# Computing Bounds for Second Zagreb Coindex of Sum Graphs

Authors: Muhammad Javaid; Muhammad Ibraheem; Uzma Ahmad; Q. Zhu
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

Topological indices or coindices are one of the graph-theoretic tools which are widely used to study the different structural and chemical properties of the under study networks or graphs in the subject of computer science and chemistry, respectively. For these investigations, the operations of graphs always played an important role for the study of the complex networks under the various topological indices or coindices. In this paper, we determine bounds for the second Zagreb coindex of a well-known family of graphs calledF-sum (S-sum, R-sum, Q-sum, and T-sum) graphs in the form of Zagreb indices and coindices of their factor graphs, where these graphs are obtained by using four subdivision-related operations and Cartesian product of graphs. At the end, we illustrate the obtained results by providing the exact and bonded values of some specific F-sum graphs.


---

## Body

## 1. Introduction

A topological index (TI) is a function from the set of graphs to the set of real numbers that assigns the different numerical values to the different graphs unless the graphs are isomorphic. Moreover, TIs are essential tools to discuss various physical and chemical properties of the graphs such as volume, density, connectivity, boiling point, freezing point, and heat of formation and evaporation [1, 2]. TIs are also used to study the quantitative structure property relationships QSPRs, quantitative structure activity relationships QSARs, and clinical practices of various medications in the subject of cheminformatics and pharmaceutical industries, respectively (see [3–5]). Mainly TIs have three types such as degree, distance, and polynomial based but the degree-based TIs are more studied than others (see the most recent review [6]).Firstly, an American Chemist Harry Wiener (1947) used a distance-based TI to calculate the boiling point of paraffin (see [7]). First and second Zagreb indices are introduced by Gutman and Trinajsti in 1972; these indices are used to calculate total π-electron energy of alternant hydrocarbons [8]. Kinkar and Gutman calculated different relations between the second Zagreb index of a graph and its complement (see [9]). Yan et al. computed sharp bounds for the second Zagreb index of different unicyclic graphs [10]. Carlos et al. calculated the second Zagreb index of the graphs with minimum and maximum vertex degrees. They also investigated trees with the maximum value of the second Zagreb index among all trees with maximum vertex degree [11].Recently, Zagreb coindices are introduced by Ashrafi et al., and they studied them for the derived graphs obtained by the operations of joining, union, disjunction, Cartesian product, and corona product (see [12, 13]). Kinkar et al. calculated the first Zagreb index and multiplicative Zagreb coindices of tree (see [14]). Gutman obtained coindices of graphs and their complements (see [15]). Nilanjan et al. calculated F-coindex of some graph operations (see [16]). Javaid et al. calculated the first Zagreb connection index and coindex of some derived graphs [17]. Ramane et al. calculated coindices for the transmission and reciprocal transmission-based graphs (see [18]). Mansour and Song computed a and a,b-analogs of Zagreb indices and coindices of graphs [19]. For further studies of Zagreb indices, see [20].There are various operations on graphs such as union, intersection, complement, product, and subdivision. These operations on graphs are useful to obtain the new graphs from the old ones. Yan et al. listed five new graphsLG, SG, QG, RG, and TG with the help of five operations L, S, Q, R, and T on a graph G, respectively, and studied the behavior of Wiener index of these graphs (see [4]). Eliasi and Taeri computed the Wiener indices of the F-sum graphs obtained by the Cartesian product of FG1 and G2, where F∈S,R,Q,T [21]. Later on, many researchers worked on these F-sum graphs such as Deng et al. [22] computed first and second Zagreb indices, Akhtar and Imran calculated the forgotten index [23], Liu et al. computed first general Zagreb index [24], Ahmad et al. calculated sharp bounds of general sum-connectivity index [11], and Alanazi et al. calculated Gutman indices [25].In this paper, we compute the bounds for the second Zagreb coindex ofF-sum graphs in the form of Zagreb indices and coindices of their factor graphs. At the end, the obtained results are additionally illustrated with the assistance of examples of the exact and bonded values for some specific F-sum graphs. The rest of the paper is settled as follows: Section 2 contains the basic definitions and notions, Section 3 covers the main results, and Section 4 presents conclusion with specific examples related to the derived results.


