# Computing Bounds for General Randic Coindex of Sum Graphs

Authors: Muhammad Javaid; Muhammad Ibraheem; Ebenezer Bonyah
Journal: Journal of Mathematics
 (2021)

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## Abstract

The physical and structural properties of molecular structure or graph such as boiling point, melting point, surface tension, or solubility are studied using topological index (TI). Topological index is a mathematical formula that can be applied to any graph which models some molecular structures. The various operations play an important role in graph theory such as joining, union, intersection, products, and subdivision. In this paper, we computed the bounds for general Randic coindex ofF-sum graphs such as (S-sum, R-sum, Q-sum, and T-sum) in the form of their factor graphs. At the end, results are illustrated by numerical table for the particular F-sum graphs.


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## Body

## 1. Introduction

Graph theory is playing an important role in various sciences particularly in computer science and mathematical chemistry. The branch of mathematics which combines chemistry and graph theory is called chemical graph theory. The molecular structure descriptors have been used for quantifying information on molecules. This relates to characterizing physico-chemical, toxicologic, pharmacologic, biological, and other properties of chemical compounds by utilizing topological index (TI). TI is a mathematical formula that can be applied to any graph which models some molecular structures. It is an efficient mathematical method in avoiding laboratory experiments and time consumption [1, 2].Actually, various TIs are introduced in order to describe physical and chemical properties of molecules. These indices are divided into different classes, namely, degree-based, distance-based, and polynomial-based, but the degree-based class is studied more than others, see the latest survey [3]. In 1947, Wiener calculated the boiling point of paraffin using a degree-based TI, see [4]. Gutman and Trinajstic calculated total π-electron energy of hydrocarbons using degree-based first and second Zagreb indices [5]. Li and Zheng provided the idea of first general Zagreb index (FGZI) [6].The Randic index was proposed by Randic in 1975 and has been widely studied in different areas. Li and Shi [7] calculated the extremal values of Randic index and its higher-order, zeroth-order, and general form for the extremal graphs. Delorme et al. [8] proved a best-possible lower bound for triangle-free graph with minimum degree of graph and Gutman et al. [9] point out a hitherto unnoticed feature of a molecular graph for Randic index. Arizmendi and Arizmendi proved that graph energy is twice of the Randic index and investigated that the equality holds iff graph is the union of complete bipartite graphs [10]. Li and Yang calculated the bounds for graphs whose general Randic indices reach the maximum and minimum [11]. Furtula and Gutman calculated Randic energy value of the connected graph with a fixed number of vertices [12]. Gao and Lu calculated the sharp bounds for the unicyclic graphs [13] and Li et al. [14] computed the bounds for of chemical n−m graphs for general Randic index.Ma et al. [15] gave a brief review for the Randic from 1975 to date such as zeroth-order Randic indices, sum-connectivity indices, geometric-arithmetic indices, Randic spectrum and energy, harmonic index, Randic matrix, D-L-S generalization, Balaban index, and atom-bond connectivity index. Milovanovic et al. [16] wrote a note and calculated some mathematical properties of the general zeroth-order Randic coindex of graphs in [17].In the development of new graphs, the various operations play an important role in graph theory such as joining, union, intersection, products, and subdivision. Yan et al. [18] listed five different operations 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; further, they computed Wiener index of these graphs. Eliasi and Taeri introduced the F-sum graphs such as G1+FG2, where F∈S,R,Q,T, and calculated the Wiener indices of graphs in [19]. Later on, many researchers worked on these F-sum graphs such as Sarala et al. computed first and second Zagreb indices [20], Imran et al. [21] investigated the bounds of degree-based topological indices such as bounds of Zagreb indices, multiple Zagreb indices, the atom-bond connectivity (ABC) index, the forgotten topological index, the geometric-arithmetic (GA) index, and the Narumi-Katayama index, and Li et al. [22] computed bounds on general Randic indexes. Javaid et al. [23] calculated bounds for second Zagreb coindex, Akhter and Imran [24] calculated the forgotten topological index, Liu et al. [25] computing first general Zagreb index of operations on graphs, and Javaid et al. [26] calculated the Zagreb coindex and connection index of these graphs.In this article, we investigated the sharp bounds for general Randic coindex of graphs that are obtained by using subdivisions related operations such asR¯αG1+SG2, R¯αG1+RG2, R¯αG1+QG2, and R¯αG1+TG2. The rest of the paper is organized as follows: Section 2 contains preliminaries and notations, Section 3 contains the main theorems of the work, and Section 4 contains conclusion of the work; further, the results are illustrated using examples for some particular F-sum graphs.


