# Bounds of Degree-Based Molecular Descriptors for GeneralizedF-sum Graphs

Authors: Jia Bao Liu; Sana Akram; Muhammad Javaid; Abdul Raheem; Roslan Hasni
Journal: Discrete Dynamics in Nature and Society
 (2021)

---

## Abstract

A molecular descriptor is a mathematical measure that associates a molecular graph with some real numbers and predicts the various biological, chemical, and structural properties of the underlying molecular graph. Wiener (1947) and Trinjastic and Gutman (1972) used molecular descriptors to find the boiling point of paraffin and totalπ-electron energy of the molecules, respectively. For molecular graphs, the general sum-connectivity and general Randić are well-studied fundamental topological indices (TIs) which are considered as degree-based molecular descriptors. In this paper, we obtain the bounds of the aforesaid TIs for the generalized F-sum graphs. The foresaid TIs are also obtained for some particular classes of the generalized F-sum graphs as the consequences of the obtained results. At the end, 3D-graphical presentations are also included to illustrate the results for better understanding.


---

## Body

## 1. Introduction

A molecular descriptor called by the topological index (TI) is a function from the set of (molecular) graphs to the set of real numbers. TIs are studied as a subtopic of chemical graph theory to predict the chemical reactions, biological attributes, and physical features of the compounds in theoretical chemistry, toxicology, pharmaceutical industry, and environmental chemistry, see [1]. In addition, these TIs are also used to characterize the molecular structure with respect to quantitative structure activity and property relationships which are studied in the subject of cheminformatics, see [2]. TIs have been classified into different classes but degree-based TIs play a significant part in the theory of chemical structures or networks. Firstly, Wiener [3] used the TI to find boiling point of paraffin. Gutman and Trinajstic calculated total π-electron energy of the molecules by a TI that is recognized as the first Zagreb index in the literature, see [4]. In 2009, Zhou and Trinajstic [5] suggested the sum-connectivity index, which was subsequently generalized in 2010 [6]. The Randić index was defined in 1975 [7]; later on the idea was subsequently extended to the generalized Randić connectivity index by Li and Gutman, see [8]. For more studies, we refer to [9–12].In chemical graph theory, operations of graphs are frequently used to find the new families of graphs. Yan et al. [13] defined the four subdivision-related operations S11,S12,S13,andS14 on a molecular graph M and obtained the Wiener indices of the resultant graphs S11M,S12M,S13M, and S14M. After that, Eliasi and Taeri [14] defined the F-sum graph M1+F1M2 with the help of Cartesian product of M1 and FM2, where F1ϵS11,S12,S13,S14. Deng et al. [15] and Akhter and Imran [16] also computed the 1st and 2nd Zagreb and general sum-connectivity indices of the F-sum graphs, respectively. Recently, Liu et al. [17] generalized these subdivision-related operations and defined the generalized F-sum graphs M1+FkM2 for FkεSk1,Sk2,Sk3,Sk4, where k≥1 is an integral value. They also computed the 1st and 2nd Zagreb indices for these newly obtained graphs. For further studies of F-sum and generalized F-sum graphs, see [18–27].Now, we extend this study by computing the bounds (upper and lower) of the general sum-connectivity and general Randić indices for the generalizedF-sum graphs. In the remaining paper, Section 2 consists of main results of bounds and Section 3 has conclusion and applications of the obtained results.