## 2. Preliminaries

A graph denoted byG=VG,EG is formed by set of vertices VG and edges EG, where edge set is subset of the Cartesian product of set of vertices, i.e., EG⊆VG×VG. In a simple connected graph G=VG,EG, total number of vertices is called its order (denoted by VG) and total number of edges is called its size (presented by EG). The degree of a vertex u∈VG is number of its neighborhood vertices that is denoted by du. The complement of G is denoted by G¯ and defined as VG¯=VG, and any two vertices (say u and v) imply that uv∈G¯ iff uv∉G. Gutman and Trinajsti in 1972 [8] introduced the first and second Zagreb indices (denoted by M1 and M2) as follows:(1)M1G=∑p1p2∈EGdGp1+dGp2,M2G=∑p1p2∈EGdGp1dGp2.The second Zagreb coindexM¯2G is defined in [13] as follows:(2)M¯2G=∑p1p2∉EGdGp1dGp2.It is important to note that the above defined coindex uses degrees ofG but run over EG¯.LetG be a graph, then(i)
SG is a graph obtained by inserting one vertex in every edge of G(ii)
RG is a graph obtained from SG by joining the adjacent vertices of G(iii)
QG is a graph formed from SG by joining the pairs of new vertices which are on the adjacent edges (the edges with one common vertex) of G(iv)
TG is obtained by performing both operations of RG and QG on SG, respectivelyLetG1 and G2 be two simple connected graphs, then their F-sum graphs are denoted by G1+FG2 having vertex set VG1+FG2=VG1∪EG1×VG2 and u1,u2v1,v2∈EG1+FG2 iff(i)
u1=v1∈VG1 and u2∽v2∈G2(ii)
u2=v2∈VG2 and u1∽v1∈FG1, where F∈S,R,Q,TFor details, see Figures1–3.Figure 1
(a)G≅C3; (b) SG≅SC3; (c) QG≅QC3; (d) RG≅RC3; (e) TG≅TC3.
(a)(b)(c)(d)(e)Figure 2
G≅C3; H≅P2; C3+SP2; C3+RP2.
(a)(b)(c)Figure 3
C3+QP2 and C3+TP2.


## 3. Main Results

In this section, main results of the second Zagreb coindex for the F-sum graphs are discussed.Theorem 1.
LetG1 and G2 be two simple connected graphs, then second Zagreb coindex of G1+SG2 is given as follows:(3)α1≤M¯2G1+SG2≤α2,where(4)α1=2n2e12n1−2+n1n2−1+2n22e12−n2e1+4e2e1n1−2+n1n2−1+2e2+e2¯M1G1+2e2M¯1G1+n2+2e2+e2¯M2G1M¯2G1+e1+e1¯M1G2+2e1M¯1G2+2e1+e1¯M2G2+n1+2e1+e1¯M¯2G2+M1G2+M¯1G2M1G1+M¯1G1,α2=4n2e1ESG1n2−1+n2n1−2+2n22e12−n2e1+4e2e1n1−2+n1n2−1+2e2+e2¯M1G1+2e2M¯1G1+n2+2e2+e2¯M2G1M¯2G1+e1+e1¯M1G2+2e1M¯1G2+2e1+e1¯M2G2+n1+2e1+e1¯M¯2G2+M1G2+M¯1G2M1G1+M¯1G1.Proof.