## 2. Preliminaries

LetVG be nonempty set vertices and EG⊆VG⊗VG be the set of edges, then by combining both VGandEG, a graph is formed that is denoted by G=VG,EG. The cardinality of vertex set is called order of graph and cardinality of edge set is known as size of the graph which are denoted by VG=n and EG=m, respectively. Let v∈VG, then its degree is denoted by dv and defined as number of edges incident on it. Let G be a graph; its maximum and minimum degrees are denoted by ΔG and δG, respectively. For any graph G, its complement is denoted by G¯ and defined as uv∈EG¯ iff uv∉EG¯. Gutman and Trinajstic [5] introduced degree-based TIs known as Zagreb indices. Now, we define first and second Zagreb indices and coindices for any G,(1)M1G=∑x1x2∈EGdGx1+dGx2,M2G=∑x1x2∈EGdGx1dGx2,M¯1G=∑x1x2∉EGdGx1+dGx2,M¯2G=∑x1x2∉EGdGx1dGx2.Li and Zheng [6] introduced the first general Zagreb index that is defined as(2)M1αG=∑x1x2∈EGdGx1α−1+dGx2α−1.By puttingα=2 and α=3, we obtained the first Zagreb index and forgotten index, respectively.Zhou and Trinajstic [27] introduced the general sum-connectivity index (GSCI) denoted by χkG after that general sum-connectivity coindex was introduced which is denoted by χ¯kG; these are defined as(3)χkG=∑x1x2∈EGdGx1+dGx2k,χ¯kG=∑x1x2∉EGdGx1+dGx2k.Bollobas and Erdos [28] introduced the concept of general Randic coindex denoted RαG; its coindex is denoted by R¯αG which is defined as(4)RαG=∑x1x2∈EHdGx1dGx2α,R¯αG=∑x1x2∉EGdGx1dGx2α.The binomial and trinomial theorems are very important while expanding expression of those described as(5)y1+y2n=∑i=0nniy1n−iy2i,(6)y1+y2+y3n=∑m,k,lm,k,lm+k+l=nPm,k,ly1my2ky3l,where Pm,k,l=m+k+l/m!k!l!.LetG be a graph, then SG is known as edge subdivision graph that is obtained inserting a vertex in each edge of G, RG is called triangle parallel graph that is obtained from SG by joining an edge between the adjacent vertices of G, QG is called superposition graph obtained from SG by joining an edge between the pairs of new vertices which are on the adjacent edges of G, and TG is called total graph obtained by performing both operations of RG and QG on SG.LetG1 and G2 be two simple connected graphs, then their F−sum graph with vertex set VG1+FG2=VG1∪EG1×VG2 and s1,s2x1,x2∈EG1+FG2 iff s1=x1∈VG1 and s2∽x2∈G2, s2=x2∈VG2 and s1∽x1∈FG1, where F∈S,R,Q,T, For details, see Figures 1 and 2.Figure 1
GraphP4 and its subdivision.Figure 2
GraphsG≅P4, H≅P2, and G+FH≅P4+FP2.


## 3. Main Results

This section contains results of the bounds for general Randic coindex.(7)α1=∑s1s2∉ESG1s1∈VG1s2∈VSG1−VG1dSG1s1dSs2,α2=∑s1,s2∈VSG1−VG1∑x1,x2∈VG2ds1,x1s2,x2α.Theorem 1.
LetG1+SG2 be an S-sum graph, then its general Randic coindex R¯αG1+SG2 is given as(8)LB≤R¯αG1+SG2≤UB,where(9)LB=∑m,k,lm,k,lm+k+l=αPm,k,lM12m+kG1χ¯kG2δG22l+δG12mM12l+kG2χkG1+χ¯kG1+δG22lM1kG1+M¯1kG1M1kG2+M¯1kG2+∑i=0ααiδiSG1M2n−iSG1M1iG2+M¯1iG2+α1n−iM1iG2+M¯1iG2+22α−1n22e12−n2e1,UB=∑m,k,lm,k,lm+k+l=αPm,k,lM12m+kG1χ¯kG2ΔG22l+ΔG12mM12l+kG2χkG1+χ¯kG1+ΔG22lM1kG1+M¯1kG1M1kG2+M¯1kG2+∑i=0ααiΔiSG1M2n−iSG1M1iG2+M¯1iG2+α1n−iM1iG2+M¯1iG2+22α−1n22e12−n2e1.Proof.
