# Some More Results on Reciprocal Degree Distance Index andℱ-Sum Graphs

Authors: Asima Razzaque; Saima Noor; Salma Kanwal; Ayesha Manzoor
Journal: Journal of Mathematics
 (2022)

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## Abstract

A chemical invariant of graphical structureZ is a unique value characteristic that remains unchanged under graph automorphisms. In the study of QSAR/QSPR, like many other chemical invariants, reciprocal degree distance has played a significant role to estimate the bioactivity of several compounds in chemistry. Reciprocal degree distance is a chemical invariant, which is the degree weighted version of Harary index, i.e., ℛDDZ=1/2∑μ,ν∈VZ(dZμ+dZν/dZμ,ν). Eliasi and Taeri proposed four new graphic unary operations: SZ,ℛZ,QZ, and TZ, frequently implemented in sum of graphs, symbolized as Z1+Z2ℱ, i.e., sum of two graphs ℱZ1, Z2;ℱ is one of the unary graphic operations S,ℛ,Q,T. This work provides constraints for the above-mentioned invariant for this binary graphic operation F-sum of graphs.


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## Body

## 1. Introduction

Graphic structures in this work are all simple, finite, and directionless. The collection of nodes and edges in a graphZ is denoted by VZ and ΕZ, respectively. Let dZμ,ν be the distance between two vertices μ and ν in Z, given as the cardinality of lines in shortest path between the two nodes. dZμ is the degree of a vertex μ. Chemical invariants have been proven to be beneficial in determining the correlation between a molecule’s structure and its features. Quantitative structure-property relationships (QSPRs) and quantitative structure-activity relationships (QSARs) [1] are two different sorts of topological invariants. Many chemical invariants are focused on degrees, while others are focused on distances, eccentricity, connectivity, and so on.Under these parameters, numerous graph processes are executed immediately on basic graphs to examine their features. For such composite graphs, multiple researchers estimated topological invariants and put forward a big collection of results regarding this concept as can be seen below. Another chemical invariant symbolized byWZ, that comprises the length of shortest paths between the two nodes, and specified as the total of these lengths across all node combinations for Z is the Wiener index [2–4].(1)WZ=∑μ,ν⊆VZdZμ,ν.The degree-weighted variant of the Wiener index, thedegree distance index, was proposed in 1994. Dobrynin and Kochetova [5] were the first two to explore it. DDZ is the degree distance invariant of a graph Z, which is specified as(2)DDZ=∑μ,ν⊆VZdZμ,νdZμ+dZν,where dZμ,ν is the length of the shortest route among μ and ν and dZμ is the degree of vertex μ [6, 7]. Plavsi’ et al. [8] proposed the Harary index in 1993, symbolized by ℋZ and given by(3)ℋZ=12∑μ,ν∈VZ1dZμ,ν.Das et al. [9] came up with generalized version of Harary index, i.e., τ-Harary index; τ is some positive real number, given as(4)ℋZ=12∑μ,ν∈VZ1dZμ,ν+τ.A new graph invariant named asreciprocal degree distance index was introduced by Xiong and An [10] in 2015, defined as(5)ℛDZ=12∑μ,ν∈VZdZμ+dZνdZμ,ν.The degree distance invariant is the weighted version of Wiener index; similarly, the reciprocal degree distance is the weighted version of Harary index. In 2015, Vijayaragavan [11] introduced reformulated reciprocal degree distance index, which is defined as(6)ℛDDτZ=12∑μ,ν∈VZdZμ+dZνdZμ,ν+τ.Many scientists and researchers have focused on composite graphs from the past thirty years. Several binary graphic operations like productΠ, join +, composition °, corona product, cluster, wreath product, and so on were put forward by many researchers. Now we enlist briefly the work done by several researchers for the above-mentioned binary graphic processes. Yeh and Gutman [3] computed sum of distances for every pair of nodes in the above-mentioned binary graphic operations. Paulraja and Agnes [7] made computations of accurate values of degree distance invariant for Cartesian and wreath products. Moreover, the authors determined the above-mentioned invariant for certain commonly known graphic structures like torus, Hamming, hypercube, and fence structures [12], and Khalifeh et al. brought into consideration the accurate expressions for the two Zagreb invariants and computed the same invariants for certain renowned chemical structures like nano tubes and nano torus with q multi walled.Deng et al. [13] determined first two Zagreb invariants for the four graph operations, i.e., ℱ−sum of graphs. Shirdel et al. [14] proposed a new invariant named as hyper-Zagreb invariant and provided computations for these invariants for the above-mentioned binary graphic invariants.Das et al. [9] made computations of Harary invariants for same binary processes considered in [14]. During this work, the authors encountered an expression similar to Harary invariant, and they named it as second and third Harary invariants.Eliasi and Taeri [4] introduced four new binary graphic operations and termed them as ℱ-sum of structures, which gave a new direction to researchers to work with. Here we give a brief overview of work done by researchers for this new kind of operation. Imran and Akhter [15, 16] provided exact expressions for maximum and minimum value of general sum connectivity invariant and forgotten invariant, respectively. Metsidik et al. [17] provided the exact expressions for hyper and reverse Wiener invariant for F-sum of structures, and further they presented exact formulas for later mentioned invariant under certain conditions on the parameters involved. Basavangoud and Ptail [18] studied hyper-Zagreb invariant for these four graphic processes termed as F-sum. Alizadeh et al. [19] put forward a new invariant obtained by deforming Harary invariant and termed it as additive weighted Harary invariant. They also investigated this newly defined invariant for many well-known graphic operations. Pattabiraman and Vijayaraganan [20] provided exact expressions in terms of other invariants for reciprocal degree invariant of several graphic binary operations like join, strong, wreath, and tensor product. Further, applying their computed results, they computed the above-mentioned invariant for fan, open, and closed fence and wheel graph. Xiong and An [10] provided exact formulas of multiplicative weighted Harary invariant for certain well-known graphic operations like join, symmetric difference, composition, and disjunction and join. Pattabiraman and Vijayaraganan [21] put forward formulas of reformulated reciprocal degree distance invariant for many existing graphic operations like strong, wreath, and tensor products and also considered join and composition. Pattabiraman [22] presented mathematical formulas for the above-mentioned invariant and its multiplicative version for tensor product of graphic structures. Su et al. [23] observed the monotonic attitude of reformulated reciprocal degree distance invariant and determined its maximum values for graphic structures with one cycle. Novelty of this work lies in the fact that binary operations give rise to new structures displaying characteristics of the factors in a more convincible way. So, determining the bounds for reciprocal degree distance invariant for the binary operation ℱ− sum enabled us to evaluate the chemical properties of more complex structures using their binary components. In this work, we will determine lower and upper bounds for reciprocal degree distance invariant for ℱ−sum of any two graphs, where ℱ− is one of S,ℛ,Q,T.