Using equation (2), we have(5)M¯2G1+SG2=∑p1,p2q1,q2∉EG1+SG2dp1,q1dp2,q2,M¯2G1+SG2=∑p1,p2q1,q2∉EG1+SG2dp1,q1dp2,q2=∑A+∑B+∑C,(6)∑A=∑p1,p2∈VSG1−VG1∑q1,q2∈VG2dp1,q1p2,q2=∑p1,p2∈VSG1−VG1∑q1,q2∈VG2dSG1p1dSG1p2=∑p1,p2∈VSG1−VG1∑q1,q2∈VG22×2,∑A=2n22e12−n2e1,(7)∑B=∑B1+∑B2+∑B3+∑B4+∑B5+∑B6,∑B1=∑p∈VG1∑q1q2∉EG2dt,q1dt,q2=∑p∈VG1∑q1q2∉EG2dG1p+dG2q1dG1p+dG2q2=∑p∈VG1∑q1q2∉EG2dG1pdG1p+dG1pdG2q2+dG1pdG2q1+dG2q1dG2q2=M1G1e¯2+2e1M¯1G2+n1M¯2G2,∑B2=∑p1,p2∈VG1∑q∈VG2dp1,qdp2,q=∑q∈VG2∑p1p2∈EG1dp1,qdp2,q+∑q∈VG2∑p1p2∉EG1dp1,qdp2,q=∑q∈VG2∑p1p2∈EG1dG1p1+dG2qdG1p2+dG2q+∑q∈VG2∑p1p2∉EG1dG1p1+dG2qdG1p2+dG2q=∑q∈VG2∑p1p2∈EG1dG1p1dG1t2+dG1p1dG2q+dG1p2dG2q+dG2q2+∑q∈VG2∑p1p2∉EG1dG1p1dG1p2+dG1p1dG2q+dG1p2dG2q+dG2q2=n2M2G1+2e2M1G1+e1M1G2+n2M¯2G1+2e2M¯1G1+e¯1M1G2,∑B3=∑p1p2∈EG1∑q1q2∈EG2dp1,q1dp2,q2=∑p1p2∈EG1∑q1q2∈EG2dG1p1+dG2q1dG1p2+dG2q2=∑p1p2∈EG1∑q1q2∈EG2dG1p1dG1p2+dG1p1dG2q2+dG1p2dG2q1+dG2q1dG2q2=2e2M2G1+M1G1M1G2+2e1M2G2,∑B4=∑p1p2∉EG1∑q1q2∈EG2dp1,q1dp2,q2=∑p1p2∉EG1∑q1q2∈EG2dG1p1+dG2q1dG1p2+dG2q2=∑p1p2∉EG1∑q1q2∈EG2dG1p1dG1p2+dG1p1dG2q2+dG1p2dG2q1+dG2q1dG2q2=2e2M¯2G1+M¯1G1M1G2+2e¯1M2G2,(8)∑B5=∑p1p2∉EG1∑q1q2∉EG2dp1,q1dp2,q2=∑p1p2∉EG1∑q1q2∉EG2dG1p1+dG2q1dG1p2+dG2q2=∑p1p2∉EG1∑q1q2∉EG2dG1p1dG1p2+dG1p1dG2q2+dG1p2dG2q1+dG2q1dG2q2=2e¯2M¯2G1+M¯1G1M¯1G2+2e¯1M¯2G2,∑B6=∑p1p2∈EG1∑q1q2∉EG2dp1,q1dp2,q2=∑p1p2∈EG1∑q1q2∉EG2dG1p1+dG2q1dG1p2+dG2q2=∑p1p2∈EG1∑q1q2∉EG2dG1p1dG1p2+dG1p1dG2q2+dG1p2dG2q1+dG2q1dG2q2=2e¯2M2G1+M1G1M¯1G2+2e1M¯2G2,∑B=2e2+e2¯M1G1+e2M¯1G1+n2+2e2+e2¯M2G1M¯2G1+e1+e1¯M1G2+2e1M¯1G2+e1+e1¯M2G2+n1+2e1+e1¯M¯2G2+M1G2+M¯1G2M1G1+M¯1G1,∑C=∑C1+∑C2+∑C3,∑C1=∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1∑q∈VG2dp1,qdp2,q=∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1∑q∈VG2dG1p1+dG2qdG1p2=∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1∑q∈VG2dG1p1+dG2q2+∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1∑q∈VG22dG1p1+2dG2q=2n2∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1dp1+4e2e1n1−2.
Note that(9)e1≤∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1dp1≤2e1n1−2ESG1,2n2e1+4e2e1n1−2≤∑C1≤4n2e1n1−2ESG1+4e2e1n1−2,∑C2=∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1∑q1,q2∈VG2dp1,q1dp2,q2=∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1∑q1,q2∈VG2dG1p1+dq1dSG1p2=∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1∑q1,q2∈VG2dG1p1+dq12=∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1∑q1,q2∈VG22dG1p1+2dq1=2n2n2−1∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1dG1p1+22e2e1n1−2n2−1.
