# Sharp Bounds of First Zagreb Coindex forF-Sum Graphs

Authors: Muhammad Javaid; Muhammad Ibraheem; Uzma Ahmad; Jia-Bao Liu
Journal: Journal of Mathematics
 (2021)

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## Abstract

LetG=VE,EG be a connected graph with vertex set VG and edge set EG. For a graph G, the graphs S(G), R(G), Q(G), and T(G) are obtained by applying the four subdivisions related operations S, R, Q, and T, respectively. Further, for two connected graphs G1 and G2, G1+FG2 are F-sum graphs which are constructed with the help of Cartesian product of FG1 and G2, where F∈S,R,Q,T. In this paper, we compute the lower and upper bounds for the first Zagreb coindex of these F-sum (S-sum, R-sum, Q-sum, and T-sum) graphs in the form of the first Zagreb indices and coincides of their basic graphs. At the end, we use linear regression modeling to find the best correlation among the obtained results for the thirteen physicochemical properties of the molecular structures such as boiling point, density, heat capacity at constant pressure, entropy, heat capacity at constant time, enthalpy of vaporization, acentric factor, standard enthalpy of vaporization, enthalpy of formation, octanol-water partition coefficient, standard enthalpy of formation, total surface area, and molar volume.


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## Body

## 1. Introduction

The subject of graph theory is growing and moving into mainstream of mathematics, playing a basic role in various disciplines of science especially in chemistry and computer science. A topological index (TI) is a function from a set of graphs to the set of real numbers that assigns a different numerical value to each graph unless the graphs are isomorphic. Moreover, this numeric value predicts various physical and chemical properties of the involved organic compounds in the molecular graphs such as volume, density, pressure, weight, boiling point, freezing point, vaporization point, heat of formation, and heat of evaporation [1, 2]. TIs are also used to study the quantitative structure-activity relationship QSAR and quantitative structure-property relationship QSPR and medical behaviors of different drugs in the subject of cheminformatics and pharmaceutical industries, respectively [3–5].There are various types of TIs based on degree, distance, and polynomial, but degree-based TIs are more studied than others (see the latest survey [6]). Wiener calculated boiling point of paraffin by using a distance-based TI (see [7]). Gutman and Trinajsti defined the TIs known as first and second Zagreb indices to calculate total π-electron energy of alternant hydrocarbons [8].There are different operations on graphs such as joining, deleting, union, intersection, product, complement, switching, and subdivision. These operations on graphs play an important role to obtain the new molecular graphs from the old ones. Yan et al. listed five graphsLG, SG, QG, RG, and TG with the help of five operations L, S, Q, R, and T, respectively. They also studied the behavior of Wiener index of graphs under these five decorations (see [4]). Eliasi and Taeri introduced the four subdivision-related operations using the concept of Cartesian product and obtained four sum graphs (G1+SG2, G1+RG2, G1+QG2, and G1+TG2). They also computed the Wiener indices of these newly defined F-sum graphs (see [9]). Akhtar and Imran calculated the forgotten index [10], Deng et al. [11] computed first and second Zagreb indices, and Liu et al. computed first general Zagreb index of these graphs (see [12]).Ashrafi et al. computed Zagreb coindices of composite graph operations such as joining, union, disjunction, Cartesian product, corona product, and composition (see [13]). Mansour and Song computed a and a,b-analogs of Zagreb indices and coindices of graphs [14]. Ashrafi et al. worked on extremal graphs with respect to the Zagreb coindices [15]. Ramane et al. calculated coindices for the transmission- and reciprocal transmission-based graphs (see [16]). Javaid et al. computed the first Zagreb connection index and coindex of some derived graphs [17].In this paper, we compute the lower and upper bounds for the first Zagreb coindex ofF-sum (S-sum, R-sum, Q-sum, and T-sum) graphs in the form of the first Zagreb indices and coincides of their basic graphs. At the end, the obtained results are also illustrated with the help of the examples of the exact and bonded values for some particular F-sum graphs. The reset of the paper is organized as follows: Section 2 includes the basic definitions and notions, Section 3 contains the main results of our work, and Section 4 presents particular examples related to the main results.


