# Bounds on General Randić Index forF-Sum Graphs

Authors: Xu Li; Maqsood Ahmad; Muhammad Javaid; Muhammad Saeed; Jia-Bao Liu
Journal: Journal of Mathematics
 (2020)

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## Abstract

A topological invariant is a numerical parameter associated with molecular graph and plays an imperative role in the study and analysis of quantitative structure activity/property relationships (QSAR/QSPR). The correlation between the entireπ-electron energy and the structure of a molecular graph was explored and understood by the first Zagreb index. Recently, Liu et al. (2019) calculated the first general Zagreb index of the F-sum graphs. In the same paper, they also proposed the open problem to compute the general Randić index RαΓ=∑uv∈EΓdΓu×dΓvα of the F-sum graphs, where α∈R and dΓu denote the valency of the vertex u in the molecular graph Γ. Aim of this paper is to compute the lower and upper bounds of the general Randić index for the F-sum graphs when α∈N. We present numerous examples to support and check the reliability as well as validity of our bounds. Furthermore, the results acquired are the generalization of the results offered by Deng et al. (2016), who studied the general Randić index for exactly α=1.


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## Body

## 1. Introduction

Suppose the ordered pairVΓ,EΓ denotes a finite, simple, and connected molecular graph Γ. The set represented by VΓ is the vertex set and the set denoted by EΓ, disjoint from VΓ, is the edge set. Vertices of Γ correspond to atoms, whereas edges represent bonding between atoms. For any vertex v∈VΓ, the number of vertices adjacent with v is called the degree of vertex v and is denoted by dΓv. The smallest and the largest degree of Γ are symbolized by δΓ and ΔΓ, respectively. Two primary parameters known as the order (total number of vertices) and the size (total number of edges) of graph Γ are denoted by n and e. A path Pk is a simple graph having order k and size k−1 with the property that exactly two vertices have degree 1, and rest of the vertices have degree 2. A cycle Cm is a simple graph with same order and size m in such a way that each vertex has degree 2.Graph theory is playing a remarkable role in various domains of science, especially in mathematical chemistry, computer science, and chemical graph theory since the middle of last century. LetΩ be a collection of simple graphs and R be a set of real numbers; then, a topological index (TI) is considered to be a function Φ: Ω⟶R that associates a graph to a real number. It is worth noting that all the TIs are invariant for the isomorphic structures. To probe and study the chemical, structural, and physical properties of the molecular graphs within the subject of chemical graph theory, several TIs are proposed and intensely investigated. These TIs helped to study the chemical reactivities and physical features such as heat of evaporation and formation, boiling, melting and freezing point, volume of air and vapor pressure, surface tension and density, and critical temperature of the chemical compounds that are involved in the molecular graphs. Moreover, medical behaviors of the drugs, nanomaterials, and crystalline materials which are very important for the chemical industries including pharmaceutical are studied using TIs. For further reading regarding development and applications of TIs, the readers are referred to [1–10].


