy2dHiz2,=∑i=0θθiM1γ−iGM1i+1H,∑B4=∑y1y2∉EG∑z1z2∈EHdG+SHθy1,z1+dG+SHθy2,z2=∑y1y2∉EG∑z1z2∈EH∑i=0θθidGθ−iy1dHiz1+dGθ−iy2dHiz2,=∑i=0θθiM1¯γ−iGM1i+1H,∑B5=∑y1y2∈EG∑z1z2∉EHdG+SHθy1,z1+dG+SHθy2,z2=∑y1y2∈EG∑z1z2∉EH∑i=0θθidGθ−iy1dHiz1+dGθ−iy2dHiz2,=∑i=0θθiM1γ−iGM1¯i+1H,∑B6=∑y1y2∉EG∑z1z2∉EHdG+SHθy1,z1+dG+SHθy2,z2=∑y1y2∉EG∑z1z2∉EH∑i=0θθidGθ−iy1dHiz1+dGθ−iy2dHiz2=∑i=0θθiM1¯γ−iGM1¯i+1H,∑B=∑i=0θθiM1θ−iGM1¯i+1H+M1γ−iGM1iH+M1i+1HM1γ−iG+M1¯γ−iG+M1¯i+1HM1γ−iG+M1¯γ−iG,∑C=∑C1+∑C2+∑C3+∑C4+∑C5,∑C1=∑y1y2∉ESGy1∈VGy2∈VSG−G∑z∈HdG+SHθy1,z+dG+SHθy2,z=∑y1y2∉ESGy1∈VGy2∈VSG−G∑z∈H∑i=0θθidGθ−iy1dHiz+2θ=∑i=0θθiM1¯θ−iGM1iH+2θn2e1n1−2,∑C2=∑y1y2∈ESGy1∈VGy2∈VSG−G∑z1z2∈EHdG+SHθy1,z1+dG+SHθy2,z2=∑y1y2∈ESGy1∈VGy2∈VSG−G∑z1z2∈EH∑i=0θθidGθ−iy1dHiz1+2θ,=∑i=0θθiM1γ−iGM1iH+2θ+2e1e2,∑C3=∑y1y2∈ESGy1∈VGy2∈VSG−G∑z1z2∉EHdG+SHθy1,z1+dG+SHθy2,z2=∑y1y2∈ESGy1∈VGy2∈VSG−G∑z1z2∉EH∑i=0θθidGθ−iy1dHiz1+2θ,=∑i=0θθiM1γ−iGM1¯iH+2θ+2e1e2¯,∑C4=∑y1y2∉ESGy1∈VGy2∈VSG−G∑z1z2∈EHdG+SHθy1,z1+dG+SHθy2,z2=∑y1y2∉ESGy1∈VGy2∈VSG−G∑z1z2∈EH∑i=0θθidGθ−iy1dHiz1+2θ,=∑i=0θθiM1¯γ−iGM1iH+2θ+1e1e2n1−2,∑C5=∑y1y2∉ESGy1∈VGy2∈VSG−G∑z1z2∉EHdG+SHθy1,z1+dG+SHθy2,z2=∑y1y2∉ESGy1∈VGy2∈VSG−G∑z1z2∉EH∑i=0θθidGθ−iy1dHiz1+2θ,=∑i=0θθiM1¯γ−iGM1¯iH+2θ+1e1e2¯n1−2,∑C=∑i=0θθiM1iHM1¯θ−iG+M1γ−iG+M1γ−iGM1¯iH+M1¯γ−iGM1iH+M1¯iH+2θ+1e1n1e2+e2¯.
We arrived at desired result by putting the values in equation (4).Theorem 2.
LetG+RH be R-sum graph; then, its first general Zagreb co-index is given as(5)M¯1γG+RH=2θn22e12−n2e1+2θe12n1e2+e2¯+n2n1−22∑i=0θθiM1θ−iGM1¯i+1H+M1γ−iG+M1¯γ−iG+M1¯i+1HM1γ−iG+M1¯γ−iG+M1¯θ−iGM1iH+M1γ−iGM1iH+M1¯iH+M1¯γ−iGM1iH+M1¯iH,where γ=θ−1.Proof.
