# Exact Formulae for Degree Distance Indices of Sum Graphs

Authors: Muhammad Javaid; Faiz Farid; Jia-Bao Liu; Abdulaziz Mohammed Alanazi
Journal: Journal of Mathematics
 (2022)

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## Abstract

The degree distance index (DDI) is a vertex-degree weighted version of a well-known index that is called by Wiener index (WI). In extremal theory of graphs, improving the bounds with best possible values is a worth investigating problem. In this note, the exact formulae of theDDI for the four different types of the sum graphs in the terms of various indices of their factor graphs are computed. Moreover, a comparison is also presented between the obtained exact and already existing bounded values for the particular sum graphs.


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## Body

## 1. Introduction

Molecular structures can be treated as a graph in which atoms are symbolized by nodes and bounds are presented by edges. For these molecular graphs, the graph-related combinatorial methodologies are widely applied to predict the various structural and physicochemical properties of their organic compounds such as molecular weight, density, volume, vaporization refraction, pressure, freezing, and boiling point. Topological indices (TIs) are frequently used in the last two decades for the studies of the molecular graphs. The very first distance-based topological index called by path number or Wiener index (WI) that came into existence when Wiener was working on boiling point of Paraffin [1]. After that, degree-related indices (Zagreb indices) are defined to find total π-electron energy of the molecules [2]. Moreover, TIs play an important role in analysing and modeling of the quantitative structures property and activity relationships (QSPR and QSAR) [3–5]. These all phenomena are studied with the combination (chemoinformatics) of major subjects mathematics, information sciences, and chemistry. There are many TIs; among them, important distance-based TIs are degree distance index (DDI) [6, 7], Gutman index [8], average distance index [9], and Harary index [10, 11]. More importantly, two graphs have equal TIs if they are isomorphic to each other. For further studies of the distance-based TIs, we refer to [12–15] and for degree-related TIs, see [16–18].In computing graph theory, new graphs are obtained by using the various operations of graphs. For a connected molecular graphH, the graphs F1H (subdivided graph), F2H (triangle parallel graph), F3H (line superposition graph), and F4H (total graph) are constructed with the help of the operations F1, F2, F3, and F4, respectively, see [19]. Later on, Eliasi and Taeri [20] defined F-sum graphs using product (Cartesian) on the graph FH1 and H2, where FH1 is obtained by applying the operation F∈F1,F2,F3,F4. These F-sum graphs create hexagonal chains, which are isomorphic to various chemical compounds. They also computed the Wiener indices (WIs) of these F-sum graphs H1+F1H2, H1+F2H2, H1+F3H2, and H1+F4H2. First, general Zagreb indices [21] and the general sum-connectivity index [22, 23] are also computed for these F-sum graphs. For Gutman index of these graphs, see [24, 25].Recently, the upper bounds of the DDIs for all theF-sum graphs are computed [26]. In this note, DDIs are computed for the family of F-sum (H1+F1H2,H1+F2H2, H1+F3H2, and H1+F4H2) graphs in the form of their exact formulae. Moreover, a comparison is also included to show the sharpness of the obtained results with the already existing bounds. The remaining paper is written as follows: Section 2 covers some basic terminologies and related lemmas (statements), Section 3 describes the main results, and Section 4 includes the DDIs for the particular families of the F-sum graphs as the consequence of the main results obtained in Section 3.