## 2. Preliminaries and Background

Some convinced and significant degree-based TIs closely related to our work are defined below.Definition 1.
LetΓ be a molecular graph; then, the first Zagreb index and second Zagreb index are defined as(1)M1Γ=∑v∈VΓdΓv2=∑uv∈EΓdΓu+dΓv,M2Γ=∑uv∈EdΓu⋅dΓv.
Gutman and Trinajstić [11] defined the first and second Zagreb indices to establish the relationship between the entire π-electron energy and a structure of a molecular graph.Definition 2.
LetΓ be a molecular graph; then, the first general Zagreb index (FGZI) is defined as(2)MαΓ=∑v∈VΓdΓvα=∑vw∈EΓdΓα−1v+dΓα−1w,where α∈R, α≠0, and α≠1. Li and Zheng [12] provided the idea of FGZI. It is evident that we achieve M1Γ (first Zagreb index) and FΓ (forgotten index) by putting α=2, and α=3 in FGZI, respectively [11, 13].Definition 3.
LetΓ be a molecular graph; then, the general Randić index (GRI) is defined as(3)RαΓ=∑vw∈EΓdΓvdΓwα,α∈R,α≠0.
Bollobás and Erdös originated the concept of GRI [14]. It is clear that α=−1/2 gives the classical Randić connectivity index R−1/2Γ [15] and α=1 provides second Zagreb index M2Γ [11].Definition 4.
LetΓ be a molecular graph; then, the general sum-connectivity index (GSCI) is defined as(4)χαΓ=∑vw∈EΓdΓv+dΓwα,α∈R.
Zhou and Trinajstić initiated the idea of GSCI [16]. For α=−1/2, we attain the classical sum-connectivity index (SCI) χ−1/2Γ [17].
Lučić et al. [18] studied and observed that there is good correlation between indices R−1/2Γ and χ−1/2Γ themselves besides their correlation with EπΓ (π-electron energy) of benzenoid hydrocarbons. For further insight and applications related to all above TIs, the readers are referred to [19–23].
The operations on graphs, in the construction of new graphs, also play an important role in graph theory, where the old graphs are called the factors of the new graph. Cartesian product (binary operation) is an elegant technique to construct a broader network from two base graphs and is inevitable for design and analysis of networks [24]. In [25], Eliasi and Taeri contrived and constructed the F-sum graphs G1+FG2 (F∈S,R,Q,T) by employing the idea of Cartesian product on graphs FΓ1 and Γ2, where Γ1andΓ2 are two simple-connected graphs and FΓ1 is obtained after applying F on Γ1 which is elaborated subsequently.Definition 5.
LetΓ be a simple, connected, and finite graph; then, the four significant related graphs can be defined as follows [25, 26]:(1)
Subdivision graphSΓ is an expansion of graph Γ by introducing an additional vertex on each edge of Γ.(2)
Triangle parallel graphRΓ is derived from SΓ by joining the solid vertices of the original edges of Γ that are incident with hollow vertices.(3)
Line superposition graphQΓ is obtained from SΓ by attaching those pairs of new vertices by edges which have common adjacent (solid) vertex.(4)
Total graphTΓ is constructed from SΓ by applying RΓ and QΓ, simultaneously.
For more details, see Figure1. For further insight regarding graph operations, see [27].Figure 1
Base graphC6 along with its derived graphs.Definition 6.
LetΓ1 and Γ2 be two finite, simple, and connected graphs, F be an operation (one of S, R, Q, and T), and FΓ be a graph (derived from Γ1 by employing the operation F) with VFΓ1 as vertex set and EFΓ1 as edge set. Then, the F-sum graph Γ1+FΓ2 is a graph with vertex set VΓ1+FΓ2=VFΓ1×VΓ2=VΓ1∪EΓ1×VΓ2 such that two vertices v1,v2 and w1,w2 of VΓ1+FΓ2 are adjacent if and only if [v1=w1∈VΓ and v2,w2∈EΓ2] or [v2=w2∈VΓ2 and v1,w1∈EFΓ1].
We observe that the graphΓ1+FΓ2 has VΓ2 copies of the graph FΓ1 provided that vertices of these copies are labeled with vertices of Γ2. In graph Γ1+FΓ2, the vertices of Γ1 are referred as solid vertices, whereas the vertices EΓ1 are referred as hollow vertices. Now join only solid vertices having same label in FΓ1 such that their adjacency in Γ2 is preserved. For more clarity, see Figure 2.
Following theorems from basic mathematics are of substantial significance in order to obtain core results.Figure 2
GraphsP5 and P6 along with their F-sum graphs.Theorem 1.
Binomial and trinomial theorems provide easy and powerful way in expanding expression involving finite higher powers. The algebraic expressions of binomial and trinomial expansions are described beneath, respectively.(5)x1+x2n=∑i=0nnix1n−ix2i,(6)x1+x2+x3n=∑m,k,lm+k+l=nPm,k,lx1mx2kx3l,where Pm,k,l=m+k+l!/m!k!l!.Although valency as well as spectral based TIs are current topics of increasing interest for researchers and recently Liu et al [28–30] studied weighted edge corona networks viz a viz spectra of various matrices and valency based indices of Eulerian as well as generalized Sierpinski networks. However, among the valency-based TIs, the Randić index and its variations such as general sum-connectivity, general Randić, harmonic, geometric arithmetic, and atom bond connectivity indices have ample applications in pharmacology and medicinal chemistry [31–33]; for detailed study regarding Randić index, see survey [34]. In [35], Yan et al. computed and analyzed the changes in behavior of Wiener index [36] and enhanced the results to Hosoya polynomial for graph operations presented in Definition 1. In [25], Eliasi and Taeri not only introduced the F-sum graphs but also computed the Wiener index of these graphs. Li et al. [37], Shi [38], and Pan et al. [39] provided bounds on Randić index for chemical graphs (dv≤4), bounds on Randić index for triangle-free graph, and sharp bounds on zeroth order general Randić index for unicyclic graphs with fix diameter, respectively. Ali et al. [40], Jamil et al. [41], and Elumalai et al. [42] computed bounds on zeroth order general Randić index for certain type of graphs. Later on, Deng et al. [43], Imran et al. [44], Liu et al. [45], and Ahmad et al. computed the first Zagreb index and second Zagreb index (general Randić index for exactly α=1) of the F-sum graphs, bounds of several indices of the F-sum graphs, first generalized Zagreb indices of the F-sum graphs, and bounds on general sum-connectivity index for F-sum graphs [46], respectively. Moreover, Liu et al. [45] proposed the open problem to compute the general Randić index for any α∈R. In this paper, we solve this open problem, partially, by computing the lower and upper bounds on general Randić index for the F-sum graphs for any α∈N.The rest of the paper is put together as follows. Section2 covers the materials and methods to determine main results and Section 3 includes some applications of the main results. Section 4 covers the conclusion and further directions of the work.