Using equation (2), we have(6)M¯γG+RH=∑y1,y2z1,z2∉EG+RHdθy1,z1+dθy2,z2,M¯1γG+RH=∑z1,z2∈H∑y1,y2∈VRG−GdG+RHθy1,z1+dG+RHθy2,z2+∑y1,y2∈VGdG+RHθy1,z1+dG+RHθy2,z2+∑y1,y2∈VRG1y1∈VG1y2∈VRG−GdG+RHθy1,z1+dG+RHθy2,z2=∑A+∑B+∑C.
Using equation4, we directly have(7)∑A=2θn22e12−n2e1,∑B=∑B1+∑B2+∑B3+∑B4+∑B5,∑B1=∑y∈VG∑z1z2∉EHdG+RHθy,z1+dG+RHθy,z2=∑y∈VG∑z1z2∉EH∑i=0θθi2dGθ−iydHiz1+2dGθ−iydHiz2=2∑y∈VG∑z1z2∉EH∑i=0θθidGθ−iydHiz1+dHiz2=2∑i=0θθiM1θ−iGM1¯i+1H,∑B2=∑y1y2∈EG∑z1z2∈EHdG+RHθy1,z1+dG+RHθy2,z2=∑y1y2∈EG∑z1z2∈EH∑i=0θθi2dGθ−iy1dHiz1+2dGθ−iy2dHiz2=2∑i=0θθiM1γ−iGM1i+1H,∑B3=∑y1y2∉EG∑z1z2∈EHdG+RHθy1,z1+dG+RHθy2,z2=∑y1y2∉EG∑z1z2∈EH∑i=0θθi2dGθ−iy1dHiz1+2dGθ−iy2dHiz2=2∑i=0θθiM1¯γ−iGM1i+1H,∑B4=∑y1y2∈EG∑z1z2∉EHdG+RHθy1,z1+dG+RHθy2,z2=∑y1y2∈EG∑z1z2∉EH∑i=0θθi2dGθ−iy1dHiz1+2dGθ−iy2dHiz2=2∑i=0θθiM1γ−iGM1¯i+1H,∑B5=∑y1y2∉EG∑z1z2∉EHdG+RHθy1,z1+dG+RHθy2,z2=∑y1y2∉EG∑z1z2∉EH∑i=0θθi2dGθ−iy1dHiz1+2dGθ−iy2dHiz2=2∑i=0θθiM1¯γ−iGM1¯i+1H,∑B=2∑i=0θθiM1θ−iGM1¯i+1H+M1γ−iG+M1¯γ−iG+M1¯i+1HM1γ−iG+M1¯γ−iG,∑C=∑C1+∑C2+∑C3+∑C4+∑C5,∑C1=∑y1y2∉ERGy1∈VGy2∈VRG−G∑z∈HdG+RHθy1,z+dG+RHθy2,z=∑y1y2∉ERGy1∈VGy2∈VRG−G∑z∈H∑i=0θθi2dGθ−iy1dHiz+2θ=2∑i=0θθiM1¯θ−iGM1iH+2θn2e1n1−2,∑C2=∑y1y2∈ERGy1∈VGy2∈VRG−G∑z1z2∈EHdG+RHθy1,z1+dG+RHθy2,z2=∑y1y2∈ERGy1∈VGy2∈VRG−G∑z1z2∈EH∑i=0θθi2dGθ−iy1dHiz1+2θ=2∑i=0θθiM1γ−iGM1iH+2θ+2e1e2,∑C3=∑y1y2∈ERGy1∈VGy2∈VRG−G∑z1z2∉EHdG+RHθy1,z1+dG+RHθy2,z2=∑y1y2∈ERGy1∈VGy2∈VRG−G∑z1z2∉EH∑i=0θθi2dGθ−iy1dHiz1+2θ=2∑i=0θθiM1γ−iGM1¯iH+2θ+2e1e2¯,∑C4=∑y1y2∉ERGy1∈VGy2∈VRG−G∑z1z2∈EHdG+RHθy1,z1+dG+RHθy2,z2=∑y1y2∉ERGy1∈VGy2∈VRG−G∑z1z2∈EH∑i=0θθi2dGθ−iy1dHiz1+2θ=2∑i=0θθiM1¯γ−iGM1iH+2θ+1e1e2n1−2,∑C5=∑y1y2∉ERGy1∈VGy2∈VRG−G∑z1z2∉EHdG+RHθy1,z1+dG+RHθy2,z2=∑y1y2∉ERGy1∈VGy2∈VRG−G∑z1z2∉EH∑i=0θθi2dGθ−iy1dHiz1+2θ=2∑i=0θθiM1¯γ−iGM1¯iH+2θ+1e1e2¯n1−2,∑C=2∑i=0θθiM1¯θ−iGM1iH+M1γ−iGM1iH+M1¯iH+M1¯γ−iGM1iH+M1¯iH+2θe12n1e2+e2¯+n2n1−2.