## 2. Preliminaries

LetG=VG,EG be a connected graph, where VG is order and EG is size of the graph G. For any vertex v∈VG, its degree is denoted by dv and is defined as number of vertices incident on it. An edge is formed by joining of two vertices u,v∈VG denoted by uv. The complement of any graph is denoted by G¯ and is defined as having same number of vertices as of original graph G but for any two vertices u and v, uv∈EG¯ iff uv∉EG. The first and second Zagreb indices M1G and M2G of G are defined in [8] as follows:(1)M1G=∑t∈VGdGt2=∑t1t2∈EGdGt1+dGt2,M2G=∑t1t2∈EGdGt1dGt2.The first Zagreb coindexM¯1G is defined in [15] as follows:(2)M¯1G=∑t1t2∉EGdGt1+dGt2.It should be noted that the Zagreb coindices ofG run over EG¯, but the degrees correspond to G. For two connected graphs G1 and G2, the F-sum graphs are defined as follows.LetG be a graph; then,(i)
S(G) is a graph formed by putting one vertex in each edge of G.(ii)
R(G) is a graph obtained from S(G) by joining the adjacent vertices of G.(iii)
Q(G) is a graph formed from S(G) by joining the pairs of new vertices which are on the adjacent edges of G.(iv)
T(G) is formed by applying both operations of R(G) and Q(G) on S(G).Suppose thatG1 and G2 are two connected graphs; then, their F-sum graphs are denoted by G1+FG2 and defined with vertex set VG1+SG2=VG1∪EG1×VG2 and edge set as the vertices u1,u2 and v1,v2 of G1+FG2 are joined iff(i)
u1=v1∈VG1 and u2∽v2∈G2.(ii)
u2=v2∈VG2 and u1∽v1∈FG1, where F∈S,R,Q,T.For details, see Figures1 and 2.Figure 1
(a)G≅P4, (b) SG≅SP4, (c) QG≅P4, (d) RG≅RP4, and (e) TG≅TP4.
(a)(b)(c)(d)(e)Figure 2
GraphsG≅P4, H≅P2, and G+FH≅P4+FP2.


## 3. Main Results

In this section, we discussed the main results of the first Zagreb coindex on theF-sum graphs which will be denoted by simply M−.Theorem 1.
LetG1 and G2 be two connected graphs; then, first Zagreb coindex of G1+SG2 is given as(3)α1≤M¯G1+sG2≤α2,where(4)α1=2n22e12−n2e1+2e1e2¯+e2e1+e1¯+n2e1n1−2e2+n2+2e2+e¯2+n2M1G1+n2+2e2+2e¯2M¯1G1+2e2+2e¯1M1G2+n1+2e¯1+e1M¯1G2+2n2e12n2−1+2n2−1e2+n2,α2=2n22e12−n2e1+2e1e2¯+e2e1+e1¯+n2e1n1−2e2+n2+2e2+e¯2+n2M1G1+n2+2e2+2e¯2M¯1G1+2e2+2e¯1M1G2+n1+2e¯1+e1M¯1G2+2e1ESG1n1−2n22n2−1+2n2−1e2+n2.Proof.
Using equation (2), we have(5)M¯G1+SG2=∑t1,t2x1,x2∉EG1+SG2dt1,x1+dt2,x2a,M¯G1+SG2=∑t1,t2∈VSG1−VG1∑x1,x2∈VG2dt1,x1+dt2,x2+∑t1,t2∈VG1∑x1,x2∈VG2dt1,x1+dt2,x2∑t1,t2∈VSG1t1∈VG1t2∈VSG1−VG1∑x1,x2∈VG2dt1,x1+dt2,x2,=∑A+∑B+∑C,(6)∑A=∑t1,t2∈VSG1−VG1∑x1,x2∈VG2dSG1t1+dSG1t2=∑t1,t2∈VSG1−VG1∑x1,x2∈VG22+2=2n22e12−n2e1,∑A=2n22e12−n2e1,(7)∑B=∑B1+∑B2+∑B3+∑B4+∑B5+∑B6,∑B1=∑t∈VG1∑x1x2∉EG2dt,x1+dt,x2=∑t∈VG1∑x1x2∉EG22dG1t+dG2x1+dG2x2=4e1e2¯+n1M1¯G2,∑B2=∑x∈VG2∑t1,t2∈VG1dt1,x+dt2,x=∑x∈VG2∑t1t2∈EG1dt1,x+dt2,x+∑x∈VG2∑t1t2∉EG1dt1,x+dt2,x=∑x∈VG2∑t1t2∈EG1dG1t1+dG1t2+2dG2x+∑x∈VG2∑t1t2∉EG1dG1t1+dG1t2+2dG2x=M1G1n2+4e1e2+M1¯G1n2+4e1¯e2,∑B3=∑t1t2∈EG1∑x1x2∈EG2dt1,x1+dt2,x2=∑t1t2∈EG1∑x1x2∈EG2dG1t1+dG2x1+dG1t2+dG2x2=2e2M1G1+e1M1G2.∑B4=∑t1t2∉EG1∑x1x2∈EG2dt1,x1+dt2,x2=∑t1t2∉EG1∑x1x2∈EG2dG1t1+dG2x1+dG1t2+dG2x2=2e2M1¯G1+e1¯M1G2,∑B5=∑t1t2∉EG1∑x1x2∉EG2dt1,x1+dt2,x2=∑t1t2∉EG1∑x1x2∉EG2dG1t1+dG2x1+dG1t2+dG2x2=2e2¯M1¯G1+e1¯M1¯G2,∑B6=∑t1t2∈EG1∑x1x2∉EG2dt1,x1+dt2,x2=∑t1t2∈EG1∑x1x2∉EG2dG1t1+dG2x1+dG1t2+dG2x2=2e¯2M1G1+e1M1¯G2,(8)∑B=4e1e¯2+e2e1+e¯1+2e2+e¯2+n2M1G1+n22e2+e¯2M¯1G1+2e2+e¯1M1G2+n1+2e¯1+e1M¯1G2,(9)∑C=∑C1+∑C2+∑C3,∑C1=∑t1t2∉ESG1t1∈VG1t2∈VSG1−VG1∑x∈VG2dt1,x+dt2,x,=∑t1t2∉ESG1t1∈VG1t2∈VSG1−VG1∑x∈VG2dG1t1+dx+dSG1t2=n2∑t1t2∉ESG1t1∈VG1t2∈VSG1−VG1dt1+2e1n1−2e2+n2.
