# The Reformulated F-Index of Vertex and Edge F-Join of Graphs

Authors: Durbar Maji; Ganesh Ghorai; Faria Ahmed Shami
Journal: Journal of Chemistry
 (2022)

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## Abstract

The valency based topological indices (TIs) are defined by algebraic functions as the chemical graph theory tools in which the structural parameters are used as input and the output is related with the topology of chemical species. In theoretical chemistry, the TIs are mainly used to investigate/develop the QSAR and QSPR investigations of the molecular graphs. The Reformulated F-index or RFI is one such kind of TIs. The RFI for a molecular graph Y is defined asRFY=∑uv∈EYdYu+dYv−23, where dYu is the degree of a vertex u in Y. In this paper, we study the RFI for vertex and edge version of F-join of given general graphs which are related to subdivision and total graph.


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## Body

## 1. Introduction, Basic Definitions and Notations

In mathematical chemistry (MC)/chemical graph theory (CGT) molecules are often represented by graphical method which show relationship between the atoms and bonds in those molecules and by graphs namely “molecular graphs”. These graphs characterize the bonding topology of the molecular graphs whose vertices are assumed as atoms of the molecule and edges are the bond between them. In MC/CGT, TIs are the numerical criterion/two dimensional descriptors which consider the internal atomic setting compounds as input and the output results provide the fact in the numerical form corresponding the molecular shape, size, existence of multiple bonds, etc. The TI is one of the successful mathematical toots or a theoretical pathway by which it can be established the quantitative relationship between a chemical structure and its properties. Mathematically, a TI can be expressed by a function such thatF:T⟶ℝ defined by FY=∑u∈VYTu (T(u) is the valency-function from the vertex-set of Y to the degree-set of its vertices) for any two isomorphic graphs Y1 and Y2 such that FY1=FY2, where Y1,Y2∈T and ℝ stand for the set of all molecular graphs and set real numbers.Throughout this paper all the graphs are considered as simple, connected and finite. Every chemical graph can be modeled by simple graph to correlate with their chemical structure. ConsiderY is such a simple graph having order n and m; while the vertex and edge set are abbreviated by VY and EY, respectively. The notation dYu expressed by the number of those vertices which are adjacent to u, is called the degree of u in Y.The first time idea about the TI was propounded by the chemist, wiener in 1947. I. Gutman et al. introduced the first Zagreb indexM1Y in [1] and second Zagreb index M2Y in [2], Furtula et al. defined the F-index in [3], Milicevic proposed the reformulated first Zagreb index EM1Y in [4], J. B. Liu et al. ([5])introduced the reformulated F-index, Ranjini et al. [6] re-defined the Zagreb indices, The hyper Zagreb index was put forward in 2013 by Shirdel et al. [7], Alameri et al. [8] introduced Y-index, are defined, respectively, as:
(1)M1Y=∑u∈VYdY2u=∑uv∈EYdYu+dYvM2Y=∑uv∈EYdYudYvFY=∑u∈VYdY3u=∑uv∈EYdY2u+dY2vEM1Y=∑e∈EYdY2e;dYe=dYu+dYv−2RFY=∑e∈EYdY3e=∑e~f∈EYdY2e+dY2fReZMY=∑uv∈EYdYudYvdYu+dYvHMY=∑uv∈EYdYu+dYv2YY1=∑u∈VY1dY14u=∑uv∈EY1dY13u+dY13v.Motivated by the work of X. Liu et al. [9] and P. Liu et al. [10], in this work we propose two types of four new operations on graphs based on join of graphs related to various subdivision graphs S (Y), R (Y), Q (Y) and T (Y), respectively [11]. Aram et al. [12] introduced the reformulated F-index. This index was defined as RFY=∑uv∈EYdYu+dYv−23. For many mathematical as well as chemical studies we refer our reader to follow [13–17].


## 2. Preliminaries

The join of two graphsY1 and Y2 (see [18]), denoted by Y1⊕Y2, is the union Y1⊕Y2 together with all the edges joining VY1 and VY2. The degree distribution of join of two graphs can be expressed as dY1⊕Y2u=dY1u+n2 when u∈VY1 and dY1⊕Y2u=dY1u+n1 when u∈VY2. There exist four subdivision related graphs for a connected graph Y, which are defined as follows [19]:
(i)
The subdivision graph of a graphY, denoted by SY, is obtained from Y by inserting a new vertex in each edge of Y by a path of length 2. It contains VS=VY∪EY and ES=2EY.(ii)
The semi-total point/traingle parallel graph of a graphY, denoted by RY, is derived from Y by joining an additional vertex corresponding to each edge of Y, then joining each new vertex to the end vertices of the corresponding edge. It has VR=VY∪EY and ER=ES∪EY.(iii)
The semi-total line/line super position graph of a graphY, denoted by QY, is obtained from Y by adding a new vertex corresponding to each edge of Y and then joining with edges those pairs of new vertices on adjacent edges of Y. It contains VQ=VY∪EY and EQ=ES∪EL.(iv)
The total graph of a graphY, denoted by TY, is obtained from Y by inserting a new vertex corresponding to each edge of Y, then joining each new vertex to the end vertices of the corresponding edge and joining with edges those pairs of new vertices on adjacent edges of Y, see Figure 1. Later on Akhter et al. [20] compute the forgotten topological indices of four operations. Recently Javaid et al. [21] investigated forgotten index of of generalized operations on graphs and Peng et al. [22] Compute the FGZ index of sum graphs under strong product. Also, Maji et al. [23] investigates the reformulated second Zagreb index of graph operations. In the field of chemical graph theory for further studying topological indices the interested readers can follow [24–28].Figure 1
The graphY and its SY,RY,QY and TY.


## 3. Main Results and Discussions

LetF=SY,RY,QY,TY and JY refers to the set of vertices FY which are inserted into each edge of Y such that VFY=VY∪JY. The Vertex F-join Graph [17] of Y1 and Y2, denoted by Y1⊕FY2, is obtained from FY1 and Y2 by joining each vertex of Y1 with every vertex of Y2. It contains the vertex set VFY1∪VY2 and edge set EY1∪EY2∪uvu∈VY1,v∈VY2. The set of vertices FY, denoted by JY, are added into each edge of Y such that VFY=VY∪JY. The line graph LY ([29]) of Y is the graph whose vertices correspond to the edges of Y with two vertices being adjacent if and only if the corresponding edges in Y have a vertex in common. The edge F-join graph [17] of Y1 and Y2, denoted by Y1⊕¯FY2 is obtained from FY1 and Y2 by joining each vertex of JY1 with every vertex of Y2. It contains the vertex set VFY1∪Y2 and edge set EY1∪EY1∪uv:u∈JY1,v∈VY2. In this paper, we study the RFI of vertex and edge F-join of graphs, respectively.


### 3.1. Vertex S-Join Graph (VSG)

TheVSG ([17]) of two vertex disjoint graphs Y1 and Y2, denoted by Y1⊕SY2, is derived from SY1 and Y2 by joining every vertex of VY1 to each vertex of Y2. An example of VSG of two path graphs P4 and P3 are shown in Figure 2.Figure 2
TheVSG and ESG of P4 and P3.Also the degree distribution of the vertices ofY1⊕SY2 are given in Table 1.Table 1
The degree distribution ofY1⊕SY2.
u∈VY1v∈VY2u∈JY1dY1u+n2dY2v+n12Now we compute the expression of RFI forY1⊕SY2.Theorem 1.
LetY1 and Y2 be two vertex disjoint graphs having n1,n2 vertices and m1,m2 edges, respectively. Then RFY1⊕SY2=YY1+YY2+4n2FY1+n1FY2+3ReZMY2+6n1−1HMY2+32n22−2n2+2m22+n1n2M1Y1+35n12−10n1+n1n2+2m1+4M1Y2+2m1n23+8n1−13m2+24n1+n2−2m1m2+6n1+n2−22m1n2+n1m2+n1n2n1+n2−23.Proof.
The edge setEY1⊕SY2 can be separated by three subsets like that.
∨1=uv∈EY1⊕SY2:uv∈ESY1;∨2=uv∈EY1⊕SY2:uv∈EY2 and ∨3=uv∈EY1⊕SY2:u∈VY1,v∈VY2. From the definition of RFI and the Table 1, we have
(2)RFY1⊕SY2=∑uv∈EY1⊕SY2dY1⊕SY2u+dY1⊕SY2v−23=∑uv∈ESY1dY1v+n23+∑uv∈EY2dY2u+dY2v+2n1−23+∑u∈VY1∑v∈VY1dY1u+dY2v+n1+n2−23=∑v∈VY1dY13v+3n2dY12v+3n22dY1v+n23dY1v+∑uv∈EY2dY2u+dY2v3+3n1+n1−2dY2u+dY2v2+3n1+n1−22dY2u+dY2v+n1+n1−23+∑u∈VY1∑v∈VY2dY1u+dY2v3+3n1+n2−2dY1u+dY2v2+3n1+n2−22dY1u+dY2v+n1+n2−23=YY1+3n2FY1+3n22M1Y1+2m1n23+YY2+3ReZMY2+6n1−1HMY2+12n1−12M1Y2+8n1−13m2+n2FY1+6m2M1Y1+6m1M1Y2+n1FY2+3n1+n2−2n2M1Y1+8m1m2+n1M1Y2+6n1+n2−22m1n2+n1m2+n1n2n1+n2−23.which is the required result.