## 2.ℱ-Sum of Graphs with Four Graph Operations

To begin, consider the Cartesian productZWY of graphs Z and Y. \{a=c and bd∈EY\} or \{b=d and ac∈EZ\} for the set of vertices VZ×VY; then, a,bc,d is a line of ZWY. If the separation between the two vertices in ZWY is dZWY(a,b,c,d), where(7)dZWYa,b,c,d=dZa,c+dYb,d,ZWY contains a vertex a,b of degree(8)dZWYa,b=dZa+dYb.Eliasi and Taeri [4] pioneered idea of ℱ-sums in 2009, referred to as graphic binary operations, in which ℱZ1 and Z2 are the operands. Z1+Z2ℱ represents ℱ-sum of two graphs Z1 and Z2, and ℱ is one of the graph operations S,ℛ,Q, or T. Figure 1 likewise shows the ℱ-sum of C3 and P2. The graph operations S,ℛ,Q,T are defined as follows:(1)
SZ subdivided structure is generated by inserting a node in all lines of Z.(2)
If two original nodes are neighbors, line connecting them is drawn inSZ. By combining each combination of connected black vertices, ℛZ is generated from SZ.(3)
In the same way, two white vertices inSZ are linked if their corresponding edges are neighbors in Z. By combining each combination of associated white vertices, QZ is achieved.(4)
The combination of two graphsZ and Y is represented as Z∪Y and has the identical vertex set V and lines EZ∪EY. The total graph TZ is the combination of ℛZ and QZ in this example.Figure 1
ℱ-sums of C3 and P2, C3+P2ℱ.The above four graph operations can be visualized in Figure2, and they are used on the C3 cycle.Figure 2
ℱG1=C3 for ℱ=S,ℛ,Q,T and G2=P2.For the above-mentioned four graph processes, multiple writers generated a variety of chemical invariants. The Wiener index of all these graph operations was estimated by Eliasi and Taeri [4]. Two optimum values for degree distance-based invariant for this new type of graphic binary operation, ℱ-sums, were considered by the authors in their work [6]. Findings of Eliasi and Taeri [4] were utilized to determine length of shortest paths existing in nodes of graphic structures involved in ℱ-sums. The authors in [15, 16] investigated novel results for these four graph operations using the forgotten index and sum-connectivity index. For each of these procedures, they provided the sharp value for the ℱ invariant and its exact upper and lower values attained for the sum connectivity invariant. Khalifeh et al. computed accurate formulas for both Zagreb invariants for several of the above-mentioned graphic binary operations. The authors in [13] investigated these sums to determine exact solutions for Zagreb invariants. Investigation of hyper-Zagreb invariants and co-invariants was made in [14], for various graphic binary operations. For these four graph operations, they derived the exact formula for hyper-Zagreb invariant.TakeV2=n2 replicas of the graph ℱZ1 and mark them with the nodes of graph Z2 for the ℱ-sum of graphs Z1 and Z2. The vertices of Z1+Z2ℱ can be divided into two categories: black vertices V1 and white vertices E1. Just black nodes with the identical name in ℱZ1 and tags that are adjacent in Z2 are now joined. Here we state results needed in next section.Lemma 1 (see [4]).
For two graphsZ1, Z2, letν=ν1,ν2be a node ofZ1+Z2ℱ. Then, we have(a)
Ifν1∈V1 (if ν is a black node), ultimately for every μ=μ1,μ2∈VZ1+Z2ℱ, then(9)dZ1+Z2ℱμ,ν=dℱZ1μ1,ν1+dZ2μ2,ν2.(b)
Ifν1∈E1, then for all μ=μ1,μ2∈VZ1+Z2ℱ, with μ2≠ν2,μ1=μ11ν11∈E1 and μ11,ν11∈V1 (if ν and μ are white nodes in ℱZ1, we have(10)dZ1+Z2ℱμ,ν=1+dZ2μ2,ν2+mindℱZ1μ11,ν1,dℱZ1ν11,ν1.(c)
Ifν1∈E1, then for all μ=μ1,μ2∈VZ1+Z2ℱ, where μ2=ν2 and μ1∈E1 (that is, ν and μ are white vertices in the same copy of ℱZ1), we have(11)dZ1+Z2ℱμ,ν=dℱZ1μ1,ν1.Lemma 2 (see [4]).
LetZ1andZ2be two graphs,μ1,ν1∈E1,μ2,ν2∈V2, andℱ=Sorℛ. Then, forμ=μ1,μ2andν=ν1,ν2inZ1+Z2ℱwithμ2≠ν2, we have(12)dZ1+Z2ℱμ,ν=2+dZ2μ2,ν2,μ1=ν1,dℱZ1μ1,ν1+dZ2μ2,ν2,μ1≠ν1.Lemma 3 (see [4]).
LetZ1andZ2be two graphs,μ1,ν1∈E1,μ2,ν2∈V2, andℱ=Sorℛ. Then, forμ=μ1,μ2andν=ν1,ν2inZ1+Z2ℱwithμ2≠ν2, then(13)dZ1+Z2ℱμ,ν=2+dZ2(μ2,ν2),μ1=ν1,1+dℱ(Z1)(μ1,ν1)+dZ2(μ2,ν2),μ1≠ν1.Lemma 4 (see [6]).
LetZ1andZ2be two graphs andμ=μ1,μ2be a vertex ofG1+Z2ℱ; then,(a)
Ifμ1∈V1 and μ2∈V2 (i.e., μ is a black vertex), then we have(14)dZ1+Z2ℱμ=dℱ1μ1+dZ2μ2.(b)
Ifμ1∈E1 and μ2∈V2 (that is, μ is a white vertex), then we have(15)dZ1+Z2ℱμ=dℱZ1μ1.Lemma 5 (see [6]).
LetZbe a graph. Then,(a)
Ifμ1∈VZ, then we have(16)dℱZ1μ1=k⋅dZμ1,where(17)k=1,ℱ=SorQ,2,ℱ=ℛorT.(b)
Ifμ1=μ1′μ1″∈EZ, then(18)dSZμ1=dℛZμ1=2,dQZμ1=dTZμ1=dLZμ1+2,where(19)dLZμ1=dZμ1′+dZμ1″.