## 2. Preliminaries

Throughout, we consider undirected, connected, and simple graphs withVM=ai:1≤i≤n and EM⊆VM×VM as vertex and edge sets, respectively. In addition, VM=n and EM=m are the order and size of M. For a vertex aεVM and dMa as a degree of a in M, ΔM=maxdMa:aεVM and δMa=mindMa:aεVM are the maximum and minimum degrees of the graph M. The graphs VPn=ai:1≤i≤n&EPn=aiai+1:1≤i≤n,VCn=ai:1≤i≤n&ECn=aiai+1:1≤i≤n−1∪ana1 and VKn=ai:1≤i≤n and EKn=aiaj:1≤i,j≤n are called path Pn, cycle Cn, and complete Kn, respectively. For further basic terminologies, see [28].Definition 1.
For a real numberα and (molecular) graph M, the general sum-connectivity index (GSCI) and general Randić index (GRI) are(1)χαM=∑x1,x2εEMdMx1+dMx2α,RαM=∑x1,x2εEMdMx1dMx2α.
Moreover,α=1,2 and then χαM are known as the 1st Zagreb index and hyper-Zagreb index. If α=−1/2,1,1/2, then RαM; it is known as Randić, 2nd Zagreb, and reciprocal Randić indices.
Liu [17] defined the following graphs using the generalized subdivision-related operations:(i)
Sk1M graph is obtained by inserting k-vertices in each edge of M.(ii)
Sk2M is obtained from Sk1M by joining the old vertices which are adjacent in M.(iii)
Sk3M is obtained from Sk1M by joining the new vertices lying in an edge to the corresponding new vertices of the other edge, if these edges have some common vertexes in M.(iv)
Sk4M is the union of Sk2M and Sk3M graphs.
For more details, see Figure1 for k=4.Figure 1
(a)M≅P4, (b) S4M, (c) R4M, (d) Q4M, and (e) T4M.
(a)(b)(c)(d)(e)Definition 2.
LetM1 and M2 be two connected graphs, FkεSk1,Sk2,Sk3,Sk4 and FkM1 be a graph (obtained after applying the operation Fk on M1 with vertex set VFkM1 and edge set EFkM1. Then, the generalized F-sum graph M1+FkM2 is a graph with vertex set VM1+FkM2=VFkM1×VM2=VM1∪EM1×VM2 in such a way that x1,y1,x2,y2εVM1+FkM2 are adjacent if [x1=x2∈VM1 and y1,y2∈EM2] or [y1=y2∈VM2 and x1,x2∈EFkM1], where k≥1 is an integral value.
For more details, see Figures2 and 3.Figure 2
(a)M1≅P4, (b) M2≅P4, (c) M1+S2M2, and (d) M1+R2M2.
(a)(b)(c)(d)Figure 3
(a)M1+Q2M2. (b) M1+T2M2.
(a)(b)