Note that2e1≤∑t1t2∉ESG1t1∈VG1t2∈VSG1−VG1dt1≤2e1n1−2ESG1.(10)2n2e1n1−2+2e2e1n1−2+2n2e1n1−2≤∑C1≤2n2e1n1−2ESG1+2e2e1n1−2+2n2e1n1−2,∑C2=∑t1t2∉ESG1t1∈VG1t2∈VSG1−VG1∑x1,x2∈VG2dt1,x1+dt2,x2=∑t1t2∉ESG1t1∈VG1t2∈VSG1−VG1∑x1,x2∈VG2dG1t1+dx1+dSG1t2,=n2n2−1∑t1t2∉ESG1t1∈VG1t2∈VSG1−VG1dG1t1+2e2e1n1−2n2−1+2n2n2−1e1n1−2.
Note that2e1≤∑t1t2∉ESG1t1∈VG1t2∈VSG1−VG1dt1≤2e1n1−2ESG1.(11)2n2n2−1e1n1−2+2e2e1n1−2n2−1+2n2n2−1e1n1−2≤∑C2≤2n2n2−1e1n1−2ESG1,+2e2e1n1−2n2−1+2n2n2−1e1n1−2,∑C3=∑t1t2∈ESG1t1∈VG1t2∈VSG1−V∗G1∑x1,x2∈VG2dt1,x1+dt2,x2=∑t1t2∈ESG1t1∈VG1t2∈VSG1−VG1∑x1,x2∈VG2dG1t1+dx1+dSG1t2=n2n2−1∑t1t2∈ESG1t1∈VG1t2∈VSG1−VG1dt1+2e2n2−1|∑t1t2∈ESG1t1∈VG1t2∈VSG1−VG1dt1|+2n2n2−1|∑t1t2∈ESG1t1∈VG1t2∈VSG1−G1dt1|.
Note that2e1≤∑t1t2∈ESG1t1∈VG1t2∈VSG1−VG1dt1≤2e1ESG1.(12)2e1n2n2−1+4e1n2−1e2+n2≤∑C3≤2e1n2n2−1ESG1+4e1n2−1e2+n2.
Consequently,(13)2e1n2n2−1+4e1n2−1e2+n2+2n2n2−1e1n1−2+2e2e1n1−2n2−1+2n2n2−1e1n1−2+2n2e1n1−2+2e2e1n1−2+2n2e1n1−2≤∑C≤2e1n2n2−1ESG1+4e1n2−1e2+n2+2n2n2−1e1n1−2ESG1+2e2e1n1−2n2−1+2n2n2−1e1n1−2+n2e1n1−2ESG1+2e2e1n1−2+2n2e1n1−2.
By putting the values of∑A+∑B+∑C in equation (5), we get the required proof.Theorem 2.
LetG1 and G2 be two connected graphs; then, first Zagreb coindex of G1+RG2 is given as(14)α1≤M¯G1+RG2≤α2,where(15)α1=2n22e12−n2e1+44e1e¯2+e2e¯1+2e1n2n1−2e2+n2+4e2+4e¯2M1G1+2e1+e¯1M1G2+2n2+2e2+e¯2M¯1G1+n1+2e¯1+2e1M¯1G2+2e1n222n2−1+2n2−1e2+n2,α2=2n22e12−n2e1+44e1e¯2+e2e¯1+2e1n2n1−2e2+n2+4e2+4e¯2M1G1+2n2+2e2+e¯2M¯1G1+2e1+e¯1M1G2+n1+2e¯1+2e1M¯1G2+2ERG1e1n222n2−1n1−2+2n2−1e2+n2.Proof.
Using equation (2), we have(16)M¯G1+RG2=∑t1,t2x1,x2∉EG1+RG2dt1,x1+dt2,x2,M¯G1+RG2=∑A+∑B+∑C.