## 3. Results and Discussion

In this section, the main results regarding general Randić index on theF-sum graphs Γ1+SΓ2, Γ1+RΓ2, Γ1+QΓ2 and Γ1+TΓ2, are computed, where Γ1 and Γ2 are considered to be finite, simple, and connected graphs. Throughout n1=VΓ1, n2=VΓ2, e1=EΓ1, e2=EΓ2, VSΓ1=n1+e1, ESΓ1=2e1, M0Γ1=n1, M1Γ1=2e1, M1Γ2=2e2, M2Γ1=M1Γ1, M2Γ2=M1Γ2, R1Γ2=M2Γ2, χ0Γ2=e2, and χ1Γ1=M1Γ1.Theorem 2.
LetΓ1 and Γ2 be two simple, finite, and connected graphs. For α∈N, the general Randić index of S-sum graph is LBS≤RαΓ1+SΓ2≤UBS, where(7)LBS=∑m,k,lm+k+l=nPm,k,lM2m+kΓ1χkΓ2δΓ22l+∑i=0ααiRα−iSΓ1MiΓ2δΓ1i,UBS=∑m,k,lm+k+l=nPm,k,lM2m+kΓ1χkΓ2ΔΓ22l+∑i=0ααiRα−iSΓ1MiΓ2ΔΓ1i,and Pm,k,l=m+k+l!/m!k!l!. Equality holds if and only if Γ1 and Γ2 are regular graphs with same regularity.Proof.
Suppose thatdv,w=dΓ1+SΓ2v,w denotes the degree of vertex v,w in the S-sum graph Γ1+SΓ2. Then, general Randić index for S-sum graph is calculated as(8)RαΓ1+SΓ2=∑v1,w1v2,w2∈EΓ1+SΓ2dv1,w1dv2,w2α=∑v∈VΓ1∑w1w2∈EΓ2dv,w1dv,w2α+∑w∈VΓ2∑v1v2∈ESΓ1dv1,wdv2,wα=∑1+∑2.
Consider(9)∑1=∑v∈VΓ1∑w1w2∈EΓ2dΓ1v+dΓ2w1dΓ1v+dΓ2w2α=∑v∈VΓ1∑w1w2∈EΓ2dΓ12v+dΓ1vdΓ2w1+dΓ2w2+dΓ2w1dΓ2w2α.
Using trinomial theorem, we get(10)∑1=∑v∈VΓ1∑w1w2∈EΓ2∑m,k,lm,k,lm+k+l=αPm,k,ldΓ12mvdΓ1kvdΓ2w1+dΓ2w2kdΓ2w1dΓ2w2l,where Pm,k,l=m+k+l!/m!k!l!.(11)∑1=∑m,k,lm+k+l=αPm,k,l∑w1w2∈EΓ2dΓ2w1+dΓ2w2k∑v∈VΓ1dΓ12m+kvdΓ2w1dΓ2w2l.
Using Definitions1 and 3 and the fact δΓ2w≤dΓ2w∀w∈VΓ2, we have(12)∑1≥∑m,k,lm+k+l=αPm,k,lM2m+kΓ1χkΓ2δΓ22l,∑2=∑w∈VΓ2∑v1v2∈ESΓ1dv1,wdv2,wα=∑w∈VΓ2∑v1v2∈ESΓ1dSΓ1v1+dΓ2wdSΓ1v2α=∑w∈VΓ2∑v1v2∈ESΓ1dSΓ1v1dSΓ1v2+dΓ2wdSΓ1v2α.
Now applying binomial theorem, we get(13)∑2=∑w∈VΓ2∑SΓ1∑i=0ααidSΓ1v1dSΓ1v2α−idΓ2wdSΓ1v2i=∑i=0ααi∑v1v2∈ESΓ1dSΓ1v1dSΓ1v2α−i∑w∈VΓ2dΓ2iwdSΓ1iv2.
Using Definitions5 and 6 and property of smallest degree of graph Γ1, we have(14)∑2≥∑i=0ααiRα−iSΓ1MiΓ2δΓ1i.
Substituting the (12) and (14) in (8), we have(15)RαΓ1+SΓ2≥∑m,k,lm+k+l=αPm,k,lM2m+kΓ1χkΓ2δΓ22l+∑i=0ααiRα−iSΓ1MiΓ2δΓ1i=LBS.
Similarly,(16)RαΓ1+SΓ2≤∑m,k,lm+k+l=αPm,k,lM2m+kΓ1χkΓ2ΔΓ22l+∑i=0ααiRα−iSΓ1MiΓ2ΔΓ1i=UBS.
Equality holds if and only ifΓ1 and Γ2 are regular graphs with same regularity. This completes the proof.Example 1.
LetΓ1=P4, Γ2=C3, α=2, and F=S. Then, n1=M0P4=4, n2=M0C3=3, e1=χ0P4=3, e2=χ0C3=3, δP4=1, ΔP4=2, δC3=2, and ΔC3=2. Moreover, Table 1 contains values of some indices related to certain graphs and is crucial to figure out examples throughout.