### 3.2. Edge S-Join Graph (ESG)

The ESG ([17]) of Y1 and Y2, denoted by Y1⊕¯SY2, is obtained from SY1 and Y2 by joining each vertex of JY1 to every vertex of Y2. As an example we have depicted a graph that shown in Figure 2.Here we compute the explicit expression of RFI forY1⊕¯SY2.Theorem 2.
LetYi be the graph with ni vertices and mi edges, i∈1,2. Then RFY1⊕¯SY2=YY1+YY2+3n2FY1+m1FY2+3ReZMY2+6m1−1HMY2+3n22M1Y1+34m1−12+m1m1+n2M1Y2+2m1n23+8m2m1−13+m1m2m1+n2+6m1+n22.Proof.
ForY1⊕¯SY2, the edge set can be divided into three subsets such that
(3)S1=uv∈EY1⊕¯SY2:uv∈ESY1;S2=uv∈EY1⊕¯SY2:uv∈EY2S3=uv∈EY1⊕¯SY2:u∈JY1,v∈VY2
From the definition of RFI and by using the contribution of the edgesSii=1,2,3 followed from the Table 2, we have
(4)RFY1⊕¯SY2=∑uv∈EY1⊕¯SY2dY1⊕¯SY2u+dY1⊕¯SY2−23=∑S1dY1u+n23+∑S2dY2u+dY2v+2m1−13+∑S3dY2v+n2+m13=∑u∈VY1dY1udY13u+3n2dY12u+3n22dY1u+n23+∑uv∈EY2dY2u+dY2v3+8m1−13+6m1−1dY2u+dY2v2+12m1−12dY2u+dY2v=YY1+3n2FY1+3n22M1Y1+2m1n23+YY2+3ReZMY2+8m2m1−13+6m1−1HMY2+12m1−12M1Y2+m1FY2+m1n2m1+n23+3m1m1+n2M1Y2+6m1m2m1+n22.Table 2
The degree distribution ofY1⊕¯SY2.
u∈VY1v∈VY2u∈JY1dY1udY2v+m1n2+2


### 3.3. Vertex R-Join Graph (VRG)

The VRG ([17]) of Y1 and Y2, denoted by Y1⊕RY2, is derived from RY1 and Y2 by joining each vertex of VY1 to every vertex of Y2. An example of VRG of P4 and P3 is drawn in Figure 3. The degree distribution of the vertices of Y1⊕RY2 are in Table 3.Figure 3
TheVRG and ERG of P4 and P3.Table 3
The degree distribution ofY1⊕RY2.
u∈VY1v∈VY2u∈JY12dY1u+n2dY2v+n12Now we compute the exact expression of RFI forY1⊕RY2.Theorem 3.
IfY1 and Y2 be two graphs having n1,n2 vertices and m1,m2 edges, respectively. Then RFY1⊕RY2=16YY1+YY2+20n2FY1+n1FY2+24ReZMY1+3ReZMY2+24n2−1HMY1+6n1−1HMY2+67n22−12n2+2n1n2+4m2+4M1Y1+13n12−26n1+n1n2+12m1+12M1Y2+n1+n2−23n1n2+8n2−13m1+8n1−13m2+48m1m2n1+n2−2+6n1+n2−222m1n2+n1m2.Proof.
The edge set ofEY1⊕RY2 can be separated into four subsets such that.
E1=uv∈EY1⊕RY2:uv∈EY1;E2=uv∈EY1⊕RY2:u∈JY1,v∈VY1E3=uv∈EY1⊕RY2:uv∈EY2;E4=uv∈EY1⊕RY2:u∈VY1,v∈VY2. From the definition of RFI and the Table 3, we have
(5)RFY1⊕RY2=∑Ei=14dY1⊕RY2u+dY1⊕RY2v−23=∑uv∈EY12dY1u+dY1v+2n2−13+∑u∈JY1,v∈VY12dY1v+dJY1u+n2−23+∑uv∈EY2dY2u+dY2v+2n1−13+∑u∈VY1∑v∈VY22dY1u+dY2v+n1+n2−23=∏1+∏2+∏3+∏4∏1=∑uv∈EY12dY1u+dY1v+2n2−13=8∑uv∈EY1dY13u+dY13v+3dY1udY1vdY1u+dY1v+n2−13+3n2−1dY1u+dY1vdY1u+dY1v+n2−1=8YY1+3ReZMY1+n2−13m1+3n2−1HMY1+3n2−12M1Y1∏2=∑u∈JY1,v∈VY12dY1v+n2+dJY1u+−23=∑v∈VY18dY13v+n23+6n2dY1u2dY1v+n2dY1v=8∑v∈VY1dY14v+n23∑v∈VY1dY1v+12n2∑v∈VY1dY13v+6n22∑v∈VY1dY12v=8YY1+2m1n23+12n2FY1+6n22M1Y1.∏3=∑uv∈EY2dY2u+dY2v+2n1−13=∑uv∈EY2dY23u+dY23v+3dY2udY2vdY2u+dY2v+8n1−13+6n1−1dY2u+dY2v2+12n1−12dY2u+dY2v=YY2+3ReZMY2+8m2n1−13+6n1−1HMY2+12n1−12M1Y2.∏4=∑u∈VY1∑v∈VY22dY1u+dY2v+n1+n2−23=∑u∈VY1∑v∈VY22dY1u+dY2v3+n1+n2−23+3n1+n2−22dY1u+dY2v2dY1u+dY2v+n1+n2−2=∑u∈VY1∑v∈VY28dY13u+dY23v+6dY1udY2v2dY1u+dY2v+n1+n2−23+3n1+n2−24dY12u+dY22v+4dY1udY2v+3n1+n2−222dY1u+dY2v=8n2FY1+n1FY2+24m2M1Y1+12m1M1Y2+n1+n2−23n1n2+3n1+n2−24n2M1Y1+n1M1Y2+16m1m2+3n1+n2−224m1n2+2n1m2.
By adding the∏i, i=1,2,3,4 we arrive the desired result.