Using equation (6), we directly have(17)∑A=2n22e12−n2e1,∑B=∑B1+∑B2+∑B3+∑B4+∑B5+∑B6,∑B1=∑t∈VG1∑x1x2∉EG2dt,x1+dt,x2=∑t∈VG1∑x1x2∉EG22dRG1t+dG2x1+dG2x2,=∑t∈VG1∑x1x2∉EG24dG1t+dG2x1+dG2x2=8e1e2¯+n1M1¯G2,∑B2=∑x∈VG2∑t1,t2∈VG1dt1,x+dt2,x=∑x∈VG2∑t1t2∉EG1dt1,x+dt2,x=∑x∈VG2∑t1t2∉EG1dRG1t1+dRG1t2+2dG2x=∑x∈VG2∑t1t2∉EG12dG1t1+2dG1t2+2dG2x=2M1¯G1n2+4e1¯e2,∑B3=∑t1t2∈EG1∑x1x2∈EG2dt1,x1+dt2,x2=2∑t1t2∈EG1∑x1x2∈EG2dRG1t1+dG2x1+dRG1t2+dG2x2=2∑t1t2∈EG1∑x1x2∈EG22dG1t1+dG2x1+2dG1t2+dG2x2=22e2M1G1+e1M1G2,∑B4=∑t1t2∉EG1∑x1x2∈EG2dt1,x1+dt2,x2=2∑t1t2∉EG1∑x1x2∈EG2dRG1t1+dG2x1+dRG1t2+dG2x2=2∑t1t2∉EG1∑x1x2∈EG22dG1t1+dG2x1+2dG1t2+dG2x2=22e2M1¯G1+e1¯M1G2,∑B5=∑t1t2∉EG1∑x1x2∉EG2dt1,x1+dt2,x2=2∑t1t2∉EG1∑x1x2∉EG2dRG1t1+dG2x1+dRG1t2+dG2x2=2∑t1t2∉EG1∑x1x2∉EG2dG1t1+dG2x1+dG1t2+dG2x2=22e2¯M1¯G1+e1¯M1¯G2,∑B6=∑t1t2∈EG1∑x1x2∉EG2dt1,x1+dt2,x2=2∑t1t2∈EG1∑x1x2∉EG2dRG1t1+dG2x1+dRG1t2+dG2x2=2∑t1t2∈EG1∑x1x2∉EG22dG1t1+dG2x1+2dG1t2+dG2x2=22e¯2M1G1+e1M¯1G2,(18)∑B=42e1e¯2+4e¯+e2e¯1+4e2+4e¯2M1G1+2e1+e¯1M1G2+2n2+2e2+e¯2M¯1G1+n1+2e¯1+2e1M¯1G2,∑C=∑t1,t2∈VRG1t1∈VG1t2∈VRG1−VG1dt1,x1+dt2,x2=∑t1t2∉ERG1t1∈VG1t2∈VRG1−VG1∑x∈VG2dt1,x+dt2,x+∑t1t2∉ERG1t1∈VG1t2∈VRG1−VG1∑x1,x2∈VG2dt1,x1+dt2,x2+∑t1t2∈ERG1t1∈VG1t2∈VRG1−VG1∑x1,x2∈VG2dt1,x1+dt2,x2=∑C1+∑C2+∑C3,∑C1=∑t1t2∉ERG1t1∈VG1t2∈VRG1−VG1∑x∈VG2dt1,x+dt2,x=∑t1t2∉ERG1t1∈VG1t2∈VRG1−VG1∑x∈VG2dRG1t1+dx+dRG1t2=∑t1t2∉ERG1t1∈VG1t2∈VRG1−VG1∑x∈VG2dRG1t1+dx+2=n2∑t1t2∉ERG1t1∈VG1t2∈VRG1−VG1dRG1t1+2e2e1n2−1+2n2e1n2−1.
Note that2e1≤∑t1t2∉ERG1t1∈VG1t2∈VSG1−VG1dt1≤2ERG1e1n1−2.(19)2n2e1+2e2e1n2−1+2n2e1n2−1≤∑C1≤2n2ERG1e1n1−2+2e2e1n2−1+2n2e1n2−1,∑C2=∑t1t2∉ERG1t1∈VG1t2∈VRG1−VG1∑x1,x2∈VG2dt1,x1+dt2,x2=∑t1t2∉ERG1t1∈VG1t2∈VRG1−VG1∑x1,x2∈VG2dRG1t1+dx1+dSG1t2=∑t1t2∉ERG1t1∈VG1t2∈VRG1−VG1∑x1,x2∈VG2dRG1t1+dx1+2=n2n2−1∑t1t2∉ERG1t1∈VG1t2∈VRG1−VG1dRG1t1+2e2e1n1−2n2−1+2n2n2−1e1n1−2.