Now, we compute lower and upper bound of GRI using formulas derived in Theorem2.(17)LBSP4+SC3=∑m+k+l=2Pm,k,lM2m+kP4χkC3δΓ22l+∑i=022iR2−iSP4MiC3δP4i=P1,1,0MP4χC3+P1,0,1M2P4χ0C34+P0,1,1MP4χC34+P2,0,0M4P4χ0C3+P0,2,0M2P4χ2C3+P0,0,2M0P4χ0C316+R2SP4M0C3+2RSP4MC3+R0SP4M2C3=21812+1034+6124+343+1048+4316+723+2206+612=2550.
Likewise,UBSP4+SC3=3006.
To calculate exact value forR2P4+SC3, we require edge partition of graph P4+SC3, which is presented in Table 2.
Now, we calculate exact value of GRI ofP4+SC3 for α=2. R2P4+SC3=∑vw∈EP4+SC3dP4vdC3w2=3006.
Evidently,(18)LBS=2550≤R2P4+SC3=3006≤UBS=3006.
Additionally, we computed actual values along with corresponding bounds of GRI for various cases, and some are presented in Table3.Table 1
Some particular indices of certain graphs.
Indices/graphsMM2M3M4M5M6χχ2χ3RR2R3P4610183466130103411882480P581426509819414501821240144C361224489619212481921248192SP422823102072272SP528104400Table 2
Edge partition based upon degree of end vertices of graphP4+SC3.
dv,dw(2, 3)(3, 3)(2, 4)(4, 4)E66126Table 3
Few more cases of lower and upper bounds regardingRαΓ1+SΓ2.
Γ1Γ2αLBS≤RαΓ1+SΓ2≤UBSP4C3331926≤36390≤36390P5P423274≤3346≤3850P5P4319941≤34834≤40764Theorem 3.
LetΓ1 and Γ2 be two simple, finite, and connected graphs. For α∈N, the general Randić index of R-sum graph is LBR≤RαΓ1+RΓ2≤UBR, where(19)LBR=∑i=0ααi2α+i+1MiΓ1Mα−iΓ2+∑m,k,lm+k+l=αPm,k,l22m+kM2m+kΓ1χkΓ2δΓ22l+RmΓ1Mk+2lΓ22δΓ1k,UBR=∑i=0ααi2α+i+1MiΓ1Mα−iΓ2+∑m,k,lm+k+l=αPm,k,l22m+kM2m+kΓ1χkΓ2ΔΓ22l+RmΓ1Mk+2lΓ22ΔΓ1k,and Pm,k,l=m+k+l!/m!k!l!. Equality holds if and only if Γ1 and Γ2 are regular graphs with same regularity.Proof.
Suppose thatdv,w=dΓ1+RΓ2v,w denotes the degree of vertex v,w in the R-sum graph Γ1+RΓ2. Then, general Randić index for R-sum graph is calculated as(20)RαΓ1+RΓ2=∑v1,w1v2,w2∈EΓ1+RΓ2dv1,w1dv2,w2α=∑v∈VΓ1∑w1w2∈EΓ2dv,w1dv,w2α+∑w∈VΓ2∑v1v2∈ERΓ1dv1,wdv2,wα=∑w1w2∈EΓ2dv,w1dv,w2α+∑w∈VΓ2∑v1v2∈ERΓ1v1,v2∈VΓ1dv1,wdv2,wα+∑w∈VΓ2∑v1v2∈ERΓ1v1∈VΓ1v2∈VRΓ1−VΓ1dv1,wdv2,wα=∑3+∑4+∑5,∑3=∑v∈VΓ1∑w1w2∈EΓ2dRΓ1v+dΓ2w1dRΓ1v+dΓ2w2α=∑v∈VΓ1∑w1w2∈EΓ22dΓ1v+dΓ2w12dΓ1v+dΓ2w1α=∑v∈VΓ1∑w1w2∈EΓ24dG2v+2dΓ1vdΓ2w1+dΓ2w2+dΓ2w1dΓ2w2α.
Applying trinomial theorem, we get(21)∑3=∑v∈VΓ1∑w1w2∈EΓ2∑m,k,lm+k+l=αPm,k,l4mdΓ12mv2kdΓ1kvdΓ2w1+dΓ2w2kdΓ2w1dΓ2w2l,where Pm,k,l=m+k+l!/m!k!l!. Now applying summations on the convenient expressions, then replacing with corresponding formulas, and using smallest degree of graph Γ2, we get(22)∑3=∑m,k,lm+k+l=αPm,k,l22m+k∑v∈VΓ1dΓ12m+kv∑w1w2∈EΓ2dΓ2w1+dΓ2w2kdΓ2w1dΓ2w2l≥∑m,k,lm+k+l=αPm,k,l22m+kM2m+kΓ1χkΓ2δΓ22l.