We arrived at desired result by putting the values in equation (6).Theorem 3.
LetG+QH be Q-sum graph; then, its first general Zagreb co-index is given as(8)M¯1γG+QH=+4θn2e1n1−2+4e1n1+2e2+e2¯∑i=0θθiM1θ−iGM1¯i+1H+M1γ−iGM1iH+M1i+1HM1γ−iG+M1¯γ−iG+M1¯i+1HM1γ−iG+M1¯γ−iG+M1iHM1¯θ−iG+M1γ−iG+M1γ−iGM1¯iH+M1¯γ−iGM1iH+M1¯iH+α1,where γ=θ−1.Proof.
Using equation (2), we have(9)M¯γG+QH=∑y1,y2z1,z2∉EG+QHdθy1,z1+dθy2,z2,M¯1γG+QH=∑z1,z2∈H∑y1,y2∈VQG−GdG+SHθy1,z1+dG+QHθy2,z2+∑y1,y2∈VGdG+QHθy1,z1+dG+QHθy2,z2+∑y1,y2∈VSGy1∈VGy2∈VQG−GdG+QHθy1,z1+dG+QHθy2,z2=∑A+∑B+∑C,∑A=∑y1,y2∈VQG−G∑z1,z2∈VG2dG+QHθy1,z1+dG+QHθy2,z2=α1.
Using equation4, we directly have(10)∑B=∑i=0θθiM1θ−iGM1¯i+1H+M1γ−iGM1iH+M1i+1HM1γ−iG+M1¯γ−iG+M1¯i+1HM1γ−iG+M1¯γ−iG,∑C=∑C1+∑C2+∑C3+∑C4+∑C5,∑C1=∑y1y2∉EQGy1∈VGy2∈VQG−G∑z∈HdG+QHθy1,z+dG+QHθy2,z=∑y1y2∉EQGy1∈VGy2∈VQG−G∑z∈H∑i=0θθidGθ−iy1dHiz+4θ=∑i=0θθiM1¯θ−iGM1iH+4θn2e1n1−2,∑C2=∑y1y2∈EQGy1∈VGy2∈VQG−G∑z1z2∈EHdG+QHθy1,z1+dG+QHθy2,z2=∑y1y2∈EQGy1∈VGy2∈VQG−G∑z1z2∈EH∑i=0θθidGθ−iy1dHiz1+2θ=∑i=0θθiM1γ−iGM1iH+4θ+2e1e2,∑C3=∑y1y2∈EQGy1∈VGy2∈VQG−G∑z1z2∉EHdG+QHθy1,z1+dG+QHθy2,z2=∑y1y2∈EQGy1∈VGy2∈VQG−G∑z1z2∉EH∑i=0θθidGθ−iy1dHiz1+2θ=∑i=0θθiM1γ−iGM1¯iH+4θ+2e1e2¯,∑C4=∑y1y2∉EQGy1∈VGy2∈VQG−G∑z1z2∈EHdG+QHθy1,z1+dG+QHθy2,z2=∑y1y2∉EQGy1∈VGy2∈VQG−G∑z1z2∈EH∑i=0θθidGθ−iy1dHiz1+4θ=∑i=0θθiM1¯γ−iGM1iH+4θ+1e1e2n1−2,∑C5=∑y1y2∉EQGy1∈VGy2∈VQG−G∑z1z2∉EHdG+QHθy1,z1+dG+QHθy2,z2=∑y1y2∉EQGy1∈VGy2∈VQG−G∑z1z2∉EH∑i=0θθidGθ−iy1dHiz1+4θ=∑i=0θθiM1¯γ−iGM1¯iH+4θ+1e1e2¯n1−2,∑C=∑i=0θθiM1iHM1¯θ−iG+M1γ−iG+M1γ−iGM1¯iH+M1¯γ−iGM1iH+M1¯iH+4θn2e1n1−2+4e1n1+2e2+e2¯.