Note that2e1≤∑t1t2∉ERG1t1∈VG1t2∈VSG1−VG1dt1≤2ERG1e1n1−2.(20)2n2n2−1e1n1−2+2e2e1n1−2n2−1+2n2n2−1e1n1−2≤∑C2≤2n2n2−1ERG1e1n1−2+2e2e1n1−2n2−1+2n2n2−1e1n1−2,∑C3=∑t1t2∈ERG1t1∈VG1t2∈VRG1−VG1∑x1,x2∈VG2dt1,x1+dt2,x2=∑t1t2∈ERG1t1∈VG1t2∈VRG1−VG1∑x1,x2∈VG2dRG1t1+dx1+dRG1t2+∑t1t2∈ERG1t1∈VG1t2∈VRG1−VG1dRG1t1+dx1+2=n2n2−1∑t1t2∈ERG1t1∈VG1t2∈VRG1−VG1dRG1t1+2e2n2−1|∑t1t2∈ERG1t1∈VG1t2∈VRG1−VG1dRG1t1|+2n2n2−1|∑t1t2∈ERG1t1∈VG1t2∈VRG1−VG1dRG1t1|.
Note that2e1≤∑t1t2∈ERG1t1∈VG1t2∈VSG1−VG1dt1≤4ERG1e1. So,(21)4e1n2n2−1+2e2n2−1+2n2n2−1≤∑C3≤4e1ERG1n2n2−1+2e2n2−1+2n2n2−1.
Consequently,(22)2n2e1+2e2e1n2−1+2n2e1n2−1+2n2n2−1e1n1−2+2e2e1n1−2n2−1+2n2n2−1e1n1−2+4e1n2n2−1+2e2n2−1+2n2n2−1≤∑C≤2n2ERG1e1n1−2+2e2e1n2−1+2n2e1n2−1+2n2n2−1ERG1e1n1−2+2e2e1n1−2n2−1+2n2n2−1e1n1−2+4e1ERG1n2n2−1+2e2n2−1+2n2n2−1
By putting the values of∑A+∑B+∑C in equation (16), we get the required proof.Theorem 3.
LetG1 and G2 be two connected graphs; then, first Zagreb coindex of G1+QG2 is given as(23)α1≤M¯G1+QG2≤α2,where(24)α1=4e1e¯2+e2e1+e¯1+2n2−1+e¯22e1+e¯1+2e1n1−2e2+3n2+n22n2−1e1+22e2+3n2n2−1e1+n2+2e2+2e¯2M1G1+2e2+e¯2M¯1G1+2e1+e¯1M1G2+n1+2e¯1+2e1M¯1G2,α2=4e1e¯2+e2e1+e¯1+e1n1−22e2+3n2+2e12e2+3n2n2−1+n2+2e2+2e¯2M1G1+n2+2e2+e¯2M¯1G1+2e1+e¯1M1G2+n1+2e¯1+2e1M¯1G2+n2M¯1QG1+4n2−1+e2¯EQG1+2n22EQG1e1n1−2+2n2−1+e2¯M1QG1M¯1QG1+4n2e1EQ=G1n2−1.Proof.
Using equation (2), we have(25)M¯G1+QG2=∑t1,t2x1,x2∉EG1+QG2dt1,x1+dt2,x2,M¯G1+QG2=∑A+∑B+∑C,∑A=∑A1+∑A2+∑A3+∑A4+∑A5+∑A6+∑A7,∑A1=∑t1t2∉EQG1t1,t2∈VQG1−VG1∑x∈VG2dt1,x+dt2,x=n2∑t1t2∉EQG1t1,t2∈VQG1−VG1dQG1t1+dQG1t2.
Note that0≤∑t1t2∉EQG1t1,t2∈VQG1−VG1dQG1t1+dQG1t2≤M¯1QG1, so(26)0≤∑A1≤n2M¯1QG1∑A2=∑t∈VQG1−VG1∑x1x2∈EG2dt,x1+dt,x2=∑t∈VQG1−VG1∑x1x2∈EG2dQG1t+dQG1t=2n2−1∑t∈VQG1−G1dQG1t.
Note thate1≤∑t∈VQG1−VG1dQG1t≤2EQG1, so(27)2n2−1e1≤∑A2≤4n2−1EQG1,∑A3=∑t∈VQG1−VG1∑x1x2∉EG2dt,x1+dt,x2=∑t∈VQG1−VG1∑x1x2∉EG2dQG1t+dQG1u=2e2¯∑t∈VQG1−G1dQG1u.