Next sum involves those edges fromRΓ1 whose end vertices are in VΓ1.(23)∑4=∑w∈VΓ2∑v1v2∈ERΓ1v1,v2∈Γ1dv1,wdv2,wα=∑w∈VΓ2∑v1v2∈ERΓ1v1,v2∈VΓ1dRΓ1v1+dΓ2wdRΓ1v2+dΓ2wα=∑w∈VΓ2∑v1v2∈ERΓ1v1,v2∈VΓ12dΓ1v1+dΓ2w2dΓ1v2+dΓ2wα=∑w∈VΓ2∑v1v2∈ERΓ1v1,v2∈VΓ14dΓ1v1dΓ1v2+2dΓ2wdΓ1v1+dΓ1v2+dΓ22wα.
Now applying trinomial theorem, we have(24)∑4=∑w∈VΓ2∑v1v2∈ERΓ1v1,v2∈VΓ1∑m,k,lm+k+l=αPm,k,l4dΓ1v1dΓ1v2m2kdΓ2kwdΓ1Γ1+dΓ1v2k+dΓ22lw=∑m,k,lm+k+l=αPm,k,l∑w∈VΓ2∑v1v2∈EΓ14dΓ1v1dΓ1v2m2kdΓ2kwdΓ1v1+dΓ1v2kdΓ22lw,(25)∑4=∑m,k,lm+k+l=αPm,k,l22m+k∑v1v2∈EΓ1dΓ1v1dΓ1v2m∑w∈VΓ2dΓ2k+2lwdΓ1v1+dΓ1v2k≥∑m,k,lm+k+l=αPm,k,l22m+kRmΓ1Mk+2lΓ22δΓ1k.
Subsequent sum includes those edges fromRΓ1 whose one end vertex is in VΓ1 while the other is in VRΓ1−VΓ1.(26)∑5=∑w∈VΓ2∑v1v2∈ERΓ1v1∈VΓ1v2∈VRΓ1−VΓ1dv1,wdv2,wα=∑w∈VΓ2∑v1v2∈ERΓ1v1∈VΓ1v2∈VRΓ1−VΓ12dRΓ1v1+dΓ2wα=∑w∈VΓ2∑v1v2∈ERΓ1v1∈VΓ1v2∈VRΓ1−VΓ122dGv1+dΓ2wα=∑w∈VΓ2∑v1v2∈ERΓ1v1∈VΓ1v2∈VRΓ1−VΓ12dΓ2w+4dGv1αusing binomial theorem=∑w∈VΓ2∑v1v2∈ERΓ1v1∈VΓ1v2∈VRΓ1−VΓ1∑i=0ααi2dΓ2wα−i4dGiv1i=∑i=0ααi2α+i∑v1v2∈ERΓ1v1∈VΓ1v2∈VRΓ1−VΓ1dGiv1∑w∈VΓ2dΓ2α−iw≥∑i=0ααi2α+i∑w∈VΓ2dΓ2α−iw2∑v1∈VΓ1dGiv1≥∑i=0ααi2α+i+1Mα−iΓ2MiΓ1.
Using (22), (25), and (26) in (20), we get(27)RαΓ1+RΓ2≥∑m,k,lm+k+l=αPm,k,l22m+kM2m+kΓ1χkΓ2δΓ22l+RmΓ1Mk+2lΓ22δΓ1k+∑i=0ααi2α+i+1MiΓ1Mα−iΓ2=LBR,RαΓ1+RΓ2≤∑m,k,lm+k+l=αPm,k,l22m+kM2m+kΓ1χkΓ2ΔΓ22l+RmΓ1Mk+2lΓ22ΔΓ1k+∑i=0ααi2α+i+1MiΓ1Mα−iΓ2=UBR.
Equality holds if and only ifΓ1 and Γ2 are regular graphs with same regularity. This completes the proof.Example 2.
LetΓ1=P5, Γ2=P4, α=2, and F=R. Then, n1=M0P5=5, n2=M0P4=4, e1=χ0P5=4, e2=χ0P4=3, δP5=1, ΔP5=2, δP4=1, and ΔP4=2. Now, we compute lower and upper bound of GRI using formulas derived in Theorem 3.(28)LBRP5+RP4=∑i=022i23+iMiP5M2−iP4+∑m,k,lm+k+l=2Pm,k,l22m+kM2m+kP5χkP4δP42l+RmP5Mk+2lP42δP5k=8M0P5M2P4+32MP5MP4+32M2P5M0P4+8P1,1,0M3P5χP4+2RP5MP4+2P1,0,1M2P5χ0P4+4R0P5M2P4+2P0,1,14MP5χP4+2R0P5M3P4+16P2,0,0M4P5χ0P4+R2P5M0P4+4P0,2,0M2P5χ2P4+4R0P5M2P4+P0,0,216M0P5χ0P4+R0P5M4P4,LBRP5+RP4=8510+32810+32144+162610+1262+81434+1210+48104+4182+16503+404+41434+4104+5316+434=23256.
In similar way, we calculateUBRP5+RP4=28056.
To compute exact value forR2P5+RP4, we need edge partition of graph P5+RP4, which is given in Table 4.
Exact value of GRI ofP5+RP4 for α=2 is given by R2P5+RP4=∑vw∈EP5+RP4dP5vdP4w2=24592.
Clearly,(29)23256≤R2P5+RP4=24592≤28056.
Moreover, we computed actual values along with corresponding bounds of GRI for various cases, and some are presented in Table5.Table 4
Edge partition based upon degree of end vertices of graphP5+RP4.
dv,dw(2, 3)(2, 4)(2, 5)(2, 6)(3, 4)(3, 5)(4, 4)(4, 6)(5, 5)(5, 6)(6, 6)E4412124424467Table 5
Few more cases of lower and upper bounds regardingRαΓ1+RΓ2.