## 3. Reciprocal Degree Distance Index of Four Operations on Graphs

We present constraints for the reciprocal degree distance ofZ1+Z2ℱ within this part, assuming ℱ=S,ℛ,Q, and T.


### 3.1. Reciprocal Degree Distance forℱ-Sum of Graphs, Where ℱ=S or ℛ

First, we present bounds for the reciprocal degree distance ofZ1+Z2ℱ in terms of reciprocal degree distance and Harary index of the components Z2,ℱZ1, where ℱ=S or ℛ.Theorem 1.
For two graphsZ1,Z2, bounds forℛDDZ1+Z2ℱ;ℱ=S,ℛare as follows:(20)V22ℛDDDℱZ1+4E1ℋ2Z2+4E2V2ℋDℱZ1−4E2V2ℋDZ1ℱZ1≤ℛDDZ1+ℱZ2≤V22ℛDD1ℱZ1+4E1ℋ2Z2+4E2V2ℋ1ℱZ1−4E2V2ℋ1E1ℱZ1,where DℱZ1 is the diameter and E1 denotes subfamily of white nodes of ℱZ1. Also,(21)ℋτE1ℱZ=12∑μ,ν∈E1;μ≠ν1dℱZ1μ,ν+τ.
,where τ is a real number. In the specific case, whenτ=1,2 or DZ2, equality holds for Z2≅KZ2.Proof.
Letμ=μ1,μ2 and ν=ν1,ν2 be two nodes in Z1+Z2ℱ. We explore the following three possibilities based on the colors of μ and ν.Case 1.
Assume black color nodes areμ=μ1,μ2 and ν=ν1,ν2; it means that μ,ν∈V1×V2. Using (a) in Lemma 1 and Lemma 4, we have(22)dZ1+Z2ℱμ,ν=dℱZ1μ1,ν1+dZ2μ2,ν2,A=12∑μ,ν∈VZ1+Z2ℱdZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2)=12∑μ1,μ2,ν1,ν2∈VZ1+Z2ℱdℱZ1μ1+dZ2μ2+dℱZ1ν1+dZ2ν2dℱZ1μ1,ν1+dZ2μ2,ν2≤12∑μ1,ν1∈V1∑μ2,ν2∈V2dℱZ1μ1+dZ2μ2+dℱZ1ν1+dZ2ν2dℱZ1μ1,ν1+1(as dZ2μ2,ν2≥1 and μ2≠ν2)(23)≤12∑μ2,ν2∈V2∑μ1,ν1∈V1dℱG1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+12∑μ1,ν1∈V11dℱZ1μ1,ν1+1∑μ2,ν2∈ν2dZ2μ2+dZ2ν2,A≤V222∑μ1,ν1∈V1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+4E2V212∑μ1,ν1∈V11dℱZ1μ1,ν1+1.Case 2.
Assume thatμ=μ1,μ2 and ν=ν1,ν2 are of dissimilar colors, so μ∈V1×V2 and ν∈E1×V2. In the present condition, by (a) in Lemma 1 and Lemma 4, when black node is μ and white is ν, eventually we get(24)dZ1+Z2ℱμ,ν=dℱZ1μ1,ν1+dZ2μ2,ν2,B´=12∑μ1,μ2∈V1×V2∑ν1,ν2∈E1×V2dZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2)=12∑μ2,ν2∈V2∑μ1∈V1∑ν1∈E1dℱZ1μ1+dZ2μ2+dℱZ1ν1dℱZ1μ1,ν1+dZ2μ2,ν2≤12∑μ2,ν2∈V2∑μ1∈V1∑ν1∈E1dℱZ1μ1+dZ2μ2+dℱZ1ν1dℱZ1μ1,ν1+1(as dZ2μ2,ν2≥1 and μ2≠ν2)(25)≤12∑μ2,ν2∈V2∑μ1∈V1∑ν1∈E1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+12∑μ1∈V1∑ν1∈E11dℱZ1μ1,ν1+1∑μ2,ν2∈V2dZ2μ2,B´≤V222∑μ1∈V1∑ν1∈E1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+2V2E212∑μ1∈V1∑ν1∈E11dℱZ1μ1,ν1+1.