### 3.4. Edge R-Join Graph (ERG)

The ERG ([17]) of two vertex disjoint graphs Y1 and Y2 denoted by Y1⊕¯RY2 is derived from RY1 and Y2. It is drawn by joining each vertex of JY1 with every vertex of Y2, (Figure 3).Now we compute the RFI of ERG for two vertex disjoint graphs.Theorem 4.
The RFI of ERG([17]) for Y1 and Y2 is given by RFY1⊕¯RY2=16YY1+YY2+YY2+24RezMY1+3ReZMY2+12n2FY1+m1FY2−24HMY1+6m1−1HMY2+24+6n22M1Y1+3m1m1+n2+12m1−12M1Y2+m1m1+n22m1n2+n22+6m2+8m1−13m2+2m1n23−8m1.Proof.
The edge set ofEY1⊕¯RY2 can be divided into four subsets like as.
R1=uv∈EY1⊕¯RY2:uv∈EY1;R2=uv∈EY1⊕¯RY2:u∈JY1,v∈VY1R3=uv∈EY1⊕¯RY2:u∈JY1,v∈VY2;R4=uv∈EY1⊕¯RY2:uv∈EY2. From the definition of RFI and the Table 4, we have
(6)RFY1⊕¯RY2=∑uv∈EY1⊕¯RY2dY1⊕¯RY2u+dY1⊕¯RY2v−23=∑Ri=14dY1⊕¯RY2u+dY1⊕¯RY2v−23=8∑uv∈EY1dY1u+dY1v−13+∑u∈vY1dY1u2dY1u+n23+m1∑v∈VY2dY2v+m1+n23+∑uv∈EY2dY2u+m1+dY2v+m1−23=8YY1+3ReZMY1−3HMY1+3M1Y1−m1+8YY1+12n2FY1+6n22M1Y1+2m1n23+m1FY1+m1+n23n2+6m1+n22m2+3m1+n2M1Y2+YY2+3ReZMY2+8m1−13m2+6m1−1HMY2+12m1−12M1Y2Table 4
The degree distribution ofY1⊕¯RY2.
u∈VY1v∈VY2u∈JY12dY1udY2v+m1n2+2


### 3.5. Vertex Q-Join Graph (VQG)

TheVQG ([17]) of two vertex disjoint graphs Y1 and Y2, denoted by Y1⊕QY2, is obtained from QY1 and Y2 by joining each vertex of VY1 to every vertex of Y2, (see the Figure 4).Figure 4
TheVQG and EQG of P4 and P3.Next we calculate the expression of RFI forY1⊕QY2.Theorem 5.
LetYi be ni,mi graph, i∈1,2. Then RFY1⊕QY2=YLY1+3ReZMLY1+6HMLY1+12M1LY1+8ELY1+9YY1+18ReZMY1+3n2−25FY1+8M2Y1+9n2−22M1Y1+2m1n2−23+YY2+3ReZMY2+8m2n1−13+6n1−1HMY2+12n1−12M1Y2+n2FY1+n1FY2+6m2M1Y1+6m1M1Y2+n1n2n1+n2−23+3n1+n2−2n2M1Y1+n1M1Y2+8m1m2+6n1+n2−22m1n2+m2n1.Proof.
From the definition of RFI and the Table5, we have.
(7)RFY1⊕QY2=∑uv,vw∈EY1⊕QY2dY1⊕QY2u+dY1⊕QY2v−23=∑ℚi=14dY1⊕QY2u+dY1⊕QY2v−23,whereℚ1=uv∈EY1⊕QY2:uv∈EJY1;ℚ2=uv∈EY1⊕QY2:u∈VY1,v∈JY1;ℚ3=uv∈EY1⊕QY2:u∈VY1,v∈VY2;ℚ4=uv∈EY1⊕QY2:uv∈EY2=∑uv,vw∈EY1dY1u+dY1v+dY1v+dY1w−23+∑uv∈EY1dY1u+n2+dY1u+dY1v−23+dY1v+n2+dY1u+dY1v−23+∑uv∈EY2dY2u+n1+dY2v+n1−23+∑u∈VY1∑v∈VY2dG1u+n2+dY2v+n1−23=∑e,f∈LY1,e=uv,f=vwdLY1e+dLY1f+23+∑uv∈EY12dY1u+dY1v+n2−23+dY1u+2dY1v+n2−23+∑uv∈EY2dY2u+dY2v+2n1−13+∑u∈VY1∑v∈VY2dY1u+n2+dY2v+n1−23=∑e,f∈ELY1dLY13e+dLY13f+3dLY1edLY1fdLY1e+dLY1f+6dLY1e+dLY1f2+12dLY1edLY1f+8+∑uv∈EY19dY13u+dY13v+18dY1udY1vdY1u+dY1v+2n2−23+3n2−22dY1u+dY1v+3n2−25dY12u+dY12v+8dY1udY1v+∑uv∈EY2dY23u+dY23v+3dY2udY2vdY2u+dY2v+8n1−13+6n1−1dY2u+dY2v2+12n1−12dY2u+dY2v+∑u∈VY1∑v∈VY2dY13u+dY13v+3dY1u2dY2v+3dY1udY22v+n1+n2−23+3n1+n2−2dY1u+dY2v2+3n1+n2−22dY1u+dY2v=YLY1+3ReZMLY1+6HMLY1+12M1LY1+8ELY1+9YY1+18ReZMY1+3n2−25FY1+8M2Y1+9n2−22M1Y1+2m1n2−23+YY2+3ReZMY2+8m2n1−13+6n1−1HMY2+12n1−12M1Y2+n2FY1+n1FY2+6m2M1Y1+6m1M1Y2+n1n2n1+n2−23+3n1+n2−2n2M1Y1+n1M1Y2+8m1m2+6n1+n2−22m1n2+m2n1.Table 5
The degree distribution ofY1⊕QY2.
u∈VY1v∈VY2e=w,k∈JY1dY1u+n2dY2v+n1dY1w+dY1k


### 3.6. Edge Q-Join Graph (EQG)

The EQG ([17]) of Y1 and Y2 denoted by Y1⊕¯QY2 is a new formed graph that is derived from QY1 and Y2. It is obtained by joining each vertex of JY1 with every vertex of Y2, (see Figure 4).Now we determine the RFI of EQG for two vertex disjoint graphsY1 and Y2.Theorem 6.
LetY1 and Y2 be two vertex disjoint graphs. Then RFY1⊕¯QY2=YLY1+3ReZMLY1+8n2+13ELY1+6n2+1HMLY1+12n2+12M1LY1+n2+9YY1+YY2+3n2+6ReZMY1+3ReZMY2+3n2−2FY1+m1FY2+3n22+m1n2+2n2+2m2−8HMY1+6m1−1HMY2+3M1Y1M1Y2+33n2−22+n2n2+m1−22+4m1m2+4m2n2−8m2M1Y1+35m12−10m1+m1n2+4M1Y2+m1n2n2+m1−23+6m1m2n2+m1−22+8m2m1−13+2m1n2−23.Proof.
We have from definition of RFI and the Table6(8)RFY1⊕¯QY2=∑uv∈EY1⊕¯QY2dY1⊕¯QY2u+dY1⊕¯QY2v−23
The edge set ofY1⊕¯QY2 can be partitioned into four subsets such as
(9)Q1=uv∈EY1⊕¯QY2:uv∈EJY1;Q2=uv∈EY1⊕¯QY2:u∈VY1,v∈JY1Q3=uv∈EY1⊕¯QY2:u∈JY1,v∈VY2;Q4=uv∈EY1⊕¯QY2:uv∈EY2
Therefore, the RFI ofY1⊕¯QY2 is given by
(10)=∑Qi=14dY1⊕¯QY2u+dY1⊕¯QY2v−23=∑Q1dY1u+2dY1v+dY1w+2n2−13+∑Q2dY1u+dJY1e−23+dY1v+dJY1f−23+∑Q3dJY1u+n2+dJY2v+m1−23+∑Q4dY2u+dY2v−2m1−13=∑e,f∈ELY1dLY1e+dLY1f+2n2+13+∑uv∈EY1dY13u+dY13v+dJY1u−23+3dY1udJY1u−22+3dY12udJY1u−2+dJY1v−23+3dY12vdJY1u−22+3dY12vdJY1u−2+∑uv∈EY1∑w∈VY2dY1u+dY1v+dY2w+n2+m1−23+∑uv∈EY2dY2u+dY2v−2m1−13=∑e,f∈ELY1dLY1e+dLY1f+3dLY1edLY1fdLY1e+dLY1f+8n2+13+6n2+1dLY1e+dLY1f2+12n2+12dLY1e+dLY1f+∑uv∈EY19dY13u+dY13v+12n2−2dY1u+dY1v2+18dY1udY1vdY1u+dY1v+3n2−2dY12u+dY12v+6n2−22dY1u+dY1v+2m1n2−23+3n2−22dY1u+dY1v+∑uv∈EY1∑w∈VY2dY13u+dY13v+3dY1udY1vdY1u+dY1v+dY23w+n2+m1−23+3n2+m1−22dY2w+3n2+m1−2dY22w+3dY1u+dY1v2dY2w+n2+m1−2+3dY1u+dY1vdY22w+2n2+m1−2dY2w+n2+m1−22+∑uv∈EY2dY23u+dY23v+dY2udY2vdY2u+dY2v+6m1−1dY2u+dY2v2+12m1−12dY2u+dY2v+8m1−13=YLY1+3ReZMLY1+8n2+13ELY1+6n2+1HMLY1+12n2+12M1LY1+9YY1+12n2−2HMY1+18ReZMY1+3n2−2FY1+9n2−22M1Y1+2m1n2−23+n2YY1+3n2ReZMY1+m1FY2+3n22+m1n2+2m2−2n2HMY1+3M1Y1M1Y2+4m2n2+m1−2+n2n2+m1−22+m1n2n2+m1−23+6m1m2n2+m1−22+3m1n2+m1−2M1Y2+YY2+3ReZMY2+6m1−1HMY2+12m1−12M1Y2+8m2m1−13.Table 6
The degree distribution ofY1⊕¯QY2.
u∈VY1v∈VY2e=w,k∈JY1dY1udY2v+m1dY1w+dY1k+n2