Note that(10)e1≤∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1dp1≤2e1n1−2ESG1,2n2n2−1e1+4e2e1n1−2n2−1≤∑C2≤4n2n2−1e1n1−2ESG1+4e2e1n1−2n2−1,∑C3=∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1∑q1,q2∈VG2dp1,q1+dp2,q2=∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1∑q1,q2∈VG2dG1p1+dq1dSG1p2=∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1∑q1,q2∈VG2dG1p1+dq12=∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1∑q1,q2∈VG22dG1p1+2dq1=2n2n2−1∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1dp1+22e2n2−12e1.
Note that(11)2e1≤∑p1p2∉ESG1p1∈VG1p2∈VSG1−VG1dp1≤2e1ESG1,4e1n2n2−1+8e1e2n2−1≤∑C3≤4e1n2n2−1ESG1+8e1e2n2−1.
Consequently,(12)2n2e1+4e2e1n1−2+2n2n2−1e1+4e2e1n1−2n2−1+4e1n2n2−1+8e1e2n2−1≤∑C≤4n2e1n1−2ESG1+4e2e1n1−2+4n2n2−1e1n1−2ESG1+4e2e1n1−2n2−1+4e1n2n2−1ESG1+8e1e2n2−1.
We obtained the required result by putting the values of∑A+∑B+∑C in equation (5).Theorem 2.
LetG1 and G2 be two simple connected graphs, then second Zagreb coindex of G1+RG2 is given as follows:(13)α1≤M¯2G1+RG2≤α2,where(14)α1=4n2e13n2−2+2n22e12−n2e1+4e2e1n2n1−2+2n2−1+4e2¯M1G1+4e2M¯1G1+8e2+e2¯M2G1+4n2+2e2+e2¯M¯2G1+e1¯M1G2+4e1M¯1G2+2e1+e1¯M2G2+n1+2e1+e1¯M¯2G2+2M1G2+M¯1G2M1G1+M¯1G1,α2=8n2e1ERG1n1−2+n2−1n1−1+2n22e12−n2e1+4e2e1n2n1−2+2n2−1+4e2¯M1G1+4e2M¯1G1+8e2+e2¯M2G1+4n2+2e2+e2¯M¯2G1+e1¯M1G2+4e1M¯1G2+2e1+e1¯M2G2+n1+2e1+e1¯M¯2G2+2M1G2+M¯1G2M1G1+M¯1G1.Proof.
Using equation (2), we have(15)M¯2G1+RG2=∑p1,p2q1,q2∉EG1+RG2dp1,q1dp2,q2=∑A+∑B+∑C.
Using equation (6), we directly have(16)∑A=2n22e12−n2e1,∑B=∑B1+∑B2+∑B3+∑B4+∑B5+∑B6,∑B1=∑p∈VG1∑q1q2∉EG2dp,q1dp,q2=∑p∈VG1∑q1q2∉EG2dRp+dG2q1dRp+dG2q2=∑p∈VG1∑q1q2∉EG2dRpdRp+dRpdG2q2+dRpdG2q1+dG2q1dG2q2=∑p∈VG1∑q1q2∉EG24dG1p2+2dG1pdG2q2+dG2q1+dG2q1dG2q2=4e¯2M1G1+22e1M¯1G2+n1M¯2G2,∑B2=∑q∈VG2∑p1,p2∈VG1dp1,qdp2,q=∑q∈VG2∑p1p2∉EG1dp1,qdp2,q=∑q∈VG2∑p1p2∉EG1dRp1+dG2qdRp2+dG2q=∑q∈VG2∑p1p2∉EG12dG1p1+dG2q2dG1p2+dG2q=∑q∈VG2∑p1p2∉EG14dG1p1dG1p2+2dG1p1dG2q+2dG1p2dG2q+dG2q2=4n2M¯2G1+22e2M¯1G1+e¯1M1G2,∑B3=∑p1p2∈EG1∑q1q2∈EG2dp1,q1+dp2,q2=∑p1p2∈EG1∑q1q2∈EG2dRpdRp+dRpdG2q2+dRpdG2q1+dG2q1dG2q2=∑p1p2∈EG1∑q1q2∈EG2dRpdRp+dRpdG2q2+dRpdG2q1+dG2q1dG2q2=2∑p1p2∈EG1∑q1q2∈EG24dG1p2+2dG1pdG2q2+dG2q1+dG2q1dG2q2=24e2M2G1+e1M2G2+2M1G1M1G2,∑B4=∑p1p2∉EG1∑q1q2∈EG2dp1,q1dp2,q2=∑p1p2∉EG1∑q1q2∈EG2dRp1+dG2q1dRp2+dG2q2=∑p1p2∉EG1∑q1q2∈EG22dG1p1+dG2q12dG1p2+dG2q2=∑p1p2∉EG1∑q