Using equation (4), we have(10)R¯αG1+SG2=∑A+∑B+∑C.
Consider(11)∑A=∑s1,s2∈VSG1−VG1∑x1,x2∈VG2ds1,x1s2,x2α=∑s1,s2∈VSG1−VG1∑x1,x2∈VG2dSG1s1dSG1s2α=∑s1,s2∈VSG1−VG1∑x1,x2∈VG22×2α,∑A=22α−1n22e12−n2e1,∑B=∑i=17Bi,∑B1=∑s∈VG1∑x1x2∉EG2ds,x1ds,x2α=∑s∈VG1∑x1x2∉EG2dG1s+dG2x1dG1s+dG2x2α=∑s∈VG1∑x1x2∉EG2dG12s+dG1sdG2x2+dG2x1+dG2x1dG2x2α.
Using equation (6),(12)=∑s∈VG1∑x1x2∉EG2∑m,k,lm,k,lm+k+l=αPm,k,ldG12msdG1ksdG2x2+dG2x1kdG2x1dG2x2l=∑m,k,lm,k,lm+k+l=αPm,k,l∑s∈VG1∑x1x2∉EG2dG12m+ksdG2x2+dG2x1kdG2x1dG2x2l.
As we know,δG≤dx≤ΔG,x∈VG,(13)≤∑m,k,lm,k,lm+k+l=αPm,k,lM12m+kG1χ¯kG2ΔG22l,∑B2=∑s1s2∈EG1∑x∈VG2ds1,xds2,xα=∑s1s2∈EG1∑x∈VG2dG1s1+dG2xdG1s1+dG2xα=∑s1s2∈EG1∑x∈VG2dG1s1dG1s2+dG2xdG1s1+dG1s2+dG2xdG2xα=∑s1s2∈EG1∑x∈VG2∑m,k,lm,k,lm+k+l=αPm,k,ldG1s1dG1s2mdG2xdG1s1+dG1s2kdG2x2l=∑m,k,lm,k,lm+k+l=αPm,k,l∑s1s2∈EG1∑x∈VG2dG1s1dG1s2mdG1s1+dG1s2kdG2x2l+k≤∑m,k,lm,k,lm+k+l=αPm,k,lM12l+kG2χkG1ΔG12m,∑B3=∑s1s2∉EG1∑x∈VG2ds1,xds2,xα=∑s1s2∉EG1∑x∈VG2dG1s1+dG2xdG1s1+dG2xα=∑s1s2∉EG1∑x∈VG2dG1s1dG1s2+dG2xdG1s1+dG1s2+dG2xdG2xα=∑s1s2∉EG1∑x∈VG2dG1s1dG1s2mdG2xdG1s1+dG1s2kdG2x2l=∑s1s2∉EG1∑x∈VG2∑m,k,lm,k,lm+k+l=αPm,k,ldG1s1dG1s2mdG1s1+dG1s2kdG2x2l+k≤∑m,k,lm,k,lm+k+l=αPm,k,lM12l+kG2χ¯kG1ΔG12m,∑B4=∑s1s2∈EG1∑x1x2∈EG2ds1,x1ds2,x2α=∑s1s2∈EG1∑x1x2∈EG2dG1s1+dG2x1dG1s2+dG2x2α=∑s1s2∈EG1∑x1x2∈EG2dG1s1dG1s2+dG1s1dG2x2+dG1s2dG2x1+dG2x1dG2x2α=∑m,k,lm,k,lm+k+l=αPm,k,l∑s1s2∈EG1∑x1x2∈EG2dG1s1dG1s2mdG1s1dG2x2+dG1s2dG2x1kdG2x1dG2x2l≤∑m,k,lm,k,lm+k+l=αPm,k,lΔG12mΔG22lM1kG1M1kG2,∑B5=∑s1s2∉EG1∑x1x2∈EG2ds1,x1ds2,x2α,=∑s1s2∉EG1∑x1x2∈EG2dG1s1+dG2x1dG1s2+dG2x2α=∑s1s2∉EG1∑x1x2∈EG2dG1s1dG1s2+dG1s1dG2x2+dG1s2dG2x1+dG2x1dG2x2α=∑s1s2∉EG1∑x1x2∈EG2∑m,k,lm,k,lm+k+l=αPm,k,ldG1s1dG1s2mdG1s1dG2x2+dG1s2dG2x1kdG2x1dG2x2l≤∑m,k,lm,k,lm+k+l=αPm,k,lΔG12mΔG22lM¯1kG1M1kG2,∑B6=∑s1s2∈EG1∑x1x2∉EG2ds1,x1ds2,x2α=∑s1s2∉EG1∑x1x2∈EG2dG1s1+dG2x1dG1s2+dG2x2α=∑s1s2∉EG1∑x1x2∉EG2dG1s1dG1s2+dG1s1dG2x2+dG1s2dG2x1+dG2x1dG2x2α=∑s1s2∉EG1∑x1x2∉EG2∑m,k,lm,k,lm+k+l=αPm,k,ldG1s1dG1s2mdG1s1dG2x2+dG1s2dG2x1kdG2x1dG2x2l≤∑m,k,lm,k,lm+k+l=αPm,k,lΔG12mΔG22lM1kG1M¯1kG2,∑B7=∑s1s2∈EG1∑x1x2∈EG2ds1,x1ds2,x2α=∑s1s2∉EG1∑x1x2∈EG2dG1s1+dG2x1dG1s2+dG2x2α=∑s1s2∉EG1∑x1x2∉EG2dG1s1dG1s2+dG1s1dG2x2+dG1s2dG2x1+dG2x1dG2x2α=∑s1s2∉EG1∑x1x2∉EG2∑m,k,lm,k,lm+k+l=αPm,k,ldG1s1dG1s2mdG1s1dG2x2+dG1s2dG2x1kdG2x1dG2x2l≤∑m,k,lm,k,lm+k+l=αPm,k,lΔG12mΔG22lM¯1kG1M¯1kG2,∑B≤∑m,k,lm,k,lm+k+l=αPm,k,lM12m+kG1χ¯kG2ΔG22l+ΔG12mM12l+kG2χkG1+χ¯kG1+ΔG22lM1kG1+M¯1kG1M1kG2+M¯1kG2,∑C=∑i=15Ci,∑C1=∑s1s2∈ESG1s1∈VG1s2∈VSG1−VG1∑x1,x2∈VG2ds1,xds2,xα=∑s1s2∈ESG1s1∈VG1s2∈VSG1−VG1∑x1,x2∈VG2dSs1+dG2x1dSs2α=∑s1s2∈ESG1s1∈VG1s2∈VSG1−VG1∑x1,x2∈VG2dSG1s1dSs2+dG2x1dSG1s2α=∑s1s2∈ESG1s1∈VG1s2∈VSG1−VG1∑x1,x2∈VG2∑i=0ααidSG1s1dSs2n−idG2x1idSG1s2i≤∑i=0ααiM2n−iSG1M1iG2ΔiSG1,∑C2=∑s1s2∈ESG1s1∈VG1s2∈VSG1−VG1∑x1,x2∉VG2ds1,xds2,xα=∑s1s2∈ESG1s1∈VG1s2∉VSG1−VG1∑x1,x2∉VG2dSs1+dG2x1dSs2α=∑s1s2∈ESG1s1∈VG1s2∈VSG1−VG1∑x1,x2∉VG2dSG1s1dSs2+dG2x1dSG1s2α=∑s1s2∈ESG1s1∈VG1s2∈VSG1−VG1∑x1,x2∉VG2∑i=0ααidSG1s1dSs2n−idG2x1idSG1s2i≤∑i=0ααiM2n−iSG1M¯1iG2ΔiSG1,∑C