## 3. Main Results

In this section, we find out the sharp bounds of GSCI and GRI of generalized F-sum graphs.Theorem 1.
For a real numberα>0 and counting number k≥1, the lower and upper bounds on the GSCI and GRI of generalized F-sum graph M1+Sk1M2 are as follows:(a)
2αn1m2ΔM1+ΔM2α+2n2m1ΔM1+ΔM2+2α+4αn2m1k−1≤χαM1+Sk1M2≤2αn1m2δM1+δM2α+2n2m1δM1+δM2+2α+4αn2m1k−1,(b)
n1m2ΔM1+ΔM22α+2α+1n2m1ΔM1+ΔM2α+4αn2m1k−1≤RαM1+Sk1M2≤n1m2δM1+δM22α+2α+1n2m1δM1+δM2α+4αn2m1k−1,where equalities hold if M1 and M2 are regular graphs.Proof.
(a) By the definition of GSCI, we have(2)χαM1+Sk1M2=∑x1,y1x2,y2εEM1+Sk1M2dM1+Sk1M2x1,y1+dM1+Sk1M2x2,y2α,=∑xεVM1∑y1y2εEM2dM1+Sk1M2x,y1+dM1+Sk1M2x,y2α+∑yεVM2∑x1x2εESk1M1dM1+Sk1M2x1,y+dM1+Sk1M2x2,yα=∑xεVM1∑y1y2εEM2dM1+Sk1M2x,y1+dM1+Sk1M2x,y2α+∑yεVM2∑x1x2εESk1M1x1εVM1,x2εVSk1M1−M1dM1+Sk1M2x1,y+dM1+Sk1M2x2,yα+∑yεVM2∑x1x2εESk1M1x1,x2εVSk1M1−M1dM1+Sk1M2x1,y+dM1+Sk1M2x2,yα=∑1+∑2+∑3.
Consider(3)∑1=∑xϵVM1∑y1y2ϵEM2dM1+Sk1M2x,y1+dM1+Sk1M2x,y2α=∑xϵVM1∑y1y2ϵEM22dM1x+dM2y1+dM2y2α≤2αn1m2ΔM1+ΔM2α.
SinceES11M1=2EM1 and ΔSk1M1=ΔM1, we have(4)∑2=∑yεVM2∑x1x2εESk1M1x1εVM1,x2εVSk1M1−M1dM1+Sk1M2x1,y+dG+Sk1M2x2,yα,=∑yεVM2∑x1x2εESk1M1x1εVM1,x2εVSk1M1−VM1dM1x1+dM2y+dSk1M1x2α≤2n2m1ΔM1+ΔM2+2α∑3=∑yεVM2∑x1x2εESk1M1x1,x2εVSk1M1−M1dM1+Sk1M2x1,y+dM1+Sk1M2x2,yα=∑yεVM2∑x1x2εESk1M1x1,x2εVSk1M1−M12+2α.
Since in this caseESk1M1−ES11=k−1EM1, we have(5)=k−1EM1∑yεVM24α=4αk−1VM2EM1=4αk−1n2m1.
Consequently,(6)2αn1m2ΔM1+ΔM2α+2n2m1ΔM1+ΔM2+2α+4αn2m1k−1≤χαM1+Sk1M2≤2αn1m2δM1+δM2α+2n2m1δM1+δM2+2α+4αn2m1k−1,where equalities hold if M1 and M2 are regular graphs.Proof.
(b) By the definition of GRI, we have(7)RαM1+Sk1M2=∑x1,y1x2,y2εEM1+Sk1M2dM1+Sk1M2x1,y1+dM1+Sk1M2x2,y2α=∑xεVM1∑y1y2εEM2dM1+Sk1M2x,y1+dM1+Sk1M2x,y2α+∑yεVM2∑x1x2εESk1M1dM1+Sk1M2x1,y+dM1+Sk1M2x2,yα=∑xεVM1∑y1y2εEM2dM1+Sk1M2x,y1+dM1+Sk1M2x,y2α+∑yεVM2∑x1x2εESk1M1x1εVM1,x2εVSk1M1−M1dM1+Sk1M2x1,y+dM1+Sk1M2x2,yα+∑yεVM2∑x1x2εESk1M1x1,x2εVSk1M1−M1dM1+Sk1M2x1,y+dM1+Sk1M2x2,yα=∑1+∑2+∑3.