As a result, for vertices of multiple colors, the above expression would be(26)B=2B´≤V22∑μ1∈V1∑ν1∈E1dℱG1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+4V2E212∑μ1∈V1∑ν1∈E11dℱZ1μ1,ν1+1.Case 3.
Assume thatμ=μ1,μ2 and ν=ν1,ν2 are white, that is, μ∈E1×V2 and ν∈E1×V2. Let(27)C=12∑μ1,μ2,ν1,ν2∈E1×V2dZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2).
We divided this total into two partsC=Ç^+Çˇ, and we have(28)Ç^=12∑μ1,μ2,ν1,ν2∈E1×V2;μ1=ν1,μ2≠ν2dZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2),Çˇ=12∑μ1,μ2,ν1,ν2∈E1×V2;u1≠ν1dZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2).
By Lemmas2, 4, and 5, we have(29)Ç^=12∑μ1∈E1∑μ2,ν2∈V2;μ2≠ν2dZ1+Z2ℱμ+dZ1+Z2ℱν2+dZ2μ2,ν2=12∑μ1∈E1∑u2,ν2∈V2;μ2≠ν22+22+dZ2μ2,ν2=2∑μ1∈E1∑μ2,ν2∈V2;μ2≠ν212+dZ2μ2,ν2=4E1ℋ2Z2,Çˇ=12∑μ1,ν1∈E1;μ1≠ν1∑μ2,ν2∈V2dZ1+Z2ℱμ+dZ1+Z2ℱνdℱZ1μ1,ν1+dZ2μ2,ν2≤12∑μ1,ν1∈E1;μ1≠ν1∑μ2,ν2∈V2dZ1+Z2ℱμ+dZ1+Z2ℱνdℱZ1μ1,ν1+1(as dZ2μ2,ν2≥1 and μ2≠ν2)(30)≤V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1,so(31)C=Ç^+Çˇ≤4E1ℋ2Z2+V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1.
Therefore, by the above calculations and the definition of reciprocal degree distance,(32)ℛDDZ1+Z2ℱ=A+B+C≤V222∑μ1,ν1∈V1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+4E2V212∑μ1,ν1∈V11dℱZ1μ1,ν1+1+V22∑μ1∈V1∑ν1∈E1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+4V2E212∑μ1∈V1∑ν1∈E11dℱZ1μ1,ν1+1+4E1ℋ2Z2+V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1,where(33)V22ℛDD1ℱZ1=V222∑μ1,ν1∈V1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V22∑μ1∈V1∑ν1∈E1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1,4E2V2ℋ1ℱZ1−4E2V212∑μ1,ν1∈E1;μ1≠ν11dℱZ1μ1,ν1+1=4E2V212∑μ1,ν1∈V11dℱZ1μ1,ν1+1+4V2E212∑μ1∈V1∑ν1∈E11dℱZ1μ1,ν1+1.
Put these values in equation (32), and we obtain(34)ℛDDZ1+Z2ℱ≤V22ℛDD1ℱZ1+4E1ℋ2Z2+4E2V2ℋ1ℱZ1−4E2V212∑u1,ν1∈E1;u1≠ν11dℱZ1u1,ν1+1ℛDDZ1+Z2ℱ≤V22ℛDD1ℱZ1+4E1ℋ2Z2+4E2V2ℋ1ℱZ1−4E2V2ℋ1E1ℱZ1.
AsdZ2u2,ν2≤DZ2, in the same manner, we have(35)ℛDDZ1+Z2ℱ≥V22ℛDDDℱZ1+4E1ℋ2Z2+4E2V2ℋDℱZ1−4E2V2ℋDE1ℱZ1.