1q2∈EG24dG1p1dG1p2+2dG1p1dG2q2+dG1p2dG2q1+dG2q1dG2q2=24e2M¯2G1+e¯1M2G2+2M¯1G1M1G2,(17)∑B5=∑p1p2∉EG1∑q1q2∉EG2dp1,q1dp2,q2=∑p1p2∉EG1∑q1q2∉EG2dRp1+dG2q1dRp2+dG2q2=∑p1p2∉EG1∑q1q2∉EG22dG1p1+dG2q12dG1p2+dG2q2=∑p1p2∉EG1∑q1q2∉EG24dG1p1dG1p2+2dG1p1dG2q2+dG1p2dG2q1+dG2q1dG2q2=24e¯2M¯2G1+e¯1M¯2G2+2M¯1G1M¯1G2,∑B6=∑p1p2∈EG1∑q1q2∉EG2dp1,q1dp2,q2=∑p1p2∈EG1∑q1q2∉EG2dRp1+dG2q1dRp2+dG2q2=∑p1p2∈EG1∑q1q2∉EG22dG1p1+dG2q12dG1p2+dG2q2=∑p1p2∈EG1∑q1q2∉EG24dG1p1dG1p2+2dG1p1dG2q2+dG1p2dG2q1+dG2q1dG2q2=24e¯2M2G1+e1M¯2G2+2M1G1M¯1G2,∑B=4e2¯M1G1+4e2M¯1G1+8e2+e2¯M2G1+4n2+2e2+e2¯M¯2G1+e1¯M1G2+4e1M¯1G2+2e1+e1¯M2G2+n1++2e1+e1¯M¯2G2+2M1G2+M¯1G2M1G1+M¯1G1,∑C=∑C1+∑C2+∑C3,∑C1=∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1∑q∈VG2dp1,qdp2,q=∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1∑q∈VG2dRp1+dG2qdRp2,∑C1∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1∑q∈VG22dG1p1+dG2q2=∑p1p2∉ERG1p1∈VG1p2∈VSG1−VG1∑q∈VG24dG1p1+2dG2q=4n2∑p1p2∉ERG1p1∈VG1p2∈VSG1−VG1dp1+4e2e1n1−2.
Note that(18)e1≤∑p1p2∉ERG1p1∈VG1p2∈VSG1−VG1dp1≤2e1n1−2ESG1,so(19)4n2e1+4e2e1n1−2≤∑C1≤8n2e1n1−2ERG1+4e2e1n1−2,∑C2=∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1∑q1,q2∈VG2dp1,q1dp2,q2=∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1∑q1,q2∈VG2dRp1+dq1dRG1p2=∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1∑q1,q2∈VG22dG1p1+dq12=∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1∑q1,q2∈VG24dG1p1+2dq1=4n2n2−1∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1dG1p1+22e2e1n1−2n2−1.
Note that(20)e1≤∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1dp1≤2e1n1−2ERG1,so(21)4n2n2−1e1+4e2e1n1−2n2−1≤∑C2≤8n2n2−1e1n1−2ERG1+4e2e1n1−2n2−1,∑C3=∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1∑q1,q2∈VG2dp1,q1dp2,q2=∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1∑q1,q2∈VG2dRp1+dq1dRG1p2=∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1∑q1,q2∈VG22dG1p1+dq12=∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1∑q1,q2∈VG24dG1p1+2dq1=4n2n2−1∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1dp1+22e2n2−12e1.
Note that(22)2e1≤∑p1p2∉ERG1p1∈VG1p2∈VRG1−VG1dp1≤2e1ERG1,so(23)8e1n2n2−1+8e1e2n2−1≤∑C3≤8e1n2n2−1ERG1+8e1e2n2−1.
Consequently,(24)8e1n2n2−1+8e1e2n2−1+4n2n2−1e1+4e2e1n1−2n2−1+4n2e1+4e2e1n1−2≤∑C≤8n2e1n1−2ERG1+4e2e1n1−2+8n2n2−1e1n1−2ERG1+4e2e1n1−2n2−1+8e1n2n2−1ERG1+8e1e2n2−1.