Consider(8)∑1=∑xεVM1∑y1y2εEM2dM1+Sk1M2x,y1+dM1+Sk1M2x,y2α=∑xεVM1∑y1y2εEM2dM1x+dM2y1dM1x+dM2y2α=∑xεVM1∑y1y2εEM2dM12x+dM1x1dM2y1+dM2y2+dM2y1dM2y2α≤n1m2ΔM1+ΔM22α.
SinceES11M1=2EM1 and ΔSk1M1=ΔM1, we have(9)∑2=∑yεVM2∑x1x2εESk1M1x1εVM1,x2εVSk1M1−M1dM1+Sk1M2x1,y+dG+Sk1M2x2,yα,=∑yεVM2∑x1x2εESk1M1x1εVM1,x2εVSk1M1−M1dM1x1+dM2ydSk1M1x2α=∑yεVM2∑x1x2εESk1M1x1εVM1,x2εVSk1M1−VM1ΔM1+ΔM22α≤2α+1n2m1ΔM1+ΔM2α∑3=∑yεVM2∑x1x2εESk1M1x1,x2εVSk1M1−M1dM1+Sk1M2x1,y+dM1+Sk1M2x2,yα=∑yεVM2∑x1x2εESk1M1x1,x2εVSk1M1−M12×2α.
Since in this caseESk1M1−ES11=k−1EM1, we have =k−1EM1∑yεVM24α=4αk−1n2m1.
Consequently,(10)n1m2ΔM1+ΔM22α+2α+1n2m1ΔM1+ΔM2α+4αn2m1k−1≤RαM1+Sk1M2≤n1m2δM1+δM22α+2α+1n2m1δM1+δM2α+4αn2m1k−1,where equalities hold if M1 and M2 are regular graphs.Theorem 2.
For a real numberα>0 and counting number k≥1, the lower and upper bounds on the GSCI and GRI of the generalized F-sum graph M1+Sk2M2 are as follows:(a)
2αn1m2+n2m12ΔM1+ΔM2α+2n2m12ΔM1+ΔM2+2α+4αn2m1k−1≤χαM1+Sk1M2≤2αn1m2+n2m12δM1+δM2α+2n2m12δM1+δM2+2α+4αn2m1k−1,(b)
n1m2+n2m12ΔM1+ΔM22α+2α+1n2m12ΔM1+ΔM2α+4αn2m1k−1≤RαM1+Sk2M2≤n1m2+n2m12δM1+δM22α+2α+1n2m12δM1+δM2α+4αn2m1k−1,where equalities hold if and only if M1 and M2 are regular graphs.Proof.
(a) By the definition of the GSCI, we have(11)χαM1+Sk2M2=∑x1,y1x2,y2εEM1+Sk2M2dM1+Sk2M2x1,y1+dM1+Sk2M2x2,y2α=∑xεVM1∑y1y2εEM2dM1+Sk1M2x,y1+dM1+Sk1M2x,y2α+∑yεVM2∑x1x2εESkM1dM1+Sk2M2x1,y+dM1+Sk2M2x2,yα=∑1+∑2.
Consider(12)∑1=∑xεVM1∑y1y2εEM2dM1+Sk1M2x,y1+dM1+Sk1M2x,y2α=∑xεVM1∑y1y2εEM24dM1x+dM2y1+dM2y2α≤2αn1m22ΔM1+ΔM2α∑2=∑xεVM1∑x1x2εVM1dx1,y+dx2,yα=∑yεVM2∑x1x2εESk2M1x1,x2εVM1dx1,y+dx2,yα+∑yεVM2∑x1x2εESk2M1x1εVM1x2εVSk2M1−VM1dx1,y+dx2,yα+∑yεVM2∑x1x2εESk2M1x1,x2εVSk2M1−VM1dx1,y+dx2,yα=∑′2+∑″2+∑″′2.
Consider forx1x2εVM1, we have x1x2εESk2M1 if x1x2εEM1; for x1εVM1, we obtain dS1M12x1=2dM1x1 and for x2εVSkM12−VM1, we have dSkM12x2=2.
Now, consider(13)∑′2=∑yεVM2∑x1x2εEM1x1x2εVM1dx1,y+dx2,yα=∑yεVM2∑x1x2εEM1dx1,y+dx2,yα=∑yεVM2∑x1x2εEM12dM2y+2dM1x1+2dM1x2α≤2αn2m12ΔM1+ΔM2α.