### 3.2. Reciprocal Degree Distance forℱ-Sum of Graphs, Where ℱ=Q or T

We next present boundaries for the reciprocal degree distance invariant ofZ1+Z2ℱ in terms of other graphic invariants.Theorem 2.
For two graphsZ1,Z2, ℱ=QorT. AssumeℱZ1is of diameterDℱZ1and the collection of white nodes is symbolized byE1. Then,(36)V22ℛDDDℱZ1+4E2V2HDℱZ1−4E2V2HDE1ℱZ1+V22ℛDD2E1ℱZ1−V22ℛDD1E1ℱZ1+V2ℛDDE1ℱZ1≤ℛDDZ1+Z2ℱ≤V22ℛDD1ℱZ1+4E2V2H1ℱZ1−4E2V2H1E1ℱZ1+V22ℛDDD+1E1ℱZ1−V22ℛDDDE1ℱZ1+V2ℛDDE1ℱZ1,where(37)ℛDDτE1ℱZ=12∑μ,ν∈E1dμ+dνdℱZ1μ,ν+τ,HτE1ℱZ=12∑μ,ν∈E1;μ≠ν1dℱZ1μ,ν+τ.
In the present condition,τ=1,2,DZ2 or DZ2+1 and τ must be a real number. Bounds are attained by Z2≅KZ2.Proof.
Assume thatA, B, and C are the same as in Theorem 1. The quantities of A and B do not really change in the present situation. As a result, we merely compute the result of C. Consider(38)C=12∑μ1,μ2,ν1,ν2∈E1×V2dZ1+Z2Fμ+dZ1+Z2FνdZ1+Z2F(μ1,μ2,ν1,ν2).
We divide this total into three partsC=Ç^+Çˇ+Ç´ and(39)Ç^=12∑μ1,μ2,ν1,ν2∈E1×V2;μ1=ν1,μ2≠ν2dZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2),Çˇ=12∑μ1,μ2,ν1,ν2∈E1×V2;μ1≠ν1,μ2=ν2dZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2),Ç´=12∑μ1,μ2,ν1,ν2∈E1×V2;μ1≠ν1,μ2≠ν2dZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2).
By Lemmas3, 4, and 5, we have(40)Ç^=12∑μ1∈E1∑μ2,ν2∈V2;μ2≠ν2dZ1+Z2ℱμ+dZ1+Z2ℱν2+dZ2μ2,ν2dQZμ=dTZμ=dLZμ+2dLZμ=dZμ′+dZμ″−2=12∑μ1∈E1∑μ2,ν2∈V2;μ2≠ν2dLZ1μ1+dLZ1ν+42+dZ2μ2,ν2=12∑μ1∈E1∑μ2,ν2∈V2;μ2≠ν2dLZ1μ1+dLZ1ν2+dZ2μ2,ν2+4∑μ1∈E112∑μ2,ν2∈V2;μ2≠ν212+dZ2μ2,ν2,=12∑μ1∈E12dLZ1μ1∑μ2,ν2∈ν2;μ2≠ν212+dZ2μ2,ν2+4E1ℋ2Z2=2∑μ1∈E1dLZ1μ1ℋ2Z2+4E1ℋ2Z2∑μ∈E1dLZμ=∑μ=μ´μ´´∈E1dZμ1′+dZμ1″−2=2∑μ1=μ1′μ1″∈E1dZ1μ1′+dZ1μ1″−2ℋ2Z2+4E1ℋ2Z2=2M1Z1−2E1ℋ2Z2+4E1ℋ2Z2=2ℋ2Z2M1Z1−4E1ℋ2Z2+4E1ℋ2Z2,Ç^=2ℋ2Z2M1Z1.
Again by Lemmas3 and 4,(41)Çˇ=12∑μ2∈V2∑μ1,ν1∈E1;μ1≠ν1dFZ1μ1+dℱZ1ν1dℱZ1μ1,ν1=V22∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1=V2ℛDDE1ℱZ1,Ç´=12∑μ2,ν2∈V2;μ2≠ν2∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν11+dℱZ1μ1,ν1+dZ2μ2,ν2≤12∑μ2,ν2∈V2;μ2≠ν2∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν11+dℱZ1μ1,ν1+1≤12∑μ2,ν2∈V2;μ2≠ν2∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+12∑μ2,ν2∈V2;μ2≠ν2∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+2−12∑μ2,ν2∈V2;μ2≠ν2∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1≤V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+2−V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1≤V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V22ℛDD2E1ℱZ1−V22ℛDD2E1ℱZ1.
So, value ofC is(42)C=Ç^+Çˇ+Ç´≤V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V22ℛDD2E1ℱZ1−V22ℛDD2E1ℱZ1+V2ℛDDE1ℱZ1+2ℋ2Z2M1Z1.
As a result, by the concept of reciprocal degree distance, we get(43)ℛDDZ1+Z2ℱ=A+B+C≤V2ℛDDE1ℱZ1+2ℋ2Z2M1Z1−V22ℛDD2E1ℱZ1+V22ℛDD2E1ℱZ1+V222∑μ1,ν1∈V1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V22∑μ1∈V1∑ν1∈E1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+4E2V212∑μ1,ν1∈V11dℱZ1μ1,ν1+1+4V2E212∑μ1∈V1∑ν1∈E11dℱZ1μ1,ν1+1,where(44)V22ℛDD1ℱZ1=V222∑μ1,ν1∈V1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V22∑μ1∈V1∑ν1∈E1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1,4E2V2ℋ1ℱZ1−4E2V212∑μ1,ν1∈E1;μ1≠ν11dℱZ1μ1,ν1+1=4E2V212∑μ1,ν1∈V11dℱZ1μ1,ν1+1+4V2E212∑μ1∈V1∑ν1∈E11dℱZ1μ1,ν1+1.
When we plug these numbers into equation (43), we get(45)ℛDDZ1+Z2ℱ≤V22ℛDD1ℱZ1+4E2V2ℋ1ℱZ1−4E2V2ℋ1E1ℱZ1+V22ℛDD2E1ℱZ1−V22ℛDD1E1ℱZ1+V2ℛDDE1ℱZ1.
AsdZ2μ2,ν2≤DZ2, in the same manner, we get(46)ℛDDZ1+Z2ℱ≥V22ℛDDDℱZ1+4E2V2ℋDℱZ1−4E2V2ℋDE1ℱZ1+V22ℛDDD+1E1ℱZ1−V22ℛDDDE1ℱZ1+V2ℛDDE1ℱZ1.