3=∑s1s2∉ESG1s1∈VG1s2∈VSG1−VG1∑x∈VG2ds1,xds2,xα=∑s1s2∉ESG1s1∈VG1s2∈VSG1−VG1∑x∈VG2dSs1+dG2xdSs2α=∑s1s2∉ESG1s1∈VG1s2∈VSG1−VG1∑x∈VG2dSG1s1dSs2+dG2xdSG1s2α=∑s1s2∉ESG1s1∈VG1s2∈VSG1−VG1∑x∈VG2∑i=0ααidSG1s1dSs2n−idG2xidSG1s2i≤∑i=0ααiα1n−iM1iG2ΔiSG1,∑C4=∑s1s2∉ESG1s1∈VG1s2∈VSG1−VG1∑x1,x2∈VG2ds1,xds2,xα=∑s1s2∉ESG1s1∈VG1s2∈VSG1−VG1∑x1,x2∈VG2dSs1+dG2x1dSs2α=∑s1s2∉ESG1s1∈VG1s2∈VSG1−VG1∑x1,x2∈VG2dSG1s1dSs2+dG2x1dSG1s2α=∑s1s2∉ESG1s1∈VG1s2∈VSG1−VG1∑x1,x2∈VG2∑i=0ααidSG1s1dSs2n−idG2x1idSG1s2i≤∑i=0ααiα1n−iM1iG2ΔiSG1,∑C5=∑s1s2∉ESG1s1∈VG1s2∈VSG1−VG1∑x1,x2∉VG2ds1,xds2,xα=∑s1s2∉ESG1s1∈VG1s2∉VSG1−VG1∑x1,x2∈VG2dSs1+dG2x1dSs2α=∑s1s2∉ESG1s1∈VG1s2∈VSG1−VG1∑x1,x2∉VG2dSG1s1dSs2+dG2x1dSG1s2α=∑s1s2∉ESG1s1∈VG1s2∈VSG1−VG1∑x1,x2∉VG2∑i=0ααidSG1s1dSs2n−idG2x1idSG1s2i≤∑i=0ααiα1n−iM¯1iG2ΔiSG1.
We obtained upper bound by putting value of∑A, ∑B, and ∑C in equation (13). Similarly, lower bound can be obtained using smallest degree of graphs G1 and G2.Theorem 2.
LetG1+RG2 be a R-sum graph, then its general Randic coindex R¯αG1+RG2 is given as:(14)LB≤R¯αG1+RG2≤UB,where(15)LB=∑m,k,lm,k,lm+k+l=αPm,k,l22m+kM12m+kG1χ¯kG2δG22l+δG12mM12l+kG2χkG1+χ¯kG1+δG22lM1kG1+M¯1kG1M1kG2+M¯1kG2+∑i=0ααiδiSG1M2n−iSG1M1iG2+M¯1iG2+α1n−iM1iG2+M¯1iG2+22α−1n22e12−n2e1,UB=∑m,k,lm,k,lm+k+l=αPm,k,l22m+kM12m+kG1χ¯kG2ΔG22l+ΔG12mM12l+kG2χkG1+χ¯kG1+ΔG22lM1kG1+M¯1kG1M1kG2+M¯1kG2+∑i=0ααiΔiSG1M2n−iSG1M1iG2+M¯1iG2+α1n−iM1iG2+M¯1iG2+22α−1n22e12−n2e1.Proof.
Using equation (4), we have(16)R¯αG1+RG2=∑A+∑B+∑C.