Note thatES11M1=2EM1 and if x1εVM1, then dSk2M1x1=2dM1x1, and if x2εVSk2M1−VM1, then dSk2M1x2=2.(14)∑″2=∑yεVM2∑x1x2εESk2M1x1εVM1x2εVSk2M1−VM1dx1,y+dx2,yα=∑yεVM2∑x1x2εEM1x1εVM1x2εVM1−VM12dM1x1+dSk2M1x2+dM2yα=∑yεVM2∑x1x2εEM1x1εVM1x2εVM1−VM12dM1x1+2+dM2yα≤2n2m12ΔM1+ΔM2+2α∑″′2=∑yεVM2∑x1x2εESk2M1x1,x2εVSk2M1−VM1dx1,y+dx2,yα=k−1∑yεVM2∑x1x2εESk2M1x1,x2εVSk2M1−VM1dSk2M1x1+dSk2M1x2α=4αk−1m1n2.
Hence,(15)2αn1m2+n2m12ΔM1+ΔM2α+2n2m12ΔM1+ΔM2+2α+4αn2m1k−1≤χαM1+Sk2M2≤2αn1m2+n2m12δM1+δM2α+2n2m12δM1+δM2+2α+4αn2m1k−1.
Equality holds ifM1 and M2 are regular graphs.Proof.
(b) By the definition of GRI, we have(16)RαM1+Sk2M2=∑x1,y1x2,y2εEM1+Sk2M2dG+Sk2M2x1,y1dM1+Sk2M2x2,y2α,=∑xεVM1∑y1y2εEM2dM1+Sk2M2x,y1dM1+Sk2M2x,y2α+∑yεVM2∑x1x2εESk2M1dM1+Sk2M2x,y1dM1+Sk2M2x2,yα=∑1+∑2.
Consider(17)∑1=∑xεVM1∑y1y2εEM2dM1+Sk2M2x,y1dM1+Sk2M2x,y2α,=∑xεVM1∑y1y2εEM22dM1x+dM2y12dM1x+dM2y2α=∑xεVM1∑y1y2εEM24dM12x+2dM1x1dM2y1+dM2y2+dM2y1dM2y2α≤n1m22ΔM1+ΔM22α,∑2=∑yεVM2∑x1x2εESk2M1dx1,ydx2,yα=∑yεVM2∑x1x2εESk2M1x1,x2εVM1dx1,ydx2,yα+∑yεVM2∑x1x2εESk2M1xεVM1x2εVSk2M1−VM1dx1,ydx2,yα+∑yεVM2∑x1x2εESk2M1x1,x2εVSk2M1−VM1dx1,ydx2,yα=∑′2+∑″2+∑″′2.
Consider forx1x2εVM1, we have x1x2εESk2M1 if x1x2εEM1; for x1εVM1, we obtain dSk2M1x1=2dM1x1 and for x2εVSk2M1−VM1, we have dSk2M1x2=2. Now, consider(18)∑′2=∑yεVM2∑x1x2εESk2M1x1x2εVM1dx1,ydx2,yα=∑yεVM1∑x1x2εEM1dx1,ydx2,yα,=∑yεVM2∑x1x2εEM1dSk2M1x1+dM2ydSk2M1x2+dM2yα=∑yεVM2∑x1x2εEM14dM1x1dM1x2+2dM1x1dM2y+2dM1x2dM2y+dM22yα=∑yεVM2∑x1x2εEM14ΔM1ΔM1+2ΔM2ΔM1+ΔM1+ΔM22α≤n2m12ΔM1+ΔM22α.
Note thatES12M1=2EM1 and if x1εVM1, then dSk2M1x1=2dM1x1, and if x2εVSk2M1−VM1, then dSk2M1x2=2.(19)∑″2=∑yεVM2∑x1x2εESk2M1x1εVM1x2εVSk2M1−VM1dx1,ydx2,yα,=∑yεVM2∑x1x2εESk2M1x1εVM1x2εVSk2M1−VM1dSk2M1x1+dM2ydSk2M1x2α=∑yεVM2∑x1x2εESk2M1x1εVM1x2εVSk2M1−VM12dM1x1+dM2y2α≤2α+1n2m12ΔM1+ΔM2α∑″′2=∑yεVM2∑x1x2εESk2M1x2εVSk2M1−VM1dx1,ydx2,yα,=k−1∑yεVM2∑x1x2εESk2M1x1,x2εVSk2M1−VM1dSk2M1x1dSk2M1x2α=4αk−1m1n2.
Hence,(20)n1m2+n2m12ΔM1+ΔM22α+2α+1n2m12ΔM1+ΔM2α+4αn2m1k−1≤RαM1+Sk2M2≤n1m2+n2m12δM1+δM22α+2α+1n2m12δM1+δM2α+4αn2m1k−1.
Equality holds ifM1 and M2 are regular graphs.Theorem 3.