## 3.1. Reciprocal Degree Distance forℱ-Sum of Graphs, Where ℱ=S or ℛ

First, we present bounds for the reciprocal degree distance ofZ1+Z2ℱ in terms of reciprocal degree distance and Harary index of the components Z2,ℱZ1, where ℱ=S or ℛ.Theorem 1.
For two graphsZ1,Z2, bounds forℛDDZ1+Z2ℱ;ℱ=S,ℛare as follows:(20)V22ℛDDDℱZ1+4E1ℋ2Z2+4E2V2ℋDℱZ1−4E2V2ℋDZ1ℱZ1≤ℛDDZ1+ℱZ2≤V22ℛDD1ℱZ1+4E1ℋ2Z2+4E2V2ℋ1ℱZ1−4E2V2ℋ1E1ℱZ1,where DℱZ1 is the diameter and E1 denotes subfamily of white nodes of ℱZ1. Also,(21)ℋτE1ℱZ=12∑μ,ν∈E1;μ≠ν1dℱZ1μ,ν+τ.
,where τ is a real number. In the specific case, whenτ=1,2 or DZ2, equality holds for Z2≅KZ2.Proof.
Letμ=μ1,μ2 and ν=ν1,ν2 be two nodes in Z1+Z2ℱ. We explore the following three possibilities based on the colors of μ and ν.Case 1.
Assume black color nodes areμ=μ1,μ2 and ν=ν1,ν2; it means that μ,ν∈V1×V2. Using (a) in Lemma 1 and Lemma 4, we have(22)dZ1+Z2ℱμ,ν=dℱZ1μ1,ν1+dZ2μ2,ν2,A=12∑μ,ν∈VZ1+Z2ℱdZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2)=12∑μ1,μ2,ν1,ν2∈VZ1+Z2ℱdℱZ1μ1+dZ2μ2+dℱZ1ν1+dZ2ν2dℱZ1μ1,ν1+dZ2μ2,ν2≤12∑μ1,ν1∈V1∑μ2,ν2∈V2dℱZ1μ1+dZ2μ2+dℱZ1ν1+dZ2ν2dℱZ1μ1,ν1+1(as dZ2μ2,ν2≥1 and μ2≠ν2)(23)≤12∑μ2,ν2∈V2∑μ1,ν1∈V1dℱG1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+12∑μ1,ν1∈V11dℱZ1μ1,ν1+1∑μ2,ν2∈ν2dZ2μ2+dZ2ν2,A≤V222∑μ1,ν1∈V1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+4E2V212∑μ1,ν1∈V11dℱZ1μ1,ν1+1.Case 2.
Assume thatμ=μ1,μ2 and ν=ν1,ν2 are of dissimilar colors, so μ∈V1×V2 and ν∈E1×V2. In the present condition, by (a) in Lemma 1 and Lemma 4, when black node is μ and white is ν, eventually we get(24)dZ1+Z2ℱμ,ν=dℱZ1μ1,ν1+dZ2μ2,ν2,B´=12∑μ1,μ2∈V1×V2∑ν1,ν2∈E1×V2dZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2)=12∑μ2,ν2∈V2∑μ1∈V1∑ν1∈E1dℱZ1μ1+dZ2μ2+dℱZ1ν1dℱZ1μ1,ν1+dZ2μ2,ν2≤12∑μ2,ν2∈V2∑μ1∈V1∑ν1∈E1dℱZ1μ1+dZ2μ2+dℱZ1ν1dℱZ1μ1,ν1+1(as dZ2μ2,ν2≥1 and μ2≠ν2)(25)≤12∑μ2,ν2∈V2∑μ1∈V1∑ν1∈E1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+12∑μ1∈V1∑ν1∈E11dℱZ1μ1,ν1+1∑μ2,ν2∈V2dZ2μ2,B´≤V222∑μ1∈V1∑ν1∈E1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+2V2E212∑μ1∈V1∑ν1∈E11dℱZ1μ1,ν1+1.
As a result, for vertices of multiple colors, the above expression would be(26)B=2B´≤V22∑μ1∈V1∑ν1∈E1dℱG1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+4V2E212∑μ1∈V1∑ν1∈E11dℱZ1μ1,ν1+1.Case 3.
Assume thatμ=μ1,μ2 and ν=ν1,ν2 are white, that is, μ∈E1×V2 and ν∈E1×V2. Let(27)C=12∑μ1,μ2,ν1,ν2∈E1×V2dZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2).
We divided this total into two partsC=Ç^+Çˇ, and we have(28)Ç^=12∑μ1,μ2,ν1,ν2∈E1×V2;μ1=ν1,μ2≠ν2dZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2),Çˇ=12∑μ1,μ2,ν1,ν2∈E1×V2;u1≠ν1dZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2).
By Lemmas2, 4, and 5, we have(29)Ç^=12∑μ1∈E1∑μ2,ν2∈V2;μ2≠ν2dZ1+Z2ℱμ+dZ1+Z2ℱν2+dZ2μ2,ν2=12∑μ1∈E1∑u2,ν2∈V2;μ2≠ν22+22+dZ2μ2,ν2=2∑μ1∈E1∑μ2,ν2∈V2;μ2≠ν212+dZ2μ2,ν2=4E1ℋ2Z2,Çˇ=12∑μ1,ν1∈E1;μ1≠ν1∑μ2,ν2∈V2dZ1+Z2ℱμ+dZ1+Z2ℱνdℱZ1μ1,ν1+dZ2μ2,ν2≤12∑μ1,ν1∈E1;μ1≠ν1∑μ2,ν2∈V2dZ1+Z2ℱμ+dZ1+Z2ℱνdℱZ1μ1,ν1+1(as dZ2μ2,ν2≥1 and μ2≠ν2)(30)≤V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1,so(31)C=Ç^+Çˇ≤4E1ℋ2Z2+V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1.
Therefore, by the above calculations and the definition of reciprocal degree distance,(32)ℛDDZ1+Z2ℱ=A+B+C≤V222∑μ1,ν1∈V1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+4E2V212∑μ1,ν1∈V11dℱZ1μ1,ν1+1+V22∑μ1∈V1∑ν1∈E1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+4V2E212∑μ1∈V1∑ν1∈E11dℱZ1μ1,ν1+1+4E1ℋ2Z2+V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1,where(33)V22ℛDD1ℱZ1=V222∑μ1,ν1∈V1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V22∑μ1∈V1∑ν1∈E1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1,4E2V2ℋ1ℱZ1−4E2V212∑μ1,ν1∈E1;μ1≠ν11dℱZ1μ1,ν1+1=4E2V212∑μ1,ν1∈V11dℱZ1μ1,ν1+1+4V2E212∑μ1∈V1∑ν1∈E11dℱZ1μ1,ν1+1.
Put these values in equation (32), and we obtain(34)ℛDDZ1+Z2ℱ≤V22ℛDD1ℱZ1+4E1ℋ2Z2+4E2V2ℋ1ℱZ1−4E2V212∑u1,ν1∈E1;u1≠ν11dℱZ1u1,ν1+1ℛDDZ1+Z2ℱ≤V22ℛDD1ℱZ1+4E1ℋ2Z2+4E2V2ℋ1ℱZ1−4E2V2ℋ1E1ℱZ1.
AsdZ2u2,ν2≤DZ2, in the same manner, we have(35)ℛDDZ1+Z2ℱ≥V22ℛDDDℱZ1+4E1ℋ2Z2+4E2V2ℋDℱZ1−4E2V2ℋDE1ℱZ1.