The value of∑A follows from equation (11),(17)∑A=22α−1n22e12−n2e1,∑B=∑i=17Bi,∑B1=∑s∈VG1∑x1x2∉EG2ds,x1ds,x2α=∑s∈VG1∑x1x2∉EG2dRG1s+dG2x1dRG1s+dG2x2α=∑s∈VG1∑x1x2∉EG24dG1sdG1s+2dG1sdG2x2+dG1sdG2x1+dG2x1dG2x2α=∑s∈VG1∑x1x2∉EG24dG12s+2dG1sdG2x2+dG2x1+dG2x1dG2x2α=∑s∈VG1∑x1x2∉EG2∑m,k,lm,k,lm+k+l=αPm,k,l4mdG12ms2kdG1ksdG2x2+dG2x1kdG2x1dG2x2l=∑s∈VG1∑x1x2∉EG2∑m,k,lm,k,lm+k+l=αPm,k,l22m+kdG12m+ksdG2x2+dG2x1kdG2x1dG2x2l≤22m+k∑m,k,lm,k,lm+k+l=αPm,k,lM12m+kG1χ¯kG2ΔG22l,∑B2=∑s1s2∈EG1∑x∈VG2ds1,xds2,xα=∑s1s2∈EG1∑x∈VG2dRG1s1+dG2xdRG1s1+dG2xα=∑s1s2∈EG1∑x∈VG2dRG1s1dRG1s2+dG2xdRG1s1+dRG1s2+dG2xdG2xα=∑s1s2∈EG1∑x∈VG2∑m,k,lm,k,lm+k+l=αPm,k,l4dG1s1dG1s2m2dG2xdG1s1+dG1s2kdG2x2l=22m+k∑s1s2∈EG1∑x∈VG2∑m,k,lm,k,lm+k+l=αPm,k,ldG1s1dG1s2mdG1s1+dG1s2kdG2x2l+k≤22m+k∑m,k,lm,k,lm+k+l=αPm,k,lM12l+kG2χkG1ΔG12m,∑B3=∑s1s2∉EG1∑x∈VG2ds1,xds2,xα=∑s1s2∈EG1∑x∉VG2dRG1s1+dG2xdRG1s1+dG2xα=∑s1s2∉EG1∑x∈VG2dRG1s1dRG1s2+dG2xdRG1s1+dRG1s2+dG2xdG2xα=∑s1s2∉EG1∑x∈VG2∑m,k,lm,k,lm+k+l=αPm,k,l4dG1s1dG1s2m2dG2xdG1s1+dG1s2kdG2x2l=22m+k∑s1s2∉EG1∑x∈VG2∑m,k,lm,k,lm+k+l=αPm,k,ldG1s1dG1s2mdG1s1+dG1s2kdG2x2l+k≤22m+k∑m,k,lm,k,lm+k+l=αPm,k,lM12l+kG2χ¯kG1ΔG12m,∑B4=∑s1s2∈EG1∑x1x2∈EG2ds1,x1ds2,x2α=∑s1s2∈EG1∑x1x2∈EG2dRG1s1+dG2x1dRG1s2+dG2x2α=∑s1s2∈EG1∑x1x2∈EG2dRG1s1dRG1s2+dRG1s1dG2x2+dRG1s2dG2x1+dG2x1dG2x2α=∑s1s2∈EG1∑x1x2∈EG2∑m,k,lm,k,lm+k+l=αPm,k,l4dG1s1dG1s2m2dG1s1dG2x2+dG1s2dG2x1kdG2x1dG2x2l≤22m+k∑m,k,lm,k,lm+k+l=αPm,k,lΔG12mΔG22lM1kG1M1kG2,∑B5=∑s1s2∉EG1∑x1x2∈EG2ds1,x1ds2,x2α=∑s1s2∉EG1∑x1x2∈EG2dRG1s1+dG2x1dRG1s2+dG2x2α=∑s1s2∉EG1∑x1x2∈EG2dRG1s1dRG1s2+dRG1s1dG2x2+dRG1s2dG2x1+dG2x1dG2x2α=∑s1s2∉EG1∑x1x2∈EG2∑m,k,lm,k,lm+k+l=αPm,k,l4dG1s1dG1s2m2dG1s1dG2x2+dG1s2dG2x1kdG2x1dG2x2l≤22m+k∑m,k,lm,k,lm+k+l=αPm,k,lΔG12mΔG22lM¯1kG1M1kG2,∑B6=∑s1s2∈EG1∑x1x2∉EG2ds1,x1ds2,x2α=∑s1s2∉EG1∑x1x2∈EG2dRG1s1+dG2x1dRG1s2+dG2x2α=∑s1s2∉EG1∑x1x2∉EG2dRG1s1dRG1s2+dRG1s1dG2x2+dRG1s2dG2x1+dG2x1dG2x2α=∑s1s2∉EG1∑x1x2∉EG2∑m,k,lm,k,lm+k+l=αPm,k,l4dG1s1dG1s2m2dG1s1dG2x2+dG1s2dG2x1kdG2x1dG2x2l≤22m+k∑m,k,lm,k,lm+k+l=αPm,k,lΔG12mΔG22lM1kG1M¯1kG2,∑B7=∑s1s2∈EG1∑x1x2∈EG2ds1,x1ds2,x2α=∑s1s2∉EG1∑x1x2∈EG2dRG1s1+dG2x1dRG1s2+dG2x2α=∑s1s2∉EG1∑x1x2∉EG2dRG1s1dRG1s2+dRG1s1dG2x2+dRG1s2dG2x1+dG2x1dG2x2α=∑s1s2∉EG1∑x1x2∉EG2∑m,k,lm,k,lm+k+