### 3.7. Vertex T-Join Graph (VTG)

TheVTG ([17]) of Y1 and Y2 is obtained from TY1 and Y2 by joining each vertex of VY1 to every vertex of Y2 and is denoted by Y1⊕TY2, (see the Figure 5).Figure 5
TheVTG and ETG of P4 and P3.Here we derive the explicit formula of RFI for VTG of two graphs.Theorem 7.
LetYi be a graph with ni vertices and mi edges, i=1,2. Then
(11)RFY1⊕TY2=YLY1+8ELY1+3ReZMLY1+6HMLY1+12M1LY1+28YY1+36ReZMY1+30n2−2FY1+36n2−2M2Y1+12n2−22M1Y1+2n2−23m1+YY2+3ReZMY2+8YY1+3ReZMY1+m1n2−13+3n2−12M1Y1+3n2−1HMY1+8m2n1−13+6n1−1HMY2+12n1−12M1Y2+8n2FY1+n1FY2+24m2M1Y1+12m1M1Y2+n1n2n1+n2−23+3n1+n2−22+4m1n2+2m2n1+3n1+n2−24n2M1Y1+n1M1Y2+16m1m2.Proof.
By the definition of RFI and using the Table7, we have.
(12)RFY1⊕TY2=∑uv,vw∈EY1⊕TY2dY1⊕TY2u+dY1⊕TY2v−23=∑u,v∈JY1dJY1u+dJY1v−23+∑u,∈vY1,v∈JY1dY1u+dJY1v−23+∑uv∈EY2dY2u+dY2v+2n1−13+∑u∈VY1∑v∈VY22dY1u+dY2v+n1+n2−23+∑uv∈EY12dY1u+n2+2dY1v+n2−23=∑1+∑2+∑3+∑4+∑5say,respectively.∑1=∑u,v∈JY1dJY1u+dJY1v−23=∑e,f∈ELY1,e=uv,t=vwdLY1e+2+dLY1f3=∑e,f∈ELY1dLY13e+dLY13f+8+3dLY1edLY1fdLY1e+dLY1f+6dLY1e+dLY1f2+12dLY1e+dLY1f=YLY1+8ELY1+3ReZMLY1+6HMLY1+12M1LY1∑2=∑u∈VY1,v∈JY12dY1u+n2+dJY1v−23=∑uv∈EY13dY1u+dY1v+n2−23+dY1u+3dY1v+n2−23=∑uv∈EY128dY13u+dY13v+36dY1udY1vdY1u+dY1v+2n2−23+3n2−210dY12u+dY12v+12dY1udY1v+12n2−22dY1u+dY1v=28YY1+36ReZMY1+30n2−2FY1+36n2−2M2Y1+12n2−22M1Y1+2n2−23m1∑3=∑uv∈EY2dY2u+dY2v+2n1−13=∑uv∈EY2dY13u+dG13v+3∑uv∈EY2dY2udY2vdY2u+dY2v+8n2−13∑uv∈EY21+6n1−1∑uv∈EY2dY2u+dY2v2+12n1−12∑uv∈EY2dY2u+dY2v=YY2+3ReZMY2+8m2n1−13+6n1−1HMY2+12n1−12M1Y2∑4=∑u∈VY1∑v∈VY22dY1u+dY2v+n1+n2−23=8∑v∈VY2∑u∈VY1dY13u+∑u∈VY1∑v∈VY2dY23u+12∑v∈VY2dY2v∑u∈VY1dY12u+6∑u∈VY1dY1u∑v∈VY2dY22v+n1+n2−23∑u∈VY1∑v∈VY21+3n1+n2−22∑u∈VY1∑v∈VY22dY1u+dY2v+3n1+n2−2∑u∈VY1∑v∈VY24dY12u+dY12v+4dY1udY2v=8n2FY1+n1FY2+24m2M1Y1+12m1M1Y2+n1n2n1+n2−23+3n1+n2−224m1n2+2m2n1+3n1+n2−24n2M1Y1+n1M1Y2+16m1m2.∑5=∑uv∈EY12dY1u+n2+2dY1v+n2−23=8∑uv∈EY1dY1u+dY1v+n2−13=8∑uv∈EY1dY13u+dG13v+3dG1udY1vdY1u+dY1v+n2−13+3n2−12dY1u+dY1v+3n2−1dY1u+dY1v2=8YY1+3ReZMY1+m1n2−13+3n2−12M1Y1+3n2−1HMY1.
After simplifying∑i=15∑i, we get the required result.Table 7
The degree distribution ofY1⊕TY2.
u∈VY1v∈VY2e=w,k∈JY12dY1u+n2dY2v+n1dY1w+dY1k


### 3.8. Edge T-Join Graph (ETG)

LetY1 and Y2 be two vertex disjoint graphs. The ETG([17]) of Y1 and Y2, denoted by Y1⊕¯TY2, is derived by joining each vertex of TY1 to every vertex of Y2, (see Figure 5).Now we determine the exact expression of RFI forY1⊕¯TY2.Theorem 8.
LetY1 and Y2 be two graphs, then the RFI of Y1⊕¯TY2 is given by
(13)RFY1⊕¯TY2=36+n2YY1+YLY1+YY2+320+n2ReZMY1+3ReZMLY1+3ReZMY2+3M1Y1M1Y2+3n22+m1n2+4n2+2m2−20HMY1+6n2+1HMLY1+6m1−1HMY2+3m1+n2−2m1n2+n22−2n2+4m2+4n2−22+8M1Y1+12n2+12M1LY1+35m12−10m1+m1n2+4M1Y2+12n2−2FY1+m1FY2+8n2+13ELY1+6m1m2m1+n2−22+m1n2m1+n2−23+8m2m1−13+2m1n2−23−8m1.Proof.
By definition of RFI and using the Table8, we have
(14)RFY1⊕¯TY2=∑Ti=15dY1⊕¯TY2u+dY1⊕¯TY2v−23=∑T1dY1⊕¯TY2u+dY1⊕¯TY2v−23+∑T22dY1u+dJY1e−23+2dY1v+dJY1f−23+∑T3dJY1u+dJY1v−23+∑T4dJY1+n2+dY2v−23+∑T5dY2u+dY2v+2m1−23=∑uv∈EY12dY1u+dY1v−23+∑uv∈EY18dY13u+dY13v+2dY1u+dY1v+n2−23+12dY12u+dY12vdY1u+dY1v+n2−2+6dY1u+dY1vdY1u+dY1v2+2n2−2dY1u+dY1v+n2−22+∑e,f∈ELY1,e=uv,f=vwdLY1e+dLY1f+2n2+13+∑uv∈Y1∑w∈VY2dY1u+dY1v+dY2w+m1+n2−23+∑uv∈EY2dY2u+dY2v+2m1−13=8∑uv∈EY1dY13u+dY13v+3dY1udY1vdY1udY1v−3dY1u+dY1v2+3dY1u+dY1v−1+∑uv∈EY126dY13u+dY13v+2dY13u+dY13v+3dY1udY1vdY1u+dY1v+n2−23+3n2−22dY1udY1v+3n2−2dY1u+dY1v2+30dY1udY1vdY1u+dY1v+12n2−2dY1u+dY1v2+6n2−22dY1u+dY1v+dY1v+12n2−2dY12u+dY12v+∑e,f∈ELY1dLY13e+dLY13f+3dY1edY1f3dY1e+dY1f+8n2+13+6n2+1dLY1e+dLY1f2+12n2+12dY1e+dY1f+∑uv∈EY1∑w∈VY2dY13u+dY13v+3dY1udY1vdY1udY1v+m1+n2−2+dY2w3+3m1+n2−2+dY2wdY1udY1v2+3m1+n2−2+dY2w2dY1u+dY1v+∑uv∈EY2dY23u+dY23v+3dY2udY2vdY2u+dY2v+8m1−13+12m1−12dY2u+dY2v+6m1−1dY2u+dY2v2=8YY1+3ReZMY1−3HMY1+3M1Y1−m1+28YY1+36ReZMY1+18n2−2HMY1+12n2−22M1Y1+12n2−2FY1+2m1n2−23+YLY1+3ReZMLY1+8n2+13EJLY1+6n2+1HMLY1+12n2+12M1LY1+n2YY1+3n2ReZMY1+m1FY2+3m1m1+n2−2M1Y2+6m1m2m1+n2−22+m1n2m1+n2−23+3n22+m1n2−2n2+2m2HMY1+3M1Y1M1Y2+3m1+n2−2m1n2+n22−2n2+4m2M1Y1+YY2+3ReZMY2+8m2m1−13+12m1−12M1Y2+6m1−1HMY2.Table 8
The degree distribution ofY1⊕¯TY2.
u∈VY1v∈VY2e=w,k∈JY12dY1udY2v+m1dY1w+dY1k+n2