## 2. Preliminaries

Thoroughly, we consider two simple and connected graphsH1=H and H2. The degree of vertex X is the number of edges incident with it and is denoted by δx. The minimum length of the path between both the vertices x,y∈VH is called the distance between two vertices and is dented as dx,y. Two vertices x and y are adjacent (x∼y) if they form an edge xy. Further details can be studied in [27–29]. The more important TIs are defined as follows.Definition 1.
(see [1]). Let H be a connected graph, then its Wiener index (WI) is(1)WH=12∑y,z∈VHdy,z.Definition 2.
(see [8]). Let H be a connected graph, then the DDI of H is(2)DDH=12∑y,z∈VHdy,zδy+δz.
Yan et al. [19] defined F1H, F2H, F3H and F4H on the graph H as follows:(i)
F1H is obtained from H if every edge of graph H is replaced by a path of length 2 where old and new vertices of graph H are presented by black vertices and white vertices, respectively.(ii)
F2H is formed by assigning a new vertex to each edge of graph H, and this new vertex is connected with the end vertices of the respective edge.(iii)
F3H is formed from graph F1H if two white vertices of corresponding edges with one common end vertex are further joined together.(iv)
F4H is formed from F2H if two white vertices of corresponding edges with one common end vertex are further joined together.
Figure1 explains the graphs F1P5, F2P5, F3P5, and F4P5.Figure 1
(a)P5, (b) F1P5, (c) F2P5, (d) F3P5, and (e) F4P5.
(a)(b)(c)(d)(e)Definition 3.
LetH1+FH2 be a F-sum graph with V(H1+FH2=VH1∪EH1×VH2. For w,x,y,z∈VH1+FH2, if either w=y and xz∈EH2 or x=z and wy∈EFH1, then wy∼xz.
Figure2 explains P3+F1P2, P3+F2P2, P1+F3P2, and T-sum P3+F4P2 graphs. Now, some lemmas of distances between vertices are given as follows.Figure 2
(a)P3+F1P2. (b) P3+F2P2. (c) P3+F3P2. (d) P3+F4P2.
(a)(b)(c)(d)Lemma 1 (see [20]).
InF-sum graph H1+FH2 of two connected and simple graphs H1 and H2 where F∈F1,F2,F3,F4, if either both vertices w,x and y,z are black or one vertex w,x is black and second y,z is white, then dw,x,y,z|H1+FH2=dw,y|FH1+dx,z|H2.Lemma 2 (see [20]).
InF-sum graph H1+FH2 of two connected and simple graphs H1 and H2 where F=F1 or F2 and both w,x and y,z are white vertices, then(3)dw,x,y,z|H1+FH2=2+dx,z|H2)If w=yd(w,y|FH1+dx,z|H2Ifw≠y.Lemma 3 (see [20]).
InF-sum graph H1+FH2 of two connected and simple graphs H1 and H2 graphs where F=F3 or F4, and both w,x and y,z are white vertices, then(4)dw,x,y,z|H1+FH2=2+dx,z|H2)If w=y1+d(w,y|FH1+dx,z|H2Ifw≠y,x≠z.