For a real numberα>0 and counting number k≥1, the lower and upper bounds on the GSCI and GRI of the generalized F-sum graph M1+Sk3M2 are as follows:(a)
2αn1m2ΔM1+ΔM2α+2n2m13ΔM1+ΔM2α+2k−1n1Z1M1α+4αΔM1αn21/2Z1M1+m1α≤χαM1+Sk3M2≤2αn1m2δM1+δM1α+2n2m13δM1+δM2α+2k−1n1Z1M1α+αδM1αn21/2Z1M1+m1α,(b)
n1m2ΔM1+ΔM22α+2αn2m1ΔM12+ΔM1ΔM2α+2k−1n2Z2M1α+4αΔM12αn2≤RαM1+Sk3M2≤n1m2δM1+δM22α+2αn2m1δM12+δM1δM2α+2k−1n2Z2M1α+4αδM12αn2,where equalities hold if and only if M1 and M2 are regular graphs.Proof.
(a) By the definition of the GSCI, we have(21)χαM1+Sk3M2=∑x1,y1x2,y2εEM1+Sk3M2dM1+Sk3M2x1,y1+dM1+Sk3M2x2,y2α=∑xεVM1∑y1y2εEM2dx,y1+dx,y2α+∑yεVM2∑x1x2εESk3M2dx,y1+dx,y2α=∑1+∑2.
Now,(22)∑1=∑xεVM1∑y1y2εEM2dx,y1+dx,y2α=∑xεVM1∑y1y2εEM22dM1x+dM2y1+dM2y2α≤2αn1m2ΔM1+ΔM2α∑2=∑yεVM2∑x1x2εESk3M1dx1,y+dx2,yα=∑yεVM2∑x1x2εESk3M1x1εVM1x2εVSk3M1−VM1dx1,y+dx2,yα+∑yεVM2∑x1x2εESk3M1x1x2εVSk3M1−VM1dx1,y+dx2,yα=∑′2+∑″2.
Now,(23)∑′2=∑yεVM2∑x1x2εESk3M1x1εVM1x2εVSk3M1−VM1dx1,y+dx2,yα,=∑yεVM2∑x1x2εESk3M2x1εVM1x2εVSk3M1−VM1dSk3M1x1+dM2y+dSk3M1x2α=∑yεVM2∑x1x2εESk3M2x1εVM1x2εVSk3M1−VM1dSk3M1x1+dM2y+dSk3M1x2α.
Note thatdS3M1x2=dM1wi+dM1wj for a2εVS3M1−VM1, where a2 is the vertex inserted into the edge wiwj of M1. Then, we have(24)=∑yεVM2∑x1x2εESk3M2x1εVM1x2εVSk3M1−VM1dSk3M1x1+dM2y+dM1wi+dM1wjα≤2n2m13ΔM1+ΔM2α∑″2=∑yεVM2∑x1x2εESk3M1x1,x2εVSk3M1−VM1dx1,y+dx2,y.
We break this sum into two parts∑2=∑3+∑4, where ∑3 belong the Sk3M edges and ∑4 belong the Sk3M1.(25)∑3=∑yεVM2∑x1x2εESk3M1x1x2εVSk3M1−VM1dSk3M1x1+dSk3M1x2α,=2k−1∑yεVM2∑wiwjεEMdM1wi+dM1wjα≤2k−1n1Z1M1α∑4=∑yεVM2∑x1x2εESk3M1x1x2εVSk3M1−VM1dx1,y+dx2,yα,=∑yεVM2∑x1x2εESk3M1x1x2εVSk3M1−VM1dSk3M1x1+dSk3M1x2α.
Sincex1 is the vertex inserted into the edge wiwj of M1 and a2 is the vertex inserted into the edge wjwk of M1,(26)=∑yεVM2∑wiwjεEM1wjwkεEM1dM1wi+dM1wj+dM1wj+dM1wkα=∑yεVM2∑wiwjεEM1wjwkεEM1dM1wi+dM1wk+2dM1wjα≤4αkΔM1αn212Z1M1+m1α.
Consequently, we have(27)2αn1m2ΔM1+ΔM2α+2n2m13ΔM1+ΔM2α+2k−1n1Z1M1α+4αΔM1αn212Z1M1+m1α≤χαM1+sk3M2≤2αn1m2δM1+δM1α+2n2m13δM1+δM2α+2k−1n1Z1M1α+4αδM1αn212Z1M1+m1α.Proof.