## 3.2. Reciprocal Degree Distance forℱ-Sum of Graphs, Where ℱ=Q or T

We next present boundaries for the reciprocal degree distance invariant ofZ1+Z2ℱ in terms of other graphic invariants.Theorem 2.
For two graphsZ1,Z2, ℱ=QorT. AssumeℱZ1is of diameterDℱZ1and the collection of white nodes is symbolized byE1. Then,(36)V22ℛDDDℱZ1+4E2V2HDℱZ1−4E2V2HDE1ℱZ1+V22ℛDD2E1ℱZ1−V22ℛDD1E1ℱZ1+V2ℛDDE1ℱZ1≤ℛDDZ1+Z2ℱ≤V22ℛDD1ℱZ1+4E2V2H1ℱZ1−4E2V2H1E1ℱZ1+V22ℛDDD+1E1ℱZ1−V22ℛDDDE1ℱZ1+V2ℛDDE1ℱZ1,where(37)ℛDDτE1ℱZ=12∑μ,ν∈E1dμ+dνdℱZ1μ,ν+τ,HτE1ℱZ=12∑μ,ν∈E1;μ≠ν1dℱZ1μ,ν+τ.
In the present condition,τ=1,2,DZ2 or DZ2+1 and τ must be a real number. Bounds are attained by Z2≅KZ2.Proof.
Assume thatA, B, and C are the same as in Theorem 1. The quantities of A and B do not really change in the present situation. As a result, we merely compute the result of C. Consider(38)C=12∑μ1,μ2,ν1,ν2∈E1×V2dZ1+Z2Fμ+dZ1+Z2FνdZ1+Z2F(μ1,μ2,ν1,ν2).
We divide this total into three partsC=Ç^+Çˇ+Ç´ and(39)Ç^=12∑μ1,μ2,ν1,ν2∈E1×V2;μ1=ν1,μ2≠ν2dZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2),Çˇ=12∑μ1,μ2,ν1,ν2∈E1×V2;μ1≠ν1,μ2=ν2dZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2),Ç´=12∑μ1,μ2,ν1,ν2∈E1×V2;μ1≠ν1,μ2≠ν2dZ1+Z2ℱμ+dZ1+Z2ℱνdZ1+Z2ℱ(μ1,μ2,ν1,ν2).
By Lemmas3, 4, and 5, we have(40)Ç^=12∑μ1∈E1∑μ2,ν2∈V2;μ2≠ν2dZ1+Z2ℱμ+dZ1+Z2ℱν2+dZ2μ2,ν2dQZμ=dTZμ=dLZμ+2dLZμ=dZμ′+dZμ″−2=12∑μ1∈E1∑μ2,ν2∈V2;μ2≠ν2dLZ1μ1+dLZ1ν+42+dZ2μ2,ν2=12∑μ1∈E1∑μ2,ν2∈V2;μ2≠ν2dLZ1μ1+dLZ1ν2+dZ2μ2,ν2+4∑μ1∈E112∑μ2,ν2∈V2;μ2≠ν212+dZ2μ2,ν2,=12∑μ1∈E12dLZ1μ1∑μ2,ν2∈ν2;μ2≠ν212+dZ2μ2,ν2+4E1ℋ2Z2=2∑μ1∈E1dLZ1μ1ℋ2Z2+4E1ℋ2Z2∑μ∈E1dLZμ=∑μ=μ´μ´´∈E1dZμ1′+dZμ1″−2=2∑μ1=μ1′μ1″∈E1dZ1μ1′+dZ1μ1″−2ℋ2Z2+4E1ℋ2Z2=2M1Z1−2E1ℋ2Z2+4E1ℋ2Z2=2ℋ2Z2M1Z1−4E1ℋ2Z2+4E1ℋ2Z2,Ç^=2ℋ2Z2M1Z1.
Again by Lemmas3 and 4,(41)Çˇ=12∑μ2∈V2∑μ1,ν1∈E1;μ1≠ν1dFZ1μ1+dℱZ1ν1dℱZ1μ1,ν1=V22∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1=V2ℛDDE1ℱZ1,Ç´=12∑μ2,ν2∈V2;μ2≠ν2∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν11+dℱZ1μ1,ν1+dZ2μ2,ν2≤12∑μ2,ν2∈V2;μ2≠ν2∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν11+dℱZ1μ1,ν1+1≤12∑μ2,ν2∈V2;μ2≠ν2∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+12∑μ2,ν2∈V2;μ2≠ν2∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+2−12∑μ2,ν2∈V2;μ2≠ν2∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1≤V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+2−V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1≤V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V22ℛDD2E1ℱZ1−V22ℛDD2E1ℱZ1.