We arrived at desired result by putting the values in equation (9).Theorem 4.
LetG+TH be T-sum graph; then, its first general Zagreb co-index is given as(11)M¯1γG+TH=4θn22e12−n2e1+2∑i=0θθiM1θ−iGM1¯i+1H+M1γ−iG+M1¯γ−iG+M1¯i+1HM1γ−iG+M1¯γ−iGM1¯θ−iGM1iH+M1γ−iGM1iH+M1¯iH+M1¯γ−iGM1iH+M1¯iH+4θn2e1n1−2+4e1n1+2e2+e2¯,where γ=θ−1.Proof.
Using equation (2), we have(12)M¯1γG+TH=∑y1,y2z1,z2∉EG+THdθy1,z1+dθy2,z2,M¯1γG+TH=∑z1,z2∈H∑y1,y2∈VTG−VGdG+SHθy1,z1+dG+THθy2,z2+∑y1,y2∈VGdG+THθy1,z1+dG+THθy2,z2+∑y1,y2∈VSGy1∈VGy2∈VTG−GdG+THθy1,z1+dG+THθy2,z2=∑A+∑B+∑C.
Using equation9, we directly have(13)∑A=4θn22e12−n2e1.
The value∑A and ∑B are by equations (7) and (9)asfollows:(14)∑C=∑C1+∑C2+∑C3+∑C4+∑C5,∑C1=∑y1y2∉ETGy1∈VGy2∈VTG−G∑z∈HdG+THθy1,z+dG+THθy2,z=∑y1y2∉ETGy1∈VGy2∈VTG−G∑z∈H∑i=0θθi2dGθ−iy1dHiz+4θ=2∑i=0θθiM1¯θ−iGM1iH+4θn2e1n1−2,∑C2=∑y1y2∈ETGy1∈VGy2∈VTG−G∑z1z2∈EHdG+THθy1,z1+dG+THθy2,z2=∑y1y2∈ETGy1∈VGy2∈VTG−G∑z1z2∈EH∑i=0θθi2dGθ−iy1dHiz1+4θ,=2∑i=0θθiM1γ−iGM1iH+4θ+2e1e2,∑C3=∑y1y2∈ETGy1∈VGy2∈VTG−G∑z1z2∉EHdG+THθy1,z1+dG+THθy2,z2=∑y1y2∈ETGy1∈VGy2∈VTG−G∑z1z2∉EH∑i=0θθi2dGθ−iy1dHiz1+4θ=2∑i=0θθiM1γ−iGM1¯iH+4θ+2e1e2¯,∑C4=∑y1y2∉ETGy1∈VGy2∈VTG−G∑z1z2∈EHdG+THθy1,z1+dG+THθy2,z2=∑y1y2∉ETGy1∈VGy2∈VTG−G∑z1z2∈EH∑i=0θθi2dGθ−iy1dHiz1+4θ=2∑i=0θθiM1¯γ−iGM1iH+4θ+1e1e2n1−2,∑C5=∑y1y2∉ETGy1∈VGy2∈VTG−G∑z1z2∉EHdG+THθy1,z1+dG+THθy2,z2=∑y1y2∉ETGy1∈VGy2∈VTG−G∑z1z2∉EH∑i=0θθi2dGθ−iy1dHiz1+4θ=2∑i=0θθiM1¯γ−iGM1¯iH+4θ+1e1e2¯n1−2.
By substituting the values of∑A, ∑b, and ∑C in equation (12), we obtained required proof.


## 4. Conclusion

The study of the basic or factor graphs is an interesting problem in the theory of graphs where the original graphs becomes complex. In this study, we have computed FGZ co-index of graphs under operations such asM¯1G+SH, M¯1G+RH, M¯1G+QH, and M¯1G+TH in the terms of indices and co-indices of their basic or factor graphs.



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*Source: 5094929-2022-05-09.xml*