Note thate1≤∑t∈VQG1−VG1dQG1t≤2EQG1, so (28)2e2¯e1≤∑A3≤4e2¯EQG1,∑A4=∑t1t2∈EQG1t1,t2∈VQG1−VG1∑x1x2∈EG2dt1,x1+dt2,x2=∑t1t2∈EQG1t1,t2∈VQG1−VG1∑x1x2∈EG2dt1,x1+dt2,x2=2n2−1∑t1t2∈EQG1t1,t2∈VQG1−VG1dQG1t1+dQG1t2.
Note thate1≤∑t1t2∈EQG1t1,t2∈VQG1−VG1dQG1t1+dQG1t2≤M1QG1, so(29)2n2−1e1≤∑A4≤2n2−1M1QG1,∑A5=∑t1t2∈EQG1t1,t2∈VQG1−VG1∑x1x2∉EG2dt1,x1+dt2,x2=∑t1t2∈EQG1t1,t2∈VQG1−VG1∑x1x2∉EG2dt1,x1+dt2,x2=2e2¯∑t1t2∈EQG1t1,t2∈VQG1−VG1dQG1t1+dQG1t2.
Note thate1≤∑t1t2∈EQG1t1,t2∈VQG1−VG1dQG1t1+dQG1t2≤M1QG1, so(30)2e2¯e1≤∑A5≤2e2¯M1QG1∑A6=∑t1t2∉EQG1t1,t2∈VQG1−VG1∑x1x2∈EG2dt1,x1+dt2,x2=∑t1t2∉EQG1t1,t2∈VQG1−VG1∑x1x2∈EG2dt1,x1+dt2,x2=2n2−1∑t1t2∉EQG1t1,t2∈VQG1−VG1dQG1t1+dQG1t2.
Note thate¯1≤∑t1t2∉EQG1t1,t2∈VQG1−VG1dQG1t1+dQG1t2≤M¯1QG1, so(31)2n2−1e¯1≤∑A6≤2n2−1M¯1QG1,∑A7=∑t1t2∉EQG1t1,t2∈VQG1−VG1∑x1x2∉EG2dt1,x1+dt2,x2=∑t1t2∉EQG1t1,t2∈VQG1−VG1dt1,x1+dt2,x2=2e2¯∑t1t2∉EQG1t1,t2∈VQG1−VG1dQG1t1+dQG1t2.
Note thate¯1≤∑t1t2∉EQG1t1,t2∈VQG1−VG1dQG1t1+dQG1t2≤M¯1QG1, so(32)2e2¯e¯1≤∑A7≤2e2¯M¯1QG1.
Consequently,(33)2n2−1e1+2e2¯e1+2n2−1e1+2e2¯e1+2n2−1e¯1+2e2¯e¯1≤∑A≤+n2M¯1QG1+4n2−1EQG1+4e2¯EQG1+2n2−1M1QG1+2e2¯M1QG1+2n2−1M¯1QG1+2e2¯M¯1QG1.
Using equation (8), we directly have(34)∑B=4e1e¯2+e2e1+e¯1+2e2+e¯2+n2M1G1+n22e2+e¯2M¯1G1+2e2+e¯1M1G2+n1+2e¯1+e1M¯1G2,∑C=∑C1+∑C2+∑C3,∑C1=∑t1t2∉EQG1t1∈VG1t2∈VQG1−VG1∑x∈VG2dt1,x+dt2,x=∑t1t2∉EQG1t1∈VG1t2∈VQG1−VG1∑x∈VG2dG1t1+dx+dQG1t2=n2∑t1t2∉EQG1t1∈VG1t2∈VQG1−VG1dQG1t1+2e2e1n1−2+3n2e1n1−2.
Note that2e1≤∑t1t2∉EQG1t1∈VG1t2∈VQG1−VG1dQG1t1≤2EQG1e1n1−2, so(35)2n2e1+2e2e1n1−2+3n2e1n1−2≤∑C1≤2n2EQG1e1n1−2+2e2e1n1−2+3n2e1n1−2,∑C2=∑t1t2∉EQG1t1∈VG1t2∈VQG1−VG1∑x1,x2∈VG2dt1,x1+dt2,x2∑C2=∑t1t2∉EQG1t1∈VG1t2∈VQG1−VG1∑x1,x2∈VG2dQG1t1+dx1+dQG1t2=n2n2−1∑t1t2∉EQG1t1∈VG1t2∈VQG1−VG1dQG1t1+2e2+3n2n2−1|∑t1t2∉EQG1dQG1t1|.