We obtained the required proof by putting the values of∑A+∑B+∑C in equation (14).Theorem 3.
LetG1 and G2 be two simple connected graphs, then second Zagreb coindex of G1+QG2 is given as follows:(25)α1≤M¯2G1+QG2≤α2,where(26)α1=4e2e¯1+n2−1e¯1+e1+n22M¯2G1+n2n2−1M2G1+n2−1+e¯2M1G1+2M2G1+2e2+e2¯M1G1+2e2M¯1G1+n2+2e2+e2¯M2G1M¯2G1+e1+e1¯M1G2+2e1M¯1G2+2e1+e1¯M2G2+n1+2e1+e1¯M¯2G2+M1G2+M¯1G2M1G1+M¯1G1,α2=4e2e¯QG1+n2−1e¯QG1+eQG1+n22M¯2QG1+n2n2−1M2QG1+n2+2n2−1+e¯2M2QG1+n2−1+e¯2M1QG1+2n2−1+e¯2M2QG1+2e2+e2¯M1G1+2e2M¯1G1+n2+2e2+e2¯M2G1M¯2G1+e1+e1¯M1G2+2e1M¯1G2+2e1+e1¯M2G2+n1+2e1+e1¯M¯2G2+M1G2+M¯1G2M1G1+M¯1G1.Proof.
Using equation (2), we have(27)M¯2G1+QG2=∑p1,p2x1,x2∉EG1+QG2dp1,x1dp2,x2=∑A+∑B+∑C,∑A=∑A1+∑A2+∑A3+∑A4+∑A5+∑A6+∑A7,∑A1=∑p1p2∉EQG1p1,p2∈VQG1−G1∑x∈VG2dp1,xdp2,x=n2∑p1p2∉EQG1p1,p2∈VQG1−G1dQG1p1dQG1p2.
Note that(28)0≤∑p1p2∉EQG1p1,p2∈VQG1−G1dQG1p1dQG1p2≤M¯2QG1,so(29)0≤∑A1≤n2M¯2QG1,∑A2=∑p∈VQG1−G1∑x1x2∈EG2dp,x1dp,x2=∑p∈VQG1−G1∑x1x2∈EG2dQG1pdQG1p=n2−1∑p∈VQG1−G1dQG1p2.
Note that(30)M1G1≤∑p∈VQG1−G1dQG1p2≤M1QG1,so(31)n2−1M1G1≤∑A2≤n2−1M1QG1,∑A3=∑p∈VQG1−G1∑x1x2∉EG2dp,x1dp,x2=∑p∈VQG1−VG1∑x1x2∉EG2dQG1pdQG1p=e¯2∑t∈VQG1−VG1dQG1p2.
Note that(32)M1G1≤∑p∈VQG1−VG1dQG1p2≤M1QG1,so(33)e¯2M1G1≤∑A3≤e¯2M1QG1,∑A4=∑p1p2∈EQG1p1,p2∈VQG1−VG1∑x1x2∈EG2dp1,x1dp2,x2=∑p1p2∈EQG1p1,p2∈VQG1−VG1∑x1x2∈EG2dp1,x1dp2,x2=2n2−1∑p1p2∈EQG1p1,p2∈VQG1−VG1dQG1p1dQG1p2.
Note that(34)M2G1≤∑p1p2∈EQG1p1,p2∈VQG1−VG1dQG1p1+dQG1p2≤M2QG1,so(35)2n2−1M2G1≤∑A4≤2n2−1M2QG1,∑A5=∑p1p2∈EQG1p1,p2∈VQG1−VG1∑x1x2∈EG2dp1,x1dp2,x2=∑p1p2∈EQG1p1,p2∈VQG1−VG1∑x1x2∈EG2dp1,x1dp2,x2=2e¯2∑p1p2∈EQG1p1,p2∈VQG1−VG1dQG1p1dQG1p2.