## 3. Main Results

This section presents the degree distance indices ofF-sum graphs H1+FH2 for F∈F1,F2,F3,F4 with VH1+FH2=xi,yj:1≤i≤n,1≤j≤m∪zi,yj:1≤i≤k,1≤j≤m, where VH1=x1,x2,x3,…,xn are black vertices, Z=z1,z2,z3,…,zk are white vertices, VH2=y1,y2,y3,…,ym, EH1=k, and EH2=l.Theorem 1.
IfH1+F1H2 is the F1-sum graph of two graphs H1 and H2, then(5)DDH1+F1H2=m2DDF1H1+nn+kDDH2+8kn+kWH2+4km2−m+2lm∑i,j=1ndxi,yj|F1H1+2lm∑r=1k∑i=1ndzr,xi|F1H1.Proof.Case 1.
Between two old vertices,(6)A=12∑xi,yp,xj,yq∈H1+F1H2dxi,yp,xj,yq(δxi,yp+δxj,yq|H1+F1H2],where p,q=1 to m and i,j=1 to n, for F1-sum δxi,yp=δxi+δyp(7)=12∑i,j=1n∑p,q=1mdxi,yj|F1H1+d(yp,yq|H2δxi+δyp+δxj+δyq=12∑i,j=1n∑p,q=1mdxi,yj|F1H1δxi+δxj+12∑i,j=1n∑p,q=1mdyp,yq|H2δyp+δyq=12∑i,j=1n∑p,q=1mdxi,yj|F1H1δyp+δyq+12∑i,j=1n∑p,q=1mdyp,yq|H2δxi+δxj=m22∑i,j=1ndxi,xj|F1H1δxi+δxj+n2DDH2+12∑i,j=1ndxi,xj|F1H1∑p,q=1mδyp+δyq+12∑i,j=1nδxi+δxj∑p,q=1mdyp,yq|H2A=m22∑i,j=1ndxi,xj|F1H1δxi+δxj+n2DDH2+2lm∑i,j=1ndxi,xj|F1H1+4knWH2.Case 2.
Between old and new vertices,(8)B1=12∑zr,yp,xi,yq∈H1+F1H2dzr,yp,zi,yq(δzr,yp+δxi,yq|H1+F1H2,where p,q=1 to m, i=1 to nr=1 to k and δzr,yp=2(9)=12∑r=1k∑i=1n∑p,q=1mdzr,xi|F1H1+d(yp,yq|H22+δxi+δyq=∑r=1k∑i=1n∑p,q=1mdzr,xi|F1H1+∑r=1k∑i=1n∑p,q=1mdyp,yq|H2+12∑r=1k∑i=1n∑p,q=1mdzr,xi|F1H1δxi+12∑r=1k∑i=1n∑p,q=1mdzr,xi|F1H1δyq+12∑r=1k∑i=1n∑p,q=1mdyp,yq|H2δxi+12∑r=1k∑i=1n∑p,q=1mdyp,yq|H2δyp,=m2∑r=1k∑i=1ndzr,xi|F1H1+2knWH2+m22∑r=1k∑i=1ndzr,xi|F1H1δxi+lm∑r=1k∑i=1ndzr,xi|F1H1+2k2WH2+kn2DDH2.
The distance between old and new vertices is twice theB1, i.e.,(10)B=2m2∑r=1k∑i=1ndzr,xi|F1H1+4knWH2+m2∑r=1k∑i=1ndzr,xi|F1H1δxi,+2lm∑r=1k∑i=1ndzr,xi|F1H1+4k2WH2+knDDH2.Case 3.
Between two new vertices;(11)C=12∑zr,yp,zs,yq∈H1+F1H2dzr,yp,zs,yq(δzr,yp+δzs,yq|H1+F1H2.
Also,C=C1+C2 where(12)C1=12∑zr,yp,zs,yq∈H1+F1H2dzr,yp,zs,yq(δzr,yp+δzs,yq|H1+F1H2:r=s,p≠q,C2=12∑zr,yp,zs,yq∈H1+F1H2dzr,yp,zs,yq(δzr,yp+δzs,yq|H1+F1H2:r≠s,where δzr,yp=2, r,s=1 to k and p,q=1 to m.(13)C1=12∑p,q=1,p≠qm∑r,s=1,r=sk2+dyp,yq2+2|H1+F1H2,C1=4∑p,q=1,p≠qm∑r,s=1,r=sk+2∑p,q=1,p≠qm∑r,s=1,r=skdyp,yq|H2=m2−m4k+4kWH2,C2=12∑r,s=1,r≠sk∑p,q=1mdzr,zs|F1H1+dyp,yq|H24,C2=2∑r,s=1,r≠sk∑p,q=1mdzr,zs|F1H1+2∑r,s=1,r≠sk∑p,q=1mdyp,yq|H2,C2=2m2∑r,s=1,r≠skdzr,zs|F1H1+4k2−kWH2,C=m2−m4k+4kWH2+2m2∑r,s=1,r≠skdzr,zs|F1H1+4k2−kWH2,C=m2−m4k+2m2∑r,s=1,r≠skdzr,zs|F1H1+4k2WH2DDH1+F1H2=A+B+C,DDH1+F1H2=m22∑i,j=1ndxi,xj|F1H1δxi+δxj+n2DDH2+2lm∑i,j=1ndxi,xj|F1H1+4knWH2+2m2∑r=1k∑i=1ndzr,xi|F1H1+4knWH2+m2∑r=1k∑i=1ndzr,xi|F1H1δxi+2lm∑r=1k∑i=1ndzr,xi|F1H1+4k2WH2+knDDH2+4km2−m+2m2∑r,s=1,r≠skdzr,xi|F1H1+4k2WH2,DDH1+F1H2=m2[12∑i,j=1ndxi,xj|F1H1δxi+δxj+2∑r=1k∑i=1ndzr,xi|F1H1+∑r=1k∑i=1ndzr,xi|F1H1δxi+2∑r,s=1,r≠skdzr,zs|F1H1]+nn+kDDH2+8kn+kWH2+4km2−m+2lm∑i,j=1ndxi,xj|F1H1+2lm∑r=1k∑i=1ndzr,xi|F1H1DDH1+F1H2=m2DDF1H1+nn+kDDH2+8kn+kWH2+4km2−m+2lm∑i,j=1ndxi,xj|F1H1+2lm∑r=1k∑i=1ndzr,xi|F1H1.Theorem 2.