(b) By the definition of the GRI, we have(28)RαM1+Sk3M2=∑x1,y1x2,y2εEM1+Sk3M2dM1+Sk3M2x1,y1dM1+Sk3M2x2,y2α,=∑xεVM1∑y1y2εEM2dx,y1dx,y2α+∑yεVM2∑x1x2εESk3M2dx1,ydx2,yα=∑1+∑2.
Now,(29)∑1=∑xεVM1∑y1y2εEM2dM1+Sk3M2x,y1dM1+Sk3M2x,y2α,=∑xεVM1∑y1y2εEM2dM1x+dM2y1dM1x+dM2y2α=∑xεVM1∑y1y2εEM2dM12x+dM1x1dM2y1+dM2y2+dM2y1dM2y2α≤n1m2ΔM12+ΔM12ΔM2+ΔM22α≤n1m2ΔM1+ΔM22α,∑2=∑yεVM2∑x1,x2εESk3M1dx1,ydx2,yα,=∑yεVM2∑x1,x2εESk3M1x1εVM1x2εVSk3M1−VM1dx1,ydx2,yα+∑yεVM2∑x1,x2εESk3M1x1x2εVSk3M1−VM1dx1,ydx2,yα=∑′2+∑2.
Now,(30)∑′2=∑yεVM2∑x1x2εESk3M1x1εVM1x2εVSk3M1−VM1dx1,ydx2,yα,=∑yεVM2∑x1x2εESk3M1x1εVM1x2εVSk3M1−VM1dSk3M1x1+dM2ydSk3M1x2α.
Note thatdS3kM1x2=dM1wi+dM1wj for x2εVS3M1−VM1, where x2 is the vertex inserted into the edge wiwj of M1. Then, we have(31)=∑yεVM2∑x1x2εESk3M1x1εVM1x2εVSk3M1−VM1dM1x1+dM2ydM1wi+dM1wjα≤2αn2m1ΔM12+ΔM1ΔM2α∑″2=∑yεVM2∑x1x2εESk3M1x1,x2εVSk3M1−VM1dx1,ydx2,y.
We break this sum into two parts∑″2=∑3+∑4, where ∑3 belong the Sk3M edges and ∑4 belong the Sk3M1.(32)∑3=∑yεVM2∑x1x2εESk3M1x1,x2εVSk3M1−VM1dSk3M1x1dSk3M1x2α,=2k−1∑yεVM2∑wiwjεEMdM1widM1wjα≤2k−1n2Z2M1α∑4=∑yεVM2∑x1x2εESk3M1x1,x2εVSk3M1−VM1dx1,ydx2,yα,=dSk3M1x1dSk3M1x2αdSk3M1x1dSk3M1x2α.
Sincex1 is the vertex inserted into the edge wiwj of M1 and a2 is the vertex inserted into the edge wjwk of M1,(33)=∑yεVM2∑wiwjεEM1wjwkεEM1dM1wi+dM1wjdM1wj+dM1wkα≤4αΔM12αn2.
Consequently, we have(34)n1m2ΔM1+ΔM22α+2αn2m1ΔM12+ΔM1ΔM2α+2k−1n2Z2M1α+4αΔM12αn2≤RαM1+Sk3M2≤n1m2δM1+δM22α+2αn2m1δM12+δM1δM2α+2k−1n2Z2M1α+4αδM12αn2.Theorem 4.
For a real numberα>0 and counting number k≥1, the lower and upper bounds on the GSCI and GRI of the generalized F-sum graph M1+Sk1M2 are(a)
2αn1m2ΔM1+ΔM2α+2αn2m12ΔM1+ΔM2α+2n2m13ΔM1+ΔM2α+2k−1n1Z1M1α+4αΔM1αn21/2Z1M1+m1α≤χαM1+Sk4M2≤2αn1m2δM1+δM2α+2αn2m12δM1+δM2α,(b)
n1m2ΔM1+ΔM22α+2αn2m12ΔM1+ΔM2α+2αn2m1ΔM12+ΔM1ΔM2α+2k−1n2Z2M1α+4αΔM12αn2≤RαM1+Sk4M2≤n1m2δM1+δM22α+2αn2m12δM1+δM2α+2αn2m1δM12+δM1δM2α+2k−1n2Z2M1α+4αδM12αn2,where r is the number of common neighbors of x1 and x2 in M1.Proof.
The proof follows by Theorem2 and Theorem 3.