Note that2e1≤∑t1t2∉EQG1t1∈VG1t2∈VQG1−VG1dQG1t1≤2EQG1e1n1−2, so(36)2e1n2n2−1+e1n1−22e2+3n2n2−1≤∑C2≤2EQG1e1n1−2n2n2−1+2e2+3n2n2−1∑C3=∑t1t2∈EQG1t1∈VG1t2∈VQG1−VG1∑x1,x2∈VG2dt1,x1+dt2,x2=∑t1t2∈EQG1t1∈VG1t2∈VQG1−VG1∑x1,x2∈VG2dQG1t1+dx1+dQG1t2=n2n2−1∑t1t2∈EQG1t1∈VG1t2∈VQG1−VG1dQG1t1+2e2n2−1|∑t1t2∈EQG1t1∈VG1t2∈VQG1−VG1dQG1t1|+3n2n2−1|∑t1t2∈EQG1t1∈VG1t2∈VQG1−VG1dQG1t1|.
Note that2EG1≤∑t1t2∈EQG1t1∈VG1t2∈VQG1−VG1dQG1t1≤4EQG1e1, so(37)4n2n2−1e1+n2−12e12e22e1+3n2≤∑C3≤4n2n2−1QEG1e1+n2−12e12e2+3n2.
Consequently,(38)2n2e1+2e2e1n1−2+3n2e1n1−2+2e1n2n2−1+e1n1−22e2+3n2n2−1+4n2n2−1e1+n2−12e12e22e1+3n2≤∑C≤2n2EQG1e1n1−2+2e2e1n1−2+3n2e1n1−2+2EQG1e1n1−2n2n2−1+2e2+3n2n2−1+4n2n2−1QEG1e1+n2−12e12e2+3n2.
By putting the values of∑A+∑B+∑C in equation (25), we get the required proof.Theorem 4.
LetG1 and G2 be two connected graphs; then, first Zagreb coindex of G1+TG2 is given as(39)α1≤M¯G1+TG2≤α2,where(40)α1=42e1e¯2+e2e¯1+e1n1−22e2+3n2+2n2+2e2+e¯2M1G1+4e2+e¯2M¯1G1+2e1+e¯1M1G2+n1+2e¯1+e1M¯1G2+2n2−1+e2¯e1+n2−1+e2¯e1e¯1+2n22e1+2e1n2−1n2+2e2+3n2,α2=42e1e¯2+e2e¯1+e1n1−22e2+3n2+4e2+e¯2M1G1+2n2+4e2+e¯2M¯1G1+2e1+e¯1M1G2+n1+2e¯1+e1M¯1G2+n2M¯1TG1+4n2−1+m2¯ETG1+n2−1+m2¯M1TG1M¯1TG1+n22ETG1e1n2−1+2e12e2+3n2n2−1+4n2e1ETG1n2−1+2e1n2−12e2+3n2.Proof.
Using equation (2), we have(41)M¯G1+TG2=∑u1,u2v1,v2∉EG1+TG2du1,v1+du2,v2,M¯G1+TG2=∑t1,t2∈VTG1−VG1∑x1,x2∈VG2dt1,x1+dt2,x2+∑t1,t2∈VG1∑x1,x2∈VG2dt1,x1+dt2,x2+∑t1,t2∈VTG1−VG1∑x1,x2∈VG2dt1,x1+dt2,x2=∑A+∑B+∑C.
Using equations (30) and (17), we directly have(42)2n2−1e1+2e2¯e1+2n2−1e1+2e2¯e1+2n2−1e¯1+2e2¯e¯1≤∑A≤+n2M¯1QG1+4n2−1EQG1+4e2¯EQG1+2n2−1M1QG1+2e2¯M1QG1+2n2−1M¯1QG1+2e2¯M¯1QG1∑B=42e1e¯2+4e¯+e2e¯1+4e2+4e¯2M1G1+2e1+e¯1M1G2+2n2+2e2+e¯2M¯1G1+n1+2e¯1+2e1M¯1G2.
Now, we find(43)∑C=∑C1+∑C2+∑C3∑C1=∑t1t2∉ETG1t1∈VG1t2∈VTG1−VG1∑v∈VG2dt1,v+dt2,v2n2e¯1+2e2e1n1−23n2e1n1−2≤∑C1≤2n2TEG1e1n1−2+2e2e1n1−2+3n2e1n1−2∑C2=∑t1t2∉ETG1t1∈VG1t2∈VTG1−VG1∑x1,x2∈VG2dt1,x1+dt2,x22n2n2−1e1+22e2+3n2n2−1e1n1−2≤∑C2≤2n2n2−1ETG1e1n1−2a+22e2+3n2n2−1TEG1e1n1−2,∑C3=∑t1t2∈ETG1t1∈VG1t2∈VTG1−VG1∑x1,x2∈VG2dt1,x1+dt2,x24n2e1n2−1+4e1e2n2−1+4e13n2n2−1TEG1≤∑C3≤8n2e1n2−1TEG1+8e1e2n2−1TEG1+4e13n2n2−1TEG1.