## 3.1. Vertex S-Join Graph (VSG)

TheVSG ([17]) of two vertex disjoint graphs Y1 and Y2, denoted by Y1⊕SY2, is derived from SY1 and Y2 by joining every vertex of VY1 to each vertex of Y2. An example of VSG of two path graphs P4 and P3 are shown in Figure 2.Figure 2
TheVSG and ESG of P4 and P3.Also the degree distribution of the vertices ofY1⊕SY2 are given in Table 1.Table 1
The degree distribution ofY1⊕SY2.
u∈VY1v∈VY2u∈JY1dY1u+n2dY2v+n12Now we compute the expression of RFI forY1⊕SY2.Theorem 1.
LetY1 and Y2 be two vertex disjoint graphs having n1,n2 vertices and m1,m2 edges, respectively. Then RFY1⊕SY2=YY1+YY2+4n2FY1+n1FY2+3ReZMY2+6n1−1HMY2+32n22−2n2+2m22+n1n2M1Y1+35n12−10n1+n1n2+2m1+4M1Y2+2m1n23+8n1−13m2+24n1+n2−2m1m2+6n1+n2−22m1n2+n1m2+n1n2n1+n2−23.Proof.
The edge setEY1⊕SY2 can be separated by three subsets like that.
∨1=uv∈EY1⊕SY2:uv∈ESY1;∨2=uv∈EY1⊕SY2:uv∈EY2 and ∨3=uv∈EY1⊕SY2:u∈VY1,v∈VY2. From the definition of RFI and the Table 1, we have
(2)RFY1⊕SY2=∑uv∈EY1⊕SY2dY1⊕SY2u+dY1⊕SY2v−23=∑uv∈ESY1dY1v+n23+∑uv∈EY2dY2u+dY2v+2n1−23+∑u∈VY1∑v∈VY1dY1u+dY2v+n1+n2−23=∑v∈VY1dY13v+3n2dY12v+3n22dY1v+n23dY1v+∑uv∈EY2dY2u+dY2v3+3n1+n1−2dY2u+dY2v2+3n1+n1−22dY2u+dY2v+n1+n1−23+∑u∈VY1∑v∈VY2dY1u+dY2v3+3n1+n2−2dY1u+dY2v2+3n1+n2−22dY1u+dY2v+n1+n2−23=YY1+3n2FY1+3n22M1Y1+2m1n23+YY2+3ReZMY2+6n1−1HMY2+12n1−12M1Y2+8n1−13m2+n2FY1+6m2M1Y1+6m1M1Y2+n1FY2+3n1+n2−2n2M1Y1+8m1m2+n1M1Y2+6n1+n2−22m1n2+n1m2+n1n2n1+n2−23.which is the required result.


## 3.2. Edge S-Join Graph (ESG)

The ESG ([17]) of Y1 and Y2, denoted by Y1⊕¯SY2, is obtained from SY1 and Y2 by joining each vertex of JY1 to every vertex of Y2. As an example we have depicted a graph that shown in Figure 2.Here we compute the explicit expression of RFI forY1⊕¯SY2.Theorem 2.
LetYi be the graph with ni vertices and mi edges, i∈1,2. Then RFY1⊕¯SY2=YY1+YY2+3n2FY1+m1FY2+3ReZMY2+6m1−1HMY2+3n22M1Y1+34m1−12+m1m1+n2M1Y2+2m1n23+8m2m1−13+m1m2m1+n2+6m1+n22.Proof.
ForY1⊕¯SY2, the edge set can be divided into three subsets such that
(3)S1=uv∈EY1⊕¯SY2:uv∈ESY1;S2=uv∈EY1⊕¯SY2:uv∈EY2S3=uv∈EY1⊕¯SY2:u∈JY1,v∈VY2
From the definition of RFI and by using the contribution of the edgesSii=1,2,3 followed from the Table 2, we have
(4)RFY1⊕¯SY2=∑uv∈EY1⊕¯SY2dY1⊕¯SY2u+dY1⊕¯SY2−23=∑S1dY1u+n23+∑S2dY2u+dY2v+2m1−13+∑S3dY2v+n2+m13=∑u∈VY1dY1udY13u+3n2dY12u+3n22dY1u+n23+∑uv∈EY2dY2u+dY2v3+8m1−13+6m1−1dY2u+dY2v2+12m1−12dY2u+dY2v=YY1+3n2FY1+3n22M1Y1+2m1n23+YY2+3ReZMY2+8m2m1−13+6m1−1HMY2+12m1−12M1Y2+m1FY2+m1n2m1+n23+3m1m1+n2M1Y2+6m1m2m1+n22.Table 2
The degree distribution ofY1⊕¯SY2.
u∈VY1v∈VY2u∈JY1dY1udY2v+m1n2+2


## 3.3. Vertex R-Join Graph (VRG)

The VRG ([17]) of Y1 and Y2, denoted by Y1⊕RY2, is derived from RY1 and Y2 by joining each vertex of VY1 to every vertex of Y2. An example of VRG of P4 and P3 is drawn in Figure 3. The degree distribution of the vertices of Y1⊕RY2 are in Table 3.Figure 3
TheVRG and ERG of P4 and P3.Table 3
The degree distribution ofY1⊕RY2.
u∈VY1v∈VY2u∈JY12dY1u+n2dY2v+n12Now we compute the exact expression of RFI forY1⊕RY2.Theorem 3.
IfY1 and Y2 be two graphs having n1,n2 vertices and m1,m2 edges, respectively. Then RFY1⊕RY2=16YY1+YY2+20n2FY1+n1FY2+24ReZMY1+3ReZMY2+24n2−1HMY1+6n1−1HMY2+67n22−12n2+2n1n2+4m2+4M1Y1+13n12−26n1+n1n2+12m1+12M1Y2+n1+n2−23n1n2+8n2−13m1+8n1−13m2+48m1m2n1+n2−2+6n1+n2−222m1n2+n1m2.Proof.
The edge set ofEY1⊕RY2 can be separated into four subsets such that.
E1=uv∈EY1⊕RY2:uv∈EY1;E2=uv∈EY1⊕RY2:u∈JY1,v∈VY1E3=uv∈EY1⊕RY2:uv∈EY2;E4=uv∈EY1⊕RY2:u∈VY1,v∈VY2. From the definition of RFI and the Table 3, we have
(5)RFY1⊕RY2=∑Ei=14dY1⊕RY2u+dY1⊕RY2v−23=∑uv∈EY12dY1u+dY1v+2n2−13+∑u∈JY1,v∈VY12dY1v+dJY1u+n2−23+∑uv∈EY2dY2u+dY2v+2n1−13+∑u∈VY1∑v∈VY22dY1u+dY2v+n1+n2−23=∏1+∏2+∏3+∏4∏1=∑uv∈EY12dY1u+dY1v+2n2−13=8∑uv∈EY1dY13u+dY13v+3dY1udY1vdY1u+dY1v+n2−13+3n2−1dY1u+dY1vdY1u+dY1v+n2−1=8YY1+3ReZMY1+n2−13m1+3n2−1HMY1+3n2−12M1Y1∏2=∑u∈JY1,v∈VY12dY1v+n2+dJY1u+−23=∑v∈VY18dY13v+n23+6n2dY1u2dY1v+n2dY1v=8∑v∈VY1dY14v+n23∑v∈VY1dY1v+12n2∑v∈VY1dY13v+6n22∑v∈VY1dY12v=8YY1+2m1n23+12n2FY1+6n22M1Y1.∏3=∑uv∈EY2dY2u+dY2v+2n1−13=∑uv∈EY2dY23u+dY23v+3dY2udY2vdY2u+dY2v+8n1−13+6n1−1dY2u+dY2v2+12n1−12dY2u+dY2v=YY2+3ReZMY2+8m2n1−13+6n1−1HMY2+12n1−12M1Y2.∏4=∑u∈VY1∑v∈VY22dY1u+dY2v+n1+n2−23=∑u∈VY1∑v∈VY22dY1u+dY2v3+n1+n2−23+3n1+n2−22dY1u+dY2v2dY1u+dY2v+n1+n2−2=∑u∈VY1∑v∈VY28dY13u+dY23v+6dY1udY2v2dY1u+dY2v+n1+n2−23+3n1+n2−24dY12u+dY22v+4dY1udY2v+3n1+n2−222dY1u+dY2v=8n2FY1+n1FY2+24m2M1Y1+12m1M1Y2+n1+n2−23n1n2+3n1+n2−24n2M1Y1+n1M1Y2+16m1m2+3n1+n2−224m1n2+2n1m2.
By adding the∏i, i=1,2,3,4 we arrive the desired result.