Note that(36)M2G1≤∑p1p2∈EQG1p1,p2∈VQG1−VG1dQG1p1+dQG1p2≤M2QG1,so(37)2e¯2M2G1≤∑A5≤2e¯2M2QG1,∑A6=∑p1p2∈EQG1p1,p2∈VQG1−VG1∑x1x2∈EG2dp1,x1dp2,x2=∑p1p2∈EQG1p1,p2∈VQG1−VG1∑x1x2∈EG2dp1,x1dp2,x2=2n2−1∑p1p2∈EQG1p1,p2∈VQG1−VG1dQG1p1dQG1p2.
Note that(38)0≤∑p1p2∈EQG1p1,p2∈VQG1−VG1dQG1p1dQG1p2≤M¯2QG1,so(39)0≤∑A6≤2n2−1M¯2QG1,∑A7=∑p1p2∉EQG1p1,p2∉VQG1−VG1∑x1x2∈EG2dp1,x1dp2,x2=∑p1p2∉EQG1p1,p2∉VQG1−VG1∑x1x2∈EG2dp1,x1dp2,x2=2e¯2∑p1p2∉EQG1p1,p2∈VQG1−VG1dQG1p1dQG1p2.
Note that(40)0≤∑p1p2∉EQG1p1,p2∈VQG1−VG1dQG1p1dQG1p2≤M¯2QG1,so(41)0≤∑A7≤2e¯2M¯2QG1.
Consequently,(42)2e¯2M2G1+2n2−1M2G1+n2−1M1G1+e¯2M1G1≤∑A≤n2M¯2QG1+n2−1M1QG1+e¯2M1QG1+2n2−1M2QG1+2e¯2M2QG1+2n2−1M¯2QG1+2e¯2M¯2QG1.
Using equation (7), we directly have(43)∑B=2e2+e2¯M1G1+e2M¯1G1+n2+2e2+e2¯M2G1M¯2G1+e1+e1¯M1G2+2e1M¯1G2+e1+e1¯M2G2+n1+2e1+e1¯M¯2G2+M1G2+M¯1G2M1G1+M¯1G1,∑C=∑C1+∑C2+∑C3,∑C1=∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1∑x∈VG2dp1,xdp2,x=∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1∑x∈VG2dG1p1+dxdQG1p2=n2∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1dG1p1dQG1p2+2e2dQG1p2=n2∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1dG1p1dQG1p2+dx∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1dQG1p2.
Note that(44)M¯2G1≤∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1dQG1p1≤M¯2QG1,2e¯1≤∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1dQG1p2≤2e¯QG1,n2M¯2G1+4e2e¯1≤∑C1≤n2M¯2QG1+4e2e¯QG1,∑C2=∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1∑x1,x2∈VG2dp1,x1dp2,x2=∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1∑x1,x2∈VG2dG1p1+dx1dQG1p2=∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1∑x1,x2∈VG2dG1p1dQG1p2+dx1dQG1p2=∑x1,x2∈VG2∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1dG1p1dQG1p2+dx1∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1dQG1p2.
Note that(45)M¯2G1≤∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1dQG1p1≤M¯2QG1,2e¯1≤∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1∑dQG1p2≤2e¯QG1,n2n2−1M¯2G1+4e2n2−1e¯1≤∑C2≤n2n2−1M¯2QG1+4e2n2−1e¯QG1,∑C3=∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1∑x1,x2∈VG2dp1,x1dp2,x2=∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1∑x1,x2∈VG2dG1p1+dx1dQG1p2=∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1∑x1,x2∈VG2dG1p1dQG1p2+dx1dQG1p2=∑x1,x2∈VG2∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1dG1p1dQG1p2+dx1∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1dQG1p2.
Note that(46)M2G1≤∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1dQG1p1≤M2QG1,2e1≤∑p1p2∉EQG1p1∈VG1p2∈VQG1−VG1dQG1p2≤2eQG1,n2n2−1M2G1+4e2n2−1e1≤∑C3≤n2n2−1M2QG1+4e2n2−1eQG1.
Consequently,(47)n2M¯2G1+4e2e¯1+n2n2−1M¯2G1+4e2n2−1e¯1+n2n2−1M2G1+4e2n2−1e1≤∑C≤n2M¯2QG1+4e2e¯QG1+n2n2−1M¯2QG1+4e2n2−1e¯QG1+n2n2−1M2QG1+4e2n2−1eQG1.
We obtained the required proof by putting the values of∑A+∑B+∑C in equation (25).Theorem 4.