l=αPm,k,l4dG1s1dG1s2m2dG1s1dG2x2+dG1s2dG2x1kdG2x1dG2x2l≤22m+k∑m,k,lm,k,lm+k+l=αPm,k,lΔG12mΔG22lM¯1kG1M¯1kG2,∑B=∑m,k,lm,k,lm+k+l=αPm,k,l22m+kM12m+kG1χ¯kG2ΔG22l+ΔG12mM12l+kG2χkG1+χ¯kG1+ΔG22lM1kG1+M¯1kG1M1kG2+M¯1kG2,∑C=∑i=13Ci,∑C1=∑s1s2∈ERG1s1∈VG1s2∈VRG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2ds1,xds2,xalpha=∑s1s2∈ERG1s1∈VG1s2∉VRG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2dRs1+dG2x1dRs2α=∑s1s2∈ERG1s1∈VG1s2∉VRG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2dRs1+dG2x1dRs2α=∑s1s2∈ERG1s1∈VG1s2∈VRG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2dRG1s1dRs2+dG2x1dRG1s2α=∑s1s2∈ERG1s1∈VG1s2∈VRG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2∑i=0ααidRG1s1dRs2dG2x1idRG1s2i≤∑i=0ααiM2n−iRG1ΔiRG1M1iG2+M¯1iG2,∑C2=∑s1s2∉ERG1s1∈VG1s2∈VRG1−VG1∑x∈VG2ds1,xds2,xα=∑s1s2∉ERG1s1∈VG1s2∈VRG1−VG1∑x∈VG2dRs1+dG2xdRs2α=∑s1s2∉ERG1s1∈VG1s2∈VRG1−VG1∑x∈VG2dRG1s1dRs2+dG2xdRG1s2α=∑s1s2∉ERG1s1∈VG1s2∈VRG1−VG1∑x∈VG2∑i=0ααidRG1s1dRs2n−idG2xidRG1s2i≤∑i=0ααiα1n−iM1iG2ΔiRG1,∑C3=∑s1s2∉ERG1s1∈VG1s2∈VRG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2ds1,xds2,xα=∑s1s2∉ERG1s1∈VG1s2∈VRG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2dRs1+dG2x1dRs2α=∑s1s2∉ERG1s1∈VG1s2∈VRG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2dRs1+dG2x1dRs2α=∑s1s2∉ERG1s1∈VG1s2∈VRG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2dRG1s1dRs2+dG2x1dRG1s2α=∑s1s2∉ERG1s1∈VG1s2∈VRG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2∑i=0ααidRG1s1dRs2n−idG2x1idRG1s2i≤∑i=0ααiα1n−iΔiRG1M1iG2+M¯1iG2.
We obtained upper bound by putting value of∑A, ∑B, and ∑C in equation (16). Similarly, lower bound can be obtained using smallest degree of graphs G1 and G2.Theorem 3.
LetG1+QG2 be a Q-sum graph, then its general Randic coindex R¯αG1+QG2 is given as(18)LB≤R¯αG1+QG2≤UB,where(19)LB=∑m,k,lm,k,lm+k+l=αPm,k,lM12m+kG1χ¯kG2δG22l+δG12mM12l+kG2χkG1+χ¯kG1+δG22lM1kG1+M¯1kG1M1kG2+M¯1kG2+∑i=0ααiδiQG1M2n−iQG1M1iG2+M¯1iG2+α1n−iM1iG2+M¯1iG2+α2,UB=∑m,k,lm,k,lm+k+l=αPm,k,lM12m+kG1χ¯kG2ΔG22l+ΔG12mM12l+kG2χkG1+χ¯kG1+ΔG22lM1kG1+M¯1kG1M1kG2+M¯1kG2+∑i=0ααiΔiQG1M2n−iQG1M1iG2+M¯1iG2+α1n−iM1iG2+M¯1iG2+α2.Proof.
Using equation (4), we have(20)R¯αG1+QG2=∑A+∑B+∑C.