## 3.4. Edge R-Join Graph (ERG)

The ERG ([17]) of two vertex disjoint graphs Y1 and Y2 denoted by Y1⊕¯RY2 is derived from RY1 and Y2. It is drawn by joining each vertex of JY1 with every vertex of Y2, (Figure 3).Now we compute the RFI of ERG for two vertex disjoint graphs.Theorem 4.
The RFI of ERG([17]) for Y1 and Y2 is given by RFY1⊕¯RY2=16YY1+YY2+YY2+24RezMY1+3ReZMY2+12n2FY1+m1FY2−24HMY1+6m1−1HMY2+24+6n22M1Y1+3m1m1+n2+12m1−12M1Y2+m1m1+n22m1n2+n22+6m2+8m1−13m2+2m1n23−8m1.Proof.
The edge set ofEY1⊕¯RY2 can be divided into four subsets like as.
R1=uv∈EY1⊕¯RY2:uv∈EY1;R2=uv∈EY1⊕¯RY2:u∈JY1,v∈VY1R3=uv∈EY1⊕¯RY2:u∈JY1,v∈VY2;R4=uv∈EY1⊕¯RY2:uv∈EY2. From the definition of RFI and the Table 4, we have
(6)RFY1⊕¯RY2=∑uv∈EY1⊕¯RY2dY1⊕¯RY2u+dY1⊕¯RY2v−23=∑Ri=14dY1⊕¯RY2u+dY1⊕¯RY2v−23=8∑uv∈EY1dY1u+dY1v−13+∑u∈vY1dY1u2dY1u+n23+m1∑v∈VY2dY2v+m1+n23+∑uv∈EY2dY2u+m1+dY2v+m1−23=8YY1+3ReZMY1−3HMY1+3M1Y1−m1+8YY1+12n2FY1+6n22M1Y1+2m1n23+m1FY1+m1+n23n2+6m1+n22m2+3m1+n2M1Y2+YY2+3ReZMY2+8m1−13m2+6m1−1HMY2+12m1−12M1Y2Table 4
The degree distribution ofY1⊕¯RY2.
u∈VY1v∈VY2u∈JY12dY1udY2v+m1n2+2


## 3.5. Vertex Q-Join Graph (VQG)

TheVQG ([17]) of two vertex disjoint graphs Y1 and Y2, denoted by Y1⊕QY2, is obtained from QY1 and Y2 by joining each vertex of VY1 to every vertex of Y2, (see the Figure 4).Figure 4
TheVQG and EQG of P4 and P3.Next we calculate the expression of RFI forY1⊕QY2.Theorem 5.
LetYi be ni,mi graph, i∈1,2. Then RFY1⊕QY2=YLY1+3ReZMLY1+6HMLY1+12M1LY1+8ELY1+9YY1+18ReZMY1+3n2−25FY1+8M2Y1+9n2−22M1Y1+2m1n2−23+YY2+3ReZMY2+8m2n1−13+6n1−1HMY2+12n1−12M1Y2+n2FY1+n1FY2+6m2M1Y1+6m1M1Y2+n1n2n1+n2−23+3n1+n2−2n2M1Y1+n1M1Y2+8m1m2+6n1+n2−22m1n2+m2n1.Proof.
From the definition of RFI and the Table5, we have.
(7)RFY1⊕QY2=∑uv,vw∈EY1⊕QY2dY1⊕QY2u+dY1⊕QY2v−23=∑ℚi=14dY1⊕QY2u+dY1⊕QY2v−23,whereℚ1=uv∈EY1⊕QY2:uv∈EJY1;ℚ2=uv∈EY1⊕QY2:u∈VY1,v∈JY1;ℚ3=uv∈EY1⊕QY2:u∈VY1,v∈VY2;ℚ4=uv∈EY1⊕QY2:uv∈EY2=∑uv,vw∈EY1dY1u+dY1v+dY1v+dY1w−23+∑uv∈EY1dY1u+n2+dY1u+dY1v−23+dY1v+n2+dY1u+dY1v−23+∑uv∈EY2dY2u+n1+dY2v+n1−23+∑u∈VY1∑v∈VY2dG1u+n2+dY2v+n1−23=∑e,f∈LY1,e=uv,f=vwdLY1e+dLY1f+23+∑uv∈EY12dY1u+dY1v+n2−23+dY1u+2dY1v+n2−23+∑uv∈EY2dY2u+dY2v+2n1−13+∑u∈VY1∑v∈VY2dY1u+n2+dY2v+n1−23=∑e,f∈ELY1dLY13e+dLY13f+3dLY1edLY1fdLY1e+dLY1f+6dLY1e+dLY1f2+12dLY1edLY1f+8+∑uv∈EY19dY13u+dY13v+18dY1udY1vdY1u+dY1v+2n2−23+3n2−22dY1u+dY1v+3n2−25dY12u+dY12v+8dY1udY1v+∑uv∈EY2dY23u+dY23v+3dY2udY2vdY2u+dY2v+8n1−13+6n1−1dY2u+dY2v2+12n1−12dY2u+dY2v+∑u∈VY1∑v∈VY2dY13u+dY13v+3dY1u2dY2v+3dY1udY22v+n1+n2−23+3n1+n2−2dY1u+dY2v2+3n1+n2−22dY1u+dY2v=YLY1+3ReZMLY1+6HMLY1+12M1LY1+8ELY1+9YY1+18ReZMY1+3n2−25FY1+8M2Y1+9n2−22M1Y1+2m1n2−23+YY2+3ReZMY2+8m2n1−13+6n1−1HMY2+12n1−12M1Y2+n2FY1+n1FY2+6m2M1Y1+6m1M1Y2+n1n2n1+n2−23+3n1+n2−2n2M1Y1+n1M1Y2+8m1m2+6n1+n2−22m1n2+m2n1.Table 5
The degree distribution ofY1⊕QY2.
u∈VY1v∈VY2e=w,k∈JY1dY1u+n2dY2v+n1dY1w+dY1k