LetG1 and G2 be two graphs, then second Zagreb coindex of G1+TG2 is given as follows:(48)α1≤M¯2G1+TG2≤α2,where(49)α1=4e2e¯1+n2−1e¯1+e1+2n2M¯2G1+n2−1M¯2G1+M2G1+n2−1+e¯2M1G1+2M2G1+4e2¯M1G1+4e2M¯1G1+8e2+e2¯M2G1+4n2+2e2+e2¯M¯2G1+e1¯M1G2+4e1M¯1G2+2e1+e1¯M2G2+n1+2e1+e1¯M¯2G2+2M1G2+M¯1G2M1G1+M¯1G1,α2=4e2e¯TG1+n2−1e¯TG1+eTG1+2n2M¯2TG1+n2−1M2TG1+M¯2TG1,α2=4e2e¯TG1+n2−1e¯TG1+eTG1+2n2M¯2TG1+n2−1M2TG1+M¯2TG1+8e2+e2¯M2G1+4n2+2e2+e2¯M¯2G1+e1¯M1G2+4e1M¯1G2+2e1+e1¯M2G2+n1+2e1+e1¯M¯2G2+2M1G2+M¯1G2M1G1+M¯1G1.Proof.
Using equation (2), we have(50)M¯2G1+TG2=∑p1,p2q1,q2∉EG1+TG2dp1,q1dp2,q2=∑A+∑B+∑C.
Using equation (40), we directly have(51)2e¯2M2G1+2n2−1M2G1+n2−1M1G1+e¯2M1G1≤∑A≤n2M¯2TG1+n2−1M1TG1+e¯2M1TG1+2n2−1M2TG1+2e¯2M2TG1+2n2−1M¯2TG1+2e¯2M¯2TG1.
Using equation (15), we directly have(52)∑B=4e2¯M1G1+4e2M¯1G1+8e2+e2¯M2G1+4n2+2e2+e2¯M¯2G1+e1¯M1G2+4e1M¯1G2+2e1+e1¯M2G2+n1+2e1+e1¯M¯2G2+2M1G2+M¯1G2M1G1+M¯1G1,∑C=∑C1+∑C2+∑C3,∑C1=∑t1t2∉ETG1p1∈VG1p2∈VTG1−G1∑q∈VG2dp1,qdp2,q2n2M¯2G1+4e2e¯1≤∑C1≤2n2M¯2TG1+4e2e¯TG1,∑C2=∑p1p2∉ETG1p1∈VG1p2∈VTG1−VG1∑q1,q2∈VG2dp1,q1dp2,q22n2n2−1M¯2G1+4e2n2−1e¯1≤∑C2≤2n2n2−1M¯2TG1+4e2n2−1e¯TG1,∑C3=∑p1p2∈ETG1p1∈VG1p2∈VTG1−VG1∑q1,q2∈VG2dp1,q1dp2,q22n2n2−1M2G1+4e2n2−1e1≤∑C3≤2n2n2−1M2TG1+4e2n2−1eTG1.
Consequently,(53)2n2M¯2G1+4e2e¯1+2n2n2−1M¯2G1+4e2n2−1e¯1+2n2n2−1M2G1+4e2n2−1e1≤∑C≤2n2M¯2TG1+4e2e¯TG1+2n2n2−1M¯2TG1+4e2n2−1e¯TG1+2n2n2−1M2TG1+4e2n2−1eTG1.
We obtained required results by putting the values of∑A+∑B+∑C in equation (48).


## 4. Conclusion

In this paper, we have computed second Zagreb coindex ofF-sum graphs such as M¯2G1+SG2, M¯2G1+RG2, M¯2G1+QG2, and M¯2G1+TG2. The obtained results are illustrated with the help of specific class graphs of F-sum graphs. Let G1≅P3 and G2≅P2, then the lower and upper bounds of first Zagreb coindex for their F-sum graph are given in Table 1.Table 1
Exact and bounded values of certainF-sum graphs.
F-sum operationLower boundsExact valuesUpper boundsM¯2G1+SG2152160312M¯2G1+RG2216232728M¯2G1+QG2106220338M¯2G1+TG2150300642Now, we close our discussion that the problem is still open to compute the other generalized coindices (first general Zagreb and general Randic coindices) for theF-sum graphs.



---

*Source: 4671105-2021-07-16.xml*