Consider(21)∑A=∑s1,s2∈VQG1−VG1∑x1,x2∈VG2ds1,x1s2,x2α=∑s1,s2∈VQG1−VG1∑x1,x2∈VG2dQG1s1dQG1s2=α2.
The value of∑B follows from equation (17),(22)∑C=∑i=13Ci,∑C1=∑s1s2∈EQG1s1∈VG1s2∈VQG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2ds1,xds2,xα=∑s1s2∉EQG1s1∈VG1s2∈VQG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2dQs1+dG2x1dQs2α=∑s1s2∉EQG1s1∈VG1s2∈VQG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2dQs1+dG2x1dQs2=∑s1s2∉EQG1s1∈VG1s2∈VQG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2dQG1s1dQs2+dG2x1dQG1s2α=∑s1s2∉EQG1s1∈VG1s2∈VQG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2∑i=0ααidQG1s1dQs2n−idG2x1idQG1s2i≤∑i=0ααiM2n−iQG1ΔiQG1M1iG2+M¯1iG2,∑C2=∑s1s2∉EQG1s1∈VG1s2∈VQG1−VG1∑x∈VG2ds1,xds2,xα=∑s1s2∉EQG1s1∈VG1s2∈VQG1−VG1∑x∈VG2dQs1+dG2xdQs2α=∑s1s2∉EQG1s1∈VG1s2∈VQG1−VG1∑x∈VG2dQG1s1dQs2+dG2xdQG1s2α=∑s1s2∉EQG1s1∈VG1s2∈VQG1−VG1∑x∈VG2∑i=0ααidQG1s1dQs2n−idG2xidQG1s2i≤∑i=0ααiα1n−iM1iG2ΔiQG1,∑C3=∑s1s2∉EQG1s1∈VG1s2∈VQG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2ds1,xds2,xα=∑s1s2∉EQG1s1∈VG1s2∈VQG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2dQs1+dG2x1dQs2α=∑s1s2∉EQG1s1∈VG1s2∈VQG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2dQs1+dG2x1dQs2α=∑s1s2∉EQG1s1∈VG1s2∈VQG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2dQG1s1dQs2+dG2x1dQG1s2α=∑s1s2∉EQG1s1∈VG1s2∈VQG1−VG1∑x1,x2∈VG2+∑x1,x2∉VG2∑i=0ααidQG1s1dQs2n−idG2x1idQG1s2i≤∑i=0ααiα1n−iΔiQG1M1iG2+M¯1iG2.
We obtained upper bound by putting value of∑A, ∑B, and ∑C in equation (20). Similarly, lower bound can be obtained using smallest degree of graphs G1 and G2.Theorem 4.
LetG1+TG2 be a T-sum graph, then its general Randic coindex R¯αG1+TG2 is given as(23)LB≤R¯αG1+TG2≤UB,where(24)LB=∑m,k,lm,k,lm+k+l=αPm,k,l22m+kM12m+kG1χ¯kG2δG22l+δG12mM12l+kG2χkG1+χ¯kG1+δG22lM1kG1+M¯1kG1M1kG2+M¯1kG2+∑i=0ααiδiTG1M2n−iTG1M1iG2+M¯1iG2+α1n−iM1iG2+M¯1iG2α2,UB=∑m,k,lm,k,lm+k+l=αPm,k,l22m+kM12m+kG1χ¯kG2ΔG22l+ΔG12mM12l+kG2χkG1+χ¯kG1+ΔG22lM1kG1+M¯1kG1M1kG2+M¯1kG2+∑i=0ααiΔiTG1M2n−iTG1M1iG2+M¯1iG2+α1n−iM1iG2+M¯1iG2+α2.
It follows from Theorems2 and 3.Table 1
Bounded values of certainF-sum graphs.
F-sum operationLower boundsExact valuesUpper boundsR¯2P3+SP23247841076R¯2P3+RP2164025026784R¯2P3+QP295814321624R¯2P3+TP23604377010692


## 4. Conclusion

(i)
In this paper, we calculated the upper and lower bounds of general Randic coindex forF-sum graphs such as R¯αG1+SG2, R¯αG1+RG2, R¯αG1+QG2, and R¯αG1+TG2 in the form of Zagreb indices and coincides of their factor graphs.(ii)
We illustrate the results using two graphsG1≅Cn and G2≅Pn, then bounded values of general Randic coindex are given in Table 1.



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