Consequently,(44)2n2e¯1+2e2e1n1−2+3n2e1n1−2+2n2n2−1e1+22e2+3n2n2−1e1n1−2+4n2e1n2−1+4e1e2n2−1+4e13n2n2−1TEG1≤∑C≤2n2TEG1e1n1−2+2e2e1n1−2+3n2e1n1−2+2n2n2−1ETG1e1n1−2+22e2+3n2n2−1TEG1e1n1−2+8n2e1n2−1TEG1+8e1e2n2−1TEG1+4e13n2n2−1TEG1.
By putting the values of∑A+∑B+∑C in equation (39), we get the required proof.


## 4. Conclusion

Several topological indices (TIs) have been defined with the help of the graph-theoretic parameters of degree, distance, eccentricity, and eigenvalues, and various results are obtained by different researchers for a large number of molecular structures, graphs, or networks. This popularity is not only due to being the rich mathematical problems of the theory of graphs and networking but also due to the wide range of its applications in different disciplines of sciences, especially in chemistry and computer science such as navigation, combinatorial optimization, pattern recognition, image processing, integer programming, formation of chemical compounds, and discovery of drugs.In this paper, we have computed bounds for first Zagreb coindex ofF−sum graphs such as M¯G1+SG2, M¯G1+RG2, M¯G1+QG2, and M¯G1+TG2 in their general forms. The F-sum graphs obtained by different subdivision operations and Cartesian product of graphs present several families of molecular structures, especially graphs consisting of hexagonal chains.Table1 presents the first Zagreb coindex of the Cartesian product of certain path graphs, and Tables 2–5 show the lower and upper bounds of the first Zagreb coindex for the S-sum, R-sum, Q-sum, and T-sum graphs. Now, we compute the linear regression formula to compute correlation between exact values of M¯1Pn+Pm and bounded values of F-sum graphs M¯1PnF+Pm. In general, the linear regression formula is expressed as Y=b0+b1X, where b0 is the y-intercept (point of intersection of line and Y-axis) and b1 is slope that describes the line’s direction and inclination.Table 1
Exact value of first Zagreb coindex for Cartesian product of path graphs.
Cartesian product of pathsExact value ofM1¯P2⊗P236P3⊗P3374P4⊗P41748P5⊗P54853Table 2
Bounds of first Zagreb coindex forS-sum graph.
S-sum graphsLower bound ofM1¯Upper bound ofM1¯P2+SP23640P3+SP3374534P4+SP417483284P5+SP5485310176Table 3
Bounds of first Zagreb coindex forR-sum graph.
R-sum graphsLower bound ofM1¯Upper bound ofM1¯P2+RP23644P3+RP35101430P4+RP422087584P5+RP5731427442Table 4
Bounds of first Zagreb coindex forQ-sum graph.
Q-sum operationLower bound ofM1¯Upper bound ofM1¯P2+QP23692P3+QP34702500P4+QP417566708P5+QP5916622409Table 5
Bounds of first Zagreb coindex forT-sum graph.
T-sum operationLower bound ofM1¯Upper bound ofM1¯P2+TP238118P3+TP35564370P4+TP4200410488P5+TP5984212807The linear regression formula of the Cartesian product of pathsY in terms of lower and upper bounds of the S-sum graph X is Y=0+1X (100 percent correlation between x and y exists with 1 being the value of correlation coefficient) and Y=103.9108+0.47X (a very strong direct relationship between X and Y exists with 0.9993 being the value of correlation coefficient).The linear regression formula of the Cartesian product of pathsY in terms of lower and upper bounds of the R-sum graph X is Y=96.9504+0.6578X (a very strong direct relationship between X and Y exists with 0.9982 being the value of correlation coefficient) and Y=173.5697+0.1731X (a very strong direct relationship between X and Y exists with 0.9966 being the value of correlation coefficient).The linear regression formula of the Cartesian product of pathsY in terms of lower and upper bounds of the Q-sum graph X is Y=305.2811+0.5066X (a very strong direct relationship between X and Y exists with 0.9851 being the value of correlation coefficient) and Y=24.8959+0.218X (a very strong direct relationship between X and Y exists with 0.9962 being the value of correlation coefficient).The linear regression formula of the Cartesian product of pathsY in terms of lower and upper bounds of the T-sum graph X is Y=276.3798+0.4747X (a very strong direct relationship between X and Y exists with 0.9870 being the value of correlation coefficient) and Y=−565.4512+0.3338X (a very strong direct relationship between X and Y exists with 0.8784 being the value of correlation coefficient).Now, we close our discussion with the comment that the molecular structures which are isomorphic to sum graphs (S-sum, R-sum, Q-sum, and T-sum) have strong correlation for the thirteen physicochemical properties such as boiling point, density, heat capacity at constant pressure, entropy, heat capacity at constant time, enthalpy of vaporization, acentric factor, standard enthalpy of vaporization, enthalpy of formation, octanol-water partition coefficient, standard enthalpy of formation, total surface area, and molar volume.



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