IfH1+F2H2 is the F2-sum graph of two graphs H1 and H2, then(14)DDH1+F2H2=m2DDF2H1+nn+kDDH2+12kn+kWH2+4km2−m+2lm∑i,j=1ndxi,xj|F2H1+2lm∑r=1k∑i=1ndzr,xi|F2H1.Proof.Case 4.
Between two old vertices,(15)A=12∑xi,yp,xj,yq∈H1+F2H2dxi,yp,xj,yq(δxi,yp+δxj,yq|H1+F2H2],where p,q=1 to m and i,j=1 to n, for R sum δxi,yp=2δxi+δyp(16)=12∑i,j=1n∑p,q=1mdxi,xj|F2H1+d(yp,yq|H22δxi+δyp+2δxj+δyq=∑i,j=1n∑p,q=1mdxi,xj|F2H1δxi+δxj+12∑i,j=1n∑p,q=1mdyp,yq|H2δyp+δyq+12∑i,j=1n∑p,q=1mdxi,xj|F2H1δyp+δyq+∑i,j=1n∑p,q=1mdyp,yq|H2δxi+δxj=m2∑i,j=1ndxi,xj|F2H1δxi+δxj+n2DDH2+∑p,q=1mδyp+δyq12∑i,j=1ndxi,xj|F2H1+∑,q=1mdyp,yq|H2∑i,j=1nδxi+δxjA=m2∑i,j=1ndxi,xj|F2H1δxi+δxj+n2DDH2+2lm∑i,j=1ndxi,xj|F2H1+8knWH2.Case 5.
Between old and new vertices,(17)B1=12∑zr,yp,xi,yq∈H1+F2H2dzr,yp,xi,yq(δzr,yp+δxi,yq|H1+F2H2,where p,q=1 to m, i=1 to nr=1 to k and δzr,yp=2(18)=12∑r=1k∑i=1n∑p,q=1mdzr,xiF2H1+d(yp,yq|H2|2+2δxi+δyq=∑r=1k∑i=1n∑p,q=1mdzr,xiF2H1+∑r=1k∑i=1n∑p,q=1md(yp,yq|H2,+∑r=1k∑i=1n∑p,q=1mdzr,xiF2H1δxi+12∑r=1k∑i=1n∑p,q=1mdzr,xiF2H1δyq+∑r=1k∑i=1n∑p,q=1md(yp,yq|H2δxi+12∑r=1k∑i=1n∑p,q=1md(yp,yq|H2δyq=m2∑r=1k∑i=1ndzr,xiF2H1+2knWH2+m2∑i=1n∑r=1kdzr,xiF2H1δxi+lm∑r=1k∑i=1ndzr,xiF2H1+4k2WH2+kn2DDH2.
The distance between old and new vertices is twice theB1, i.e.,(19)B=2m2∑r=1k∑i=1ndzr,xiF2H1+4knWH2+2m2∑i=1n∑r=1kdzr,xiF2H1δxi+2lm∑r=1k∑i=1ndzr,xiF2H1+8k2WH2+knDDH2.Case 6.
Between two new vertices, C can be calculated similarly as in Theorem1 Case 3:(20)C=4km2−m+2m2∑r,s=1,r≠skdzr,zs|F2H1+4k2WH2,DDH1+F2H2=A+B+C,DDH1+F2H2=m2∑i,j=1ndxi,xj|F2H1δxi+δxj+n2DDH2+2lm∑i,j=1ndxi,xj|F2H1+8knWH2+2m2∑r=1k∑i=1ndzr,xi|F2H1+4knWH2+2m2∑r=1k∑i=1ndzr,xi|F2H1δxi+2lm∑r=1k∑i=1ndzr,xi|F2H1+8k2WH2+knDDH2+4km2−m+2m2∑r,s=1,r≠skdzr,zs|F2H1+4k2WH2.DDH1+F2H2=m2∑i,j=1ndxi,xj|F2H1δxi+δxj+2∑r=1k∑i=1ndzr,xi|F2H1+∑r=1k∑i=1ndzr,xi|F2H12δxi+2∑r,s=1,r≠skdzr,zs|F2H1+nn+kDDH2+12kn+kWH2+4km2−m+2lm∑i,j=1ndxi,xj|F2H1+2lm∑r=1k∑i=1ndzr,xi|F2H1DDH1+F2H2=m2DDF2H1+nn+kDDH2+12kn+kWH2+4km2−m+2lm∑i,j=1ndxi,xj|F2H1+2lm∑r=1k∑i=1ndzr,xi|F2H1.Theorem 3.
IfH1+F3H2 is the F3-sum graph of two graphs H1 and H2, then(21)DDH1+F3H2=m2DDF3H1+nn+kDDH2+26k−tn+kWH2+k+14k−tm2−m+2lm∑i,j=1ndxi,xj|F3H1+2lm∑r=1k∑i=1ndzr,xi|F3H1,where t is the number of pendant vertices in graph H1Proof.Case 7.
Between two old vertices, A can be calculated similarly as in Theorem1 Case 1:(22)A=m22∑i,j=1ndxi,xj|F3H1δxi+δxj+n2DDH2+2lm∑i,j=1ndxi,xj|F3H1+4knWH2.Case 8.
Between old and new vertices,(23)B1=12∑zr,yp,xi,yq∈H1+F3H2dzr,yp,xi,yq(δwr,yp+δxi,yq|H1+F3H2,where p,q=1 to m, i=1 to nr=1 to k and δxi,yq=δxi+δyj(24)=12∑r=1k∑i=1n∑p,q=1mdzr,xi|F3H1+d(yp,yq|H2δ(zr,yp)+dxi+dyq=12∑r=1k