Γ1Γ2αLBR≤RαΓ1+RΓ2≤UBRP4C3215696≤18768≤19248P4C33425952≤551232≤589632P5P43526448≤698288≤699904Theorem 4.
LetΓ1 and Γ2 be two simple, finite, and connected graphs. For α∈N, the general Randić index of Q-sum graph is LBQ≤RαΓ1+QΓ2≤UBQ, where(30)LBQ=∑m,k,lm+k+l=αPm,k,lM2m+kΓ1RlΓ22δΓ2k+∑i=0ααiMiΓ2χαΓ1δΓ1α−i+n2χαΓ12δΓ1α,UBQ=∑m,k,lm+k+l=αPm,k,lM2m+kΓ1RlΓ22ΔΓ2k+∑i=0ααiMiΓ2χαΓ1ΔΓ1α−i+n2χαΓ12ΔΓ1α,and Pm,k,l=m+k+l!/m!k!l!. Equality holds if and only if Γ1 and Γ2 are regular graphs with same regularity.Proof.
Suppose thatdv,w=dΓ1+QΓ2v,w denotes the degree of vertex v,w in the Q-sum graph Γ1+QΓ2. Then general Randić index for Q-sum graph is calculated as(31)RαΓ1+QΓ2=∑v1,w1v2,w2∈EΓ1+QΓ2dv1,w1dv2,w2α=∑v∈VΓ1∑w1w2∈EΓ2dv,w1dv,w2α+∑w∈VΓ2∑v1v2∈EQΓ1dv1,wdv2,wα=∑v∈VΓ1∑w1w2∈EΓ2dv,w1dv,w2+∑w∈VΓ2∑v1v2∈EQΓ1v1∈VΓ1v2∈VQΓ1−VΓ1dv1,wdv2,wα+∑w∈VΓ2∑v1v2∈EQΓ1v1v2∈VQΓ1−VΓ1dv1,wdv2,wαα=∑6+∑7+∑8,(32)∑6=∑v∈VΓ1∑w1w2∈EΓ2dv,w1dv,w2α=∑v∈VΓ1∑w1w2∈EΓ2dQΓ1v+dΓ2w1dQΓ1v+dΓ2w2α=∑v∈VΓ1∑w1w2∈EΓ2dΓ1v+dΓ2w1dΓ1v+dΓ2w1α,∑6=∑v∈VΓ1∑w1w2∈EΓ2dΓ12v+dΓ1vdΓ2w1+dΓ2w2+dΓ2w1dΓ2w2α.
Applying trinomial theorem, we get(33)∑6=∑v∈VΓ1∑w1w2∈EΓ2∑m,k,lm+k+l=αPm,k,ldΓ12mvdΓ1kvdΓ2w1+dΓ2w2kdΓ2w1dΓ2w2l,where Pm,k,l=m+k+l!/m!k!l!.(34)∑6=∑m,k,lm+k+l=αPm,k,l∑v∈VΓ1∑w1w2∈EΓ2dΓ2w1+dΓ2w2kdΓ12mvdΓ1kvdΓ2w1dΓ2w2l=∑m,k,lm+k+l=αPm,k,l∑w1w2∈EΓ2dΓ2w1dΓ2w2ldΓ2w1+dΓ2w2k∑v∈VΓ1dΓ12m+kv≥∑m,k,lm+k+l=αPm,k,lM2m+kΓ1RlΓ22δΓ2k.
∑7 involves those edges from QΓ1 whose one end vertex is in VΓ1 and the other is in VQΓ1−VΓ1.(35)∑7=∑w∈VΓ2∑v1v2∈EQΓ1v1∈VΓ1v2∈VQΓ1−VΓ1dv1,wdv2,wα=∑w∈VΓ2∑v1v2∈EQΓ1v1∈VΓ1v2∈VQΓ1−VΓ1dQΓ1v1+dΓ2wdQΓ1v2α=∑w∈VΓ2∑v1v2∈EQΓ1v1∈VΓ1v2∈VQΓ1−VΓ1dΓ1v1+dΓ2wαdQΓ1v2α.
Now applying binomial theorem, we have(36)∑7=∑w∈VΓ2∑v1v2∈EQΓ1v1∈VΓ1v2∈VQΓ1−VΓ1∑i=0ααidΓ1v1α−idΓ2widQΓ1v2α=∑i=0ααi∑v1v2∈EQΓ1v1∈VΓ1v2∈VQΓ1−VΓ1dQΓ1αv2∑w∈VΓ2dΓ2iwdΓ1α−iv1.
It can easily be observed thatdQΓ1v2=dΓ1us+dΓ1ut for v2∈VQΓ1−VΓ1, where v2 is the vertex inserted into the edge usut∈EΓ1. In addition, ∑v1v2∈EQΓ1v1∈VΓ1,v2∈VQΓ1−VΓ1dQΓ1v2=2∑uiuj∈EΓ1dΓ1ui+dΓ1uj.
Also, using the factδΓ1v≤dΓ1v∀v∈VΓ1, we have(37)∑7≥∑i=0ααi2∑uiuj∈EΓ1dΓ1ui+dΓ1ujαMiΓ2δΓ1α−i=∑i=0ααi2MiΓ2χαΓ1δΓ1α−i.
Next sum contains those edges fromQΓ1 whose both end vertices are in VQΓ1−VΓ1.(38)∑8=∑w∈VΓ2∑v1v2∈EQΓ1v1,v2∈VQΓ1−VΓ1dv1,wdv2,wα=∑w∈VΓ2∑v1v2∈EQΓ1v1,v2∈VQΓ1−VΓ1dQΓ1v1dQΓ1v2α=∑w∈VΓ2∑usut∈EΓ1utur∈EΓ1dΓ1us+dΓ1utdΓ1ut+dΓ1urα,where v1 and v2 are the vertices embedded into the edges usut and utur of Γ1, respectively.(39)∑8=∑w∈VΓ2∑usut∈EΓ1utur∈EΓ1dΓ1us+dΓ1utαdΓ1ut+dΓ1urα≥n2∑usut∈EΓ1dΓ1us+dΓ1utα2δΓ1α=n2χαΓ12δΓ1α.