So, value ofC is(42)C=Ç^+Çˇ+Ç´≤V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V22ℛDD2E1ℱZ1−V22ℛDD2E1ℱZ1+V2ℛDDE1ℱZ1+2ℋ2Z2M1Z1.
As a result, by the concept of reciprocal degree distance, we get(43)ℛDDZ1+Z2ℱ=A+B+C≤V2ℛDDE1ℱZ1+2ℋ2Z2M1Z1−V22ℛDD2E1ℱZ1+V22ℛDD2E1ℱZ1+V222∑μ1,ν1∈V1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V22∑μ1∈V1∑ν1∈E1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+4E2V212∑μ1,ν1∈V11dℱZ1μ1,ν1+1+4V2E212∑μ1∈V1∑ν1∈E11dℱZ1μ1,ν1+1,where(44)V22ℛDD1ℱZ1=V222∑μ1,ν1∈V1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V22∑μ1∈V1∑ν1∈E1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1+V222∑μ1,ν1∈E1;μ1≠ν1dℱZ1μ1+dℱZ1ν1dℱZ1μ1,ν1+1,4E2V2ℋ1ℱZ1−4E2V212∑μ1,ν1∈E1;μ1≠ν11dℱZ1μ1,ν1+1=4E2V212∑μ1,ν1∈V11dℱZ1μ1,ν1+1+4V2E212∑μ1∈V1∑ν1∈E11dℱZ1μ1,ν1+1.
When we plug these numbers into equation (43), we get(45)ℛDDZ1+Z2ℱ≤V22ℛDD1ℱZ1+4E2V2ℋ1ℱZ1−4E2V2ℋ1E1ℱZ1+V22ℛDD2E1ℱZ1−V22ℛDD1E1ℱZ1+V2ℛDDE1ℱZ1.
AsdZ2μ2,ν2≤DZ2, in the same manner, we get(46)ℛDDZ1+Z2ℱ≥V22ℛDDDℱZ1+4E2V2ℋDℱZ1−4E2V2ℋDE1ℱZ1+V22ℛDDD+1E1ℱZ1−V22ℛDDDE1ℱZ1+V2ℛDDE1ℱZ1.


## 4. Conclusion

In this work, we presented constraints (lower and upper) for the reciprocal degree distance ofZ1+Z2ℱ, assuming ℱ=S,ℛ in form of Theorem 1 and for ℱ=Q,T as in Theorem 2. We also observed that these bounds are expressions involving several other chemical invariants.Several tools of graph theory are being used to formulate mathematical structure of many phenomena of chemistry. Solutions of these molecular problems are being considered using some nontrivial graph theoretical ideas. Combination of mathematics, chemistry and information science is giving rise to an emerging field of research chem- informatics, that is under kind consideration of many new researchers. In the future, we are interested in computing the bounds for other degree-based chemical invariants forℱ-sums of graphs like redefined Zagreb indices. Further, we plan to order the ℱ-sums of graphs with respect to the above-mentioned chemical invariants giving first, second, and third maxima or minima.



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*Source: 3178497-2022-03-23.xml*