## 3.6. Edge Q-Join Graph (EQG)

The EQG ([17]) of Y1 and Y2 denoted by Y1⊕¯QY2 is a new formed graph that is derived from QY1 and Y2. It is obtained by joining each vertex of JY1 with every vertex of Y2, (see Figure 4).Now we determine the RFI of EQG for two vertex disjoint graphsY1 and Y2.Theorem 6.
LetY1 and Y2 be two vertex disjoint graphs. Then RFY1⊕¯QY2=YLY1+3ReZMLY1+8n2+13ELY1+6n2+1HMLY1+12n2+12M1LY1+n2+9YY1+YY2+3n2+6ReZMY1+3ReZMY2+3n2−2FY1+m1FY2+3n22+m1n2+2n2+2m2−8HMY1+6m1−1HMY2+3M1Y1M1Y2+33n2−22+n2n2+m1−22+4m1m2+4m2n2−8m2M1Y1+35m12−10m1+m1n2+4M1Y2+m1n2n2+m1−23+6m1m2n2+m1−22+8m2m1−13+2m1n2−23.Proof.
We have from definition of RFI and the Table6(8)RFY1⊕¯QY2=∑uv∈EY1⊕¯QY2dY1⊕¯QY2u+dY1⊕¯QY2v−23
The edge set ofY1⊕¯QY2 can be partitioned into four subsets such as
(9)Q1=uv∈EY1⊕¯QY2:uv∈EJY1;Q2=uv∈EY1⊕¯QY2:u∈VY1,v∈JY1Q3=uv∈EY1⊕¯QY2:u∈JY1,v∈VY2;Q4=uv∈EY1⊕¯QY2:uv∈EY2
Therefore, the RFI ofY1⊕¯QY2 is given by
(10)=∑Qi=14dY1⊕¯QY2u+dY1⊕¯QY2v−23=∑Q1dY1u+2dY1v+dY1w+2n2−13+∑Q2dY1u+dJY1e−23+dY1v+dJY1f−23+∑Q3dJY1u+n2+dJY2v+m1−23+∑Q4dY2u+dY2v−2m1−13=∑e,f∈ELY1dLY1e+dLY1f+2n2+13+∑uv∈EY1dY13u+dY13v+dJY1u−23+3dY1udJY1u−22+3dY12udJY1u−2+dJY1v−23+3dY12vdJY1u−22+3dY12vdJY1u−2+∑uv∈EY1∑w∈VY2dY1u+dY1v+dY2w+n2+m1−23+∑uv∈EY2dY2u+dY2v−2m1−13=∑e,f∈ELY1dLY1e+dLY1f+3dLY1edLY1fdLY1e+dLY1f+8n2+13+6n2+1dLY1e+dLY1f2+12n2+12dLY1e+dLY1f+∑uv∈EY19dY13u+dY13v+12n2−2dY1u+dY1v2+18dY1udY1vdY1u+dY1v+3n2−2dY12u+dY12v+6n2−22dY1u+dY1v+2m1n2−23+3n2−22dY1u+dY1v+∑uv∈EY1∑w∈VY2dY13u+dY13v+3dY1udY1vdY1u+dY1v+dY23w+n2+m1−23+3n2+m1−22dY2w+3n2+m1−2dY22w+3dY1u+dY1v2dY2w+n2+m1−2+3dY1u+dY1vdY22w+2n2+m1−2dY2w+n2+m1−22+∑uv∈EY2dY23u+dY23v+dY2udY2vdY2u+dY2v+6m1−1dY2u+dY2v2+12m1−12dY2u+dY2v+8m1−13=YLY1+3ReZMLY1+8n2+13ELY1+6n2+1HMLY1+12n2+12M1LY1+9YY1+12n2−2HMY1+18ReZMY1+3n2−2FY1+9n2−22M1Y1+2m1n2−23+n2YY1+3n2ReZMY1+m1FY2+3n22+m1n2+2m2−2n2HMY1+3M1Y1M1Y2+4m2n2+m1−2+n2n2+m1−22+m1n2n2+m1−23+6m1m2n2+m1−22+3m1n2+m1−2M1Y2+YY2+3ReZMY2+6m1−1HMY2+12m1−12M1Y2+8m2m1−13.Table 6
The degree distribution ofY1⊕¯QY2.
u∈VY1v∈VY2e=w,k∈JY1dY1udY2v+m1dY1w+dY1k+n2


## 3.7. Vertex T-Join Graph (VTG)

TheVTG ([17]) of Y1 and Y2 is obtained from TY1 and Y2 by joining each vertex of VY1 to every vertex of Y2 and is denoted by Y1⊕TY2, (see the Figure 5).Figure 5
TheVTG and ETG of P4 and P3.Here we derive the explicit formula of RFI for VTG of two graphs.Theorem 7.
LetYi be a graph with ni vertices and mi edges, i=1,2. Then
(11)RFY1⊕TY2=YLY1+8ELY1+3ReZMLY1+6HMLY1+12M1LY1+28YY1+36ReZMY1+30n2−2FY1+36n2−2M2Y1+12n2−22M1Y1+2n2−23m1+YY2+3ReZMY2+8YY1+3ReZMY1+m1n2−13+3n2−12M1Y1+3n2−1HMY1+8m2n1−13+6n1−1HMY2+12n1−12M1Y2+8n2FY1+n1FY2+24m2M1Y1+12m1M1Y2+n1n2n1+n2−23+3n1+n2−22+4m1n2+2m2n1+3n1+n2−24n2M1Y1+n1M1Y2+16m1m2.Proof.
By the definition of RFI and using the Table7, we have.
(12)RFY1⊕TY2=∑uv,vw∈EY1⊕TY2dY1⊕TY2u+dY1⊕TY2v−23=∑u,v∈JY1dJY1u+dJY1v−23+∑u,∈vY1,v∈JY1dY1u+dJY1v−23+∑uv∈EY2dY2u+dY2v+2n1−13+∑u∈VY1∑v∈VY22dY1u+dY2v+n1+n2−23+∑uv∈EY12dY1u+n2+2dY1v+n2−23=∑1+∑2+∑3+∑4+∑5say,respectively.∑1=∑u,v∈JY1dJY1u+dJY1v−23=∑e,f∈ELY1,e=uv,t=vwdLY1e+2+dLY1f3=∑e,f∈ELY1dLY13e+dLY13f+8+3dLY1edLY1fdLY1e+dLY1f+6dLY1e+dLY1f2+12dLY1e+dLY1f=YLY1+8ELY1+3ReZMLY1+6HMLY1+12M1LY1∑2=∑u∈VY1,v∈JY12dY1u+n2+dJY1v−23=∑uv∈EY13dY1u+dY1v+n2−23+dY1u+3dY1v+n2−23=∑uv∈EY128dY13u+dY13v+36dY1udY1vdY1u+dY1v+2n2−23+3n2−210dY12u+dY12v+12dY1udY1v+12n2−22dY1u+dY1v=28YY1+36ReZMY1+30n2−2FY1+36n2−2M2Y1+12n2−22M1Y1+2n2−23m1∑3=∑uv∈EY2dY2u+dY2v+2n1−13=∑uv∈EY2dY13u+dG13v+3∑uv∈EY2dY2udY2vdY2u+dY2v+8n2−13∑uv∈EY21+6n1−1∑uv∈EY2dY2u+dY2v2+12n1−12∑uv∈EY2dY2u+dY2v=YY2+3ReZMY2+8m2n1−13+6n1−1HMY2+12n1−12M1Y2∑4=∑u∈VY1∑v∈VY22dY1u+dY2v+n1+n2−23=8∑v∈VY2∑u∈VY1dY13u+∑u∈VY1∑v∈VY2dY23u+12∑v∈VY2dY2v∑u∈VY1dY12u+6∑u∈VY1dY1u∑v∈VY2dY22v+n1+n2−23∑u∈VY1∑v∈VY21+3n1+n2−22∑u∈VY1∑v∈VY22dY1u+dY2v+3n1+n2−2∑u∈VY1∑v∈VY24dY12u+dY12v+4dY1udY2v=8n2FY1+n1FY2+24m2M1Y1+12m1M1Y2+n1n2n1+n2−23+3n1+n2−224m1n2+2m2n1+3n1+n2−24n2M1Y1+n1M1Y2+16m1m2.∑5=∑uv∈EY12dY1u+n2+2dY1v+n2−23=8∑uv∈EY1dY1u+dY1v+n2−13=8∑uv∈EY1dY13u+dG13v+3dG1udY1vdY1u+dY1v+n2−13+3n2−12dY1u+dY1v+3n2−1dY1u+dY1v2=8YY1+3ReZMY1+m1n2−13+3n2−12M1Y1+3n2−1HMY1.
After simplifying∑i=15∑i, we get the required result.Table 7
The degree distribution ofY1⊕TY2.
u∈VY1v∈VY2e=w,k∈JY12dY1u+n2dY2v+n1dY1w+dY1k