Using equations (34)–(39) in (31), we get(40)RαΓ1+QΓ2≥∑m,k,lm+k+l=αPm,k,lM2m+kΓ1RlΓ22δΓ2k+∑i=0ααi2MiΓ2χαΓ1δΓ1α−i+n2χαΓ12δΓ1α=LBQ.
Similarly, we have(41)RαΓ1+QΓ2≤∑m,k,lm+k+l=αPm,k,lM2m+kΓ1RlΓ22ΔΓ2k+∑i=0ααi2MiΓ2χαΓ1ΔΓ1α−i+n2χαΓ12ΔΓ1α=UBQ.
Equality holds if and only ifΓ1 and Γ2 are regular graphs with same regularity. This completes the proof.Example 3.
To compute lower and upper bound of GRI using formulas derived in Theorem4 for the graphs and corresponding parameters presented in Example 2. we get(42)UBQP5+QP4=∑m,k,lm+k+l=2Pm,k,lM2m+kP5RlP42ΔP4k+∑i=022i2MiP4χαP5ΔP5α−i+n2χαP52ΔP5α,UBQP5+QP4=P2,0,0M4P5R0P4+P0,2,0M2P5R0P416+P0,0,2M0P5R2P4+P1,1,0M3P5R0P44+P1,0,1M2P5RP4+P0,1,1MP5RP44+2χ2P54M0P4+4MP4+M2P4+64χ2P5=503+14316+524+22634+148+884+10044+262+10+6450=10502.
In similar way, we calculateLBQP5+QP4=4630.
To compute exact value forR2P5+QP4, we need edge partition of graph P5+QP4, and observe 8 edges with end vertex degrees 2,3, 10 edges with end vertex degrees 3,3, 26 edges with end vertex degrees 3,4, and 15 edges with end vertex degrees 4,4, respectively.
Now, we compute exact value of GRI ofP5+QP4 for α=2. R2P5+QP4=∑vw∈EP5+QP4dP5vdP4w2=8682.
Obviously,(43)4630≤R2P5+QP4=8682≤10502.
Moreover, we computed actual values along with corresponding bounds of GRI for various cases, and some are presented in Table6.Table 6
Few more cases of lower and upper bounds regardingRαΓ1+QΓ2.
Γ1Γ2αLBQ≤RαΓ1+QΓ2≤UBQP4C324266≤5772≤6918P4C3350898≤78636≤96918P5P4343674≤115386≤144422Theorem 5.
LetΓ1 and Γ2 be two simple, finite, and connected graphs. For α∈N, the general Randić index of T-sum graph is LBT≤RαΓ1+TΓ2≤UBT, where(44)LBT=∑m,k,lm+k+l=αPm,k,lM2m+kΓ1χkΓ2δΓ22l+22m+kRmΓ1Mk+2lΓ22δΓ1k+∑i=0ααi2MiΓ2χαΓ1δΓ1α−i+n2χαΓ12δΓ1α,UBT=∑m,k,lm+k+l=αPm,k,lM2m+kΓ1χkΓ2ΔΓ22l+22m+kRmΓ1Mk+2lΓ22ΔΓ1k+∑i=0ααi2MiΓ2χαΓ1ΔΓ1α−i+n2χαΓ12ΔΓ1α,and Pm,k,l=m+k+l!/m!k!l!. Equality holds if and only if Γ1 and Γ2 are regular graphs with same regularity.Proof.
In total graph, we knowdΓ1+TΓ2v,w=dΓ1+RΓ2v,w for v∈VΓ1 and w∈VΓ2 and dΓ1+TΓ2v,w=dΓ1+QΓ2v,w for v∈VTΓ1−VΓ1 and w∈VΓ2. The lower and upper for total graph can be attained as a direct consequence of Theorems 3 and 4. Moreover, the results and examples illustrated in Theorems 3 and 4 implicate that the bounds provided in Theorem 5 perform significantly well, and hence examples are omitted.
Now, we propose lower and upper bounds of general Randić index ofF-sum graphs for α∈ℛ and following results are intimately tied with generalized binomial and trinomial theorems.