## 4. Applications

For a real numberα>0, the lower and upper bounds on GSCI and GRI of the generalized F-sum graphs obtained by the particular classes of graphs as the consequences of the obtained results are given in Figures 4–11.Figure 4
Graphforα=2mn6×42+42m−42n≤χαPn+S41Pm≤mn82+2×62+3×42−m2×62−3×42−82n.Figure 5
Graphforα=2mn242+232+1+3.42−m232+1−3.42−242n≤RαPn+S41Pm≤mn222+222+1+3.42−m222+1−3.42−222n.Figure 6
Graphforα=2mn2.122+2.82+3.42−m122+2.82+3.42−122n≤χαPn+S42Pm≤mn2.62+2.52+3.42−m62+2.52+3.42−62n.Figure 7
Graphforα=2mn2.622+32.222+1+3.42−m622+32.222+1+3.42−622n≤RαPn+S42Pm≤mn2.322+32.222+1+3.42−m322+32.22+1+3.42−32×2n.Figure 8
Graphforα=282nm−n+2.82mn−m+6n4n−62+82m3n−42≤χαPn+S43Pm≤42mn−n+2.42mn−m+6n4n−62+42m3n−42.Figure 9
Graphforα=2162nm−n+162mn−m+6m4m−62+162≤RαPn+S43Pm≤42nm−n+4mn−m+6m4m−62+42.Figure 10
Graphforα=282nm−n+122mn−m+2.82mn−m+6n4n−62+82m3n−42≤χαPn+S44Pm≤42nm−n+62mn−m+2.42mn−m+6n4n−62+42m3n−42.Figure 11
Graphforα=2242nm−n+222.32mn−m+242mn−m+6m4m−62+242m≤RαPn+S44Pm≤222nm−n+62mn−m+222mn−m+6m4m−62+42m.Example 1.
ForM1=Pn, M2=Pm, and k=4, we have(a)
mn6.4α+m4α−4αn≤χαPn+S41Pm≤mn8α+2.6α+3.4α−m2.6α−3.4α−8αn., and(b)
mn24α+23α+1+3.4α−m23α+1−3.4α−24αn≤RαPn+S41Pm≤mn22α+22α+1+3.4α−m22α+1−3.4α−22αn.
The graphical representation of Example1(a) is depicted in Figure 4, the lower bounds are represented by the blue graph and the upper bounds are represented by the red graph. The graphical representation of Example 1(b) is depicted in Figure 5, the lower bounds are represented by the green graph and the upper bounds are represented by the blue graph.Example 2.
ForM1=Pn, M2=Pm, and k=4, we have(a)
mn2.12α+2.8α+3.4α−m12α+2.8α+3.4α−12αn≤χαPn+S42Pm≤mn2.6α+2.5α+3.4α−m6α+2.5α+3.4α−6αn,(b)
mn2.62α+3α.22α+1+3.4α−m62α+3α.22α+1+3.4α−62αn≤RαPn+S42Pm≤mn2.32α+3α.22α+1+3.4α−m32α+3α.2α+1+3.4α−32αn.
The graphical representation of Example2(a) is depicted in Figure 6. The lower bounds are represented by the Niagara Azure graph and the upper bounds are represented by the gold graph. The graphical representation of Example 2(b)is depicted in Figure 7. The lower bounds are represented by the pink graph and the upper bounds are represented by the yellow graph.Example 3.
ForM1=Pn, M2=Pm, and k=4, we have(a)
8αnm−n+2.8αmn−m+6n4n−6α+8αm3n−4α≤χαPn+S43Pm≤4αnm−n+2.4αmn−m+6n4n−6α+4αm3n−4α,(b)
16αnm−n+16αmn−m+6m4m−6α+16α≤RαPn+S43Pm≤4αnm−n+4αmn−m+6m4m−6α+4α.
The graphical representation of Example3(a) is depicted in Figure 8. The lower bounds are represented by the blue graph and the upper bounds are represented by the green graph. The graphical representation of Example 3(b) is depicted in Figure 9. The lower bounds are represented by the yellow graph and the upper bounds are represented by the gray graph.Example 4.
ForM1=Pn, M2=Pm, and k=4, we have(a)
8αnm−n+12αmn−m+2.8αmn−m+6n4n−6α+8αm3n−4α≤χαPn+S44Pm≤4αnm−n+6αmn−m+2.4αmn−m+6n4n−6α+4αm3n−4α,(b)
24αnm−n+22α.3αmn−m+24αmn−m+6m4m−6α+24αm≤RαPn+S44Pm≤22αnm−n+6αmn−m+22αmn−m+6m4m−6α+4αm.
The graphical representation of Example4(a) is depicted in Figure 10. The lower bounds are represented by the blue graph and the upper bounds are represented by the green graph. The graphical representation of Example 4(b) is depicted in Figure 11. The lower bounds are represented by the yellow graph and the upper bounds are represented by the blue graph.


## 5. Conclusion

In this paper, the lower and upper bounds of the general sum-connectivity and general Randić indices of the generalizedF-sum graphs (Fk-sum graphs) are computed in terms of the order, size, maximum, and/or minimum degree and Zagreb indices of the factor graphs, where the Fk-sum graphs are obtained with the help of four generalized subdivision-related operations and the Cartesian product of graphs. However, the problem is still open for other types of product of graphs.



---

*Source: 8821020-2021-03-02.xml*