## 3.8. Edge T-Join Graph (ETG)

LetY1 and Y2 be two vertex disjoint graphs. The ETG([17]) of Y1 and Y2, denoted by Y1⊕¯TY2, is derived by joining each vertex of TY1 to every vertex of Y2, (see Figure 5).Now we determine the exact expression of RFI forY1⊕¯TY2.Theorem 8.
LetY1 and Y2 be two graphs, then the RFI of Y1⊕¯TY2 is given by
(13)RFY1⊕¯TY2=36+n2YY1+YLY1+YY2+320+n2ReZMY1+3ReZMLY1+3ReZMY2+3M1Y1M1Y2+3n22+m1n2+4n2+2m2−20HMY1+6n2+1HMLY1+6m1−1HMY2+3m1+n2−2m1n2+n22−2n2+4m2+4n2−22+8M1Y1+12n2+12M1LY1+35m12−10m1+m1n2+4M1Y2+12n2−2FY1+m1FY2+8n2+13ELY1+6m1m2m1+n2−22+m1n2m1+n2−23+8m2m1−13+2m1n2−23−8m1.Proof.
By definition of RFI and using the Table8, we have
(14)RFY1⊕¯TY2=∑Ti=15dY1⊕¯TY2u+dY1⊕¯TY2v−23=∑T1dY1⊕¯TY2u+dY1⊕¯TY2v−23+∑T22dY1u+dJY1e−23+2dY1v+dJY1f−23+∑T3dJY1u+dJY1v−23+∑T4dJY1+n2+dY2v−23+∑T5dY2u+dY2v+2m1−23=∑uv∈EY12dY1u+dY1v−23+∑uv∈EY18dY13u+dY13v+2dY1u+dY1v+n2−23+12dY12u+dY12vdY1u+dY1v+n2−2+6dY1u+dY1vdY1u+dY1v2+2n2−2dY1u+dY1v+n2−22+∑e,f∈ELY1,e=uv,f=vwdLY1e+dLY1f+2n2+13+∑uv∈Y1∑w∈VY2dY1u+dY1v+dY2w+m1+n2−23+∑uv∈EY2dY2u+dY2v+2m1−13=8∑uv∈EY1dY13u+dY13v+3dY1udY1vdY1udY1v−3dY1u+dY1v2+3dY1u+dY1v−1+∑uv∈EY126dY13u+dY13v+2dY13u+dY13v+3dY1udY1vdY1u+dY1v+n2−23+3n2−22dY1udY1v+3n2−2dY1u+dY1v2+30dY1udY1vdY1u+dY1v+12n2−2dY1u+dY1v2+6n2−22dY1u+dY1v+dY1v+12n2−2dY12u+dY12v+∑e,f∈ELY1dLY13e+dLY13f+3dY1edY1f3dY1e+dY1f+8n2+13+6n2+1dLY1e+dLY1f2+12n2+12dY1e+dY1f+∑uv∈EY1∑w∈VY2dY13u+dY13v+3dY1udY1vdY1udY1v+m1+n2−2+dY2w3+3m1+n2−2+dY2wdY1udY1v2+3m1+n2−2+dY2w2dY1u+dY1v+∑uv∈EY2dY23u+dY23v+3dY2udY2vdY2u+dY2v+8m1−13+12m1−12dY2u+dY2v+6m1−1dY2u+dY2v2=8YY1+3ReZMY1−3HMY1+3M1Y1−m1+28YY1+36ReZMY1+18n2−2HMY1+12n2−22M1Y1+12n2−2FY1+2m1n2−23+YLY1+3ReZMLY1+8n2+13EJLY1+6n2+1HMLY1+12n2+12M1LY1+n2YY1+3n2ReZMY1+m1FY2+3m1m1+n2−2M1Y2+6m1m2m1+n2−22+m1n2m1+n2−23+3n22+m1n2−2n2+2m2HMY1+3M1Y1M1Y2+3m1+n2−2m1n2+n22−2n2+4m2M1Y1+YY2+3ReZMY2+8m2m1−13+12m1−12M1Y2+6m1−1HMY2.Table 8
The degree distribution ofY1⊕¯TY2.
u∈VY1v∈VY2e=w,k∈JY12dY1udY2v+m1dY1w+dY1k+n2


## 4. Application

Applying the above theorems, we get the following results forn,m≥3(i)
RFPn⊕SPm=nmn+m−23+62nm−n−mn+m−22+6n+m−28nm−7n−7m+4+8m−1n3+2n−1m3+12nm2n+m−48n2−18m2+112nm−118n−108m+20(ii)
RFCn⊕SCm=nmn+m+10n+m−22+8n3m+2nm3+72n2m+60nm2+16nm+16n+8m.(iii)
RFPn⊕¯SPm=n−1m2+mn+5m−6n+m−12+8m−1n−23+2n−1m3+65n2+mn−18n−m+172m−3+62n−3m2+128mn−178n+m+206.(iv)
RFCn⊕¯SCm=nmn+m+18+8mn+5n−12+2nm3+12nm2+128nm+16n−32m.(v)
RFPn⊕RPm=8n−1m−13+482n−3m−12+2n−1m3+122n−3m2+8m−1n−13+242m−3m−12+nmn+m−23+63nm−n−2mn+m−2+3n+m−236mn−22n−40m+16+792−106n−1224m−366.(vi)
RFCn⊕RCm=8nm−13+8mn−13+nmn+m+16n+m−22+2nm3+228nm2+156n2m+288nm+352n+16m.(vii)
RFPn⊕¯RPm=2n−1m3+12m22n−3n−1mn+m−13+6m−1n+m−12+62n−3n+m−1+8n−14+8m−1n−23+242n−3n−22+164n+192mn−296m−538.(viii)
RFCn⊕¯RCm=nmn+m+6n+m2+8mn−13+2nm3+60n2m+36nm2+96nm+8n2+344n+16m.(ix)
RFPn⊕QPm=8m−1n−13+nmn+m−23+2n−1m−23+242m−3n−12+62nm−n−mn+m−22+182n−3m−22+6n+m−28nm−7n−7m+4+376mn−38n−508m−614.(x)
RFCn⊕QCm=2nm+16m−22+8n+5n−12+nmn+m+10n+m−22+48nmn+m+280nm+216n−32m.(xi)
RFPn⊕¯QPm=mn−1n+m−33+6n−1m−1n+m−32+62n−3mn+m−32+4m−1n+m−3+4m−6+6n−12m−3n+m−3+68n−15m2+nm−m−2+2n−1m−23+182n−3m−22+8n−2m+13+242n−5m+12+8m−1n−23+242m−3n−22+480nm−34n−952m−356.(xii)
RFCn⊕¯QCm=8nm+7m+12+2nm+16m−22+nmn+m+16n+m−22+108nmn+m+8mn+5n−12+216nm+160n−160m.(xiii)
RFPn⊕TPm=2n−1m−23+242n−3m−22+8m−1n−13+242m−3n−12+nmn+m−23+63nm−n−2mn+m−22+6n+m−218nm−11n−20m+8+482n−3m−12+8n−1m−13+1080nm+238n−1764m−1460.(xiv)
RFCn⊕TCm=2nm+22m−22+8mn+5n−12+nmn+m+16n+m−22+108nmn+m+480nm+536n−96m.(xv)
RFPn⊕¯TPm=62n−3n+m−3m2+nm+m−4+4m−12+8+62m−35n2+nm−20n−m+19+6n−1m−1n+m−32+mn−1n+m−33+8m−1n−23+2n−1m−23+242n−5m+12+8n−2m+13+68n−15m2+nm+5m−22+360nm+1464−694m−3210.(xvi)
RFCn⊕¯TCm=nmn+m+4n+m−22+12nmn+m+2n+m−2+60nmn+m+8nm+7m+12+8mn+5n−12+632n−32m+672nm.


## 5. Conclusion

In this paper, we have executed the explicit expressions of the RFI of graphs based on the vertex and edge F-join of two graphs which are related to various subdivision graphs like as S (Y), R (Y), Q (Y) and T (Y) of the main graph Y. We also apply the results to compute the RFI for some important classes of graphs. The new (chemical) graphs from existing graphs can be constructed by applying these types of graph operations. It is also helpful to develop for generalization of these operations. So, it has an important role to gain those which physio-chemical properties are carried from original graphs to the newly constructed graphs through these new operations. In future, the RFI of some other new kind of operations and also for different many TIs for this kind of operations can be computed. These results will be more helpful in applied chemistry.



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