## 4. Applications

### 4.1. Results for CyclesCa and Cb

LetCa and Cb be two cycle graphs with vertices a and b, respectively. Then, using Theorems 2–5, the lower and upper bounds of general Randić index of F-sum graphs Ca+SCb, Ca+RCb, Ca+QCb, and Ca+TCb, are given as follows:(45)LBS=UBS=∑m,k,lm+k+l=αPm,k,lab22m+3k+2l+∑i=0ααib222α+1,LBR=UBR=∑i=0ααiab22α+i−1+∑m,k,lm+k+l=αPm,k,lab22m+3k+2l+b2m+4k+2l+11+a−22m−1,LBQ=UBQ=∑m,k,lm+k+l=αPm,k,lab22m+3k+2l+∑i=0ααiab23α+1+ab24α,LBT=UBT=∑m,k,lm+k+l=αPm,k,lab22m+4k2k+22m+2l+∑i=0ααiab23α+1+ab24α.Note that lower and upper bounds are equal due to the reason thatCa and Cb are regular graphs with same regularity.


### 4.2. Results for PathsPa and Pb

LetPa and Pb be two path graphs with vertices a and b, respectively. Then, using Theorems 2–5, the lower and upper bounds of general Randić index of F-sum graphs Pa+SPb, Pa+RPb, Pa+QPb, and Pa+TPb, are given as(46)LBS=∑m,k,lm+k+l=αPm,k,l2k+l2+a−222m+k2+b−22l+∑i=0ααi4+b−22i+1+b2α23α+a−222α,UBS=∑m,k,lm+k+l=αPm,k,l2k+l2+a−222m+k2+b−22l+∑i=0ααi2α−i+12+b−22i+b22α23α+a−222α,LBR=∑i=0ααi4+b−22i+1+23m+2k2+a−22m+b2α23α+a−22α+∑m,k,lm+k+l=αPm,k,l2+a−222m+k23k+b−222k23m+2k+2+a−22m2+b−222k+l,UBR=∑m,k,lm+k+l=αPm,k,l22l2+a−222m+k23k+b−222k+23m+2k2+a−22m+23m+3k2+a−22m2+b−222k+l+∑i=0ααi2α−i+12+b−22i+23m+2k2+a−22m+b22α23α+a−22α,LBQ=∑m,k,lm+k+l=αPm,k,l2k+l2+a−222m+k2+b−22l+∑i=0ααi4+b−22i+1+b2α23α+a−222α,UBQ=∑m,k,lm+k+l=αPm,k,l2k+l2+a−222m+k2+b−22l+∑i=0ααi2α−i+12+b−22i+b22α23α+a−222α,LBT=∑i=0ααi4+b−22i+1+23m+2k2+a−22m+b2α23α+a−22α+∑m,k,lm+k+l=αPm,k,l2+a−222m+k23k+b−222k23m+2k+2+a−22m2+b−222k+l,UBT=∑m,k,lm+k+l=αPm,k,l22l2+a−222m+k23k+b−222k+23m+2k2+a−22m+23m+3k2+a−22m2+b−222k+l+∑i=0ααi2α−i+12+b−22i+23m+2k2+a−22m+b22α23α+a−22α.


## 4.1. Results for CyclesCa and Cb

LetCa and Cb be two cycle graphs with vertices a and b, respectively. Then, using Theorems 2–5, the lower and upper bounds of general Randić index of F-sum graphs Ca+SCb, Ca+RCb, Ca+QCb, and Ca+TCb, are given as follows:(45)LBS=UBS=∑m,k,lm+k+l=αPm,k,lab22m+3k+2l+∑i=0ααib222α+1,LBR=UBR=∑i=0ααiab22α+i−1+∑m,k,lm+k+l=αPm,k,lab22m+3k+2l+b2m+4k+2l+11+a−22m−1,LBQ=UBQ=∑m,k,lm+k+l=αPm,k,lab22m+3k+2l+∑i=0ααiab23α+1+ab24α,LBT=UBT=∑m,k,lm+k+l=αPm,k,lab22m+4k2k+22m+2l+∑i=0ααiab23α+1+ab24α.Note that lower and upper bounds are equal due to the reason thatCa and Cb are regular graphs with same regularity.


## 4.2. Results for PathsPa and Pb

LetPa and Pb be two path graphs with vertices a and b, respectively. Then, using Theorems 2–5, the lower and upper bounds of general Randić index of F-sum graphs Pa+SPb, Pa+RPb, Pa+QPb, and Pa+TPb, are given as(46)LBS=∑m,k,lm+k+l=αPm,k,l2k+l2+a−222m+k2+b−22l+∑i=0ααi4+b−22i+1+b2α23α+a−222α,UBS=∑m,k,lm+k+l=αPm,k,l2k+l2+a−222m+k2+b−22l+∑i=0ααi2α−i+12+b−22i+b22α23α+a−222α,LBR=∑i=0ααi4+b−22i+1+23m+2k2+a−22m+b2α23α+a−22α+∑m,k,lm+k+l=αPm,k,l2+a−222m+k23k+b−222k23m+2k+2+a−22m2+b−222k+l,UBR=∑m,k,lm+k+l=αPm,k,l22l2+a−222m+k23k+b−222k+23m+2k2+a−22m+23m+3k2+a−22m2+b−222k+l+∑i=0ααi2α−i+12+b−22i+23m+2k2+a−22m+b22α23α+a−22α,LBQ=∑m,k,lm+k+l=αPm,k,l2k+l2+a−222m+k2+b−22l+∑i=0ααi4+b−22i+1+b2α23α+a−222α,UBQ=∑m,k,lm+k+l=αPm,k,l2k+l2+a−222m+k2+b−22l+∑i=0ααi2α−i+12+b−22i+b22α23α+a−222α,LBT=∑i=0ααi4+b−22i+1+23m+2k2+a−22m+b2α23α+a−22α+∑m,k,lm+k+l=αPm,k,l2+a−222m+k23k+b−222k23m+2k+2+a−22m2+b−222k+l,UBT=∑m,k,lm+k+l=αPm,k,l22l2+a−222m+k23k+b−222k+23m+2k2+a−22m+23m+3k2+a−22m2+b−222k+l+∑i=0ααi2α−i+12+b−22i+23m+2k2+a−22m+b22α23α+a−22α.


## 5. Conclusion

For researchers, determining the bounds for pertinent topological index is always intriguing and attractive problem. In [45], Liu et al. proposed an open problem regarding GRI of four operations (F-sum) on graphs. This paper, potentially, addressed and figured out the bounds on GRI for the F-sum graphs Γ1+SΓ2, Γ1+RΓ2, Γ1+QΓ2, and Γ1+TΓ2 (α∈ℕ) in terms of eminent TIs of their base graphs and graph parameters. Several examples for different combinations of base graphs Γ1 and Γ2 along with different parameter α are explored. It can be observed that bounds obtained performed well when compared with exact value. It is worth mentioning that with the help of these four graph operations, one can construct the molecular graph of his own choice and GRI quantifies information of resulting molecular graph. Computing the bounds of GA-index and ABC-index (variations of GRI) for the F-sum graphs could be an interesting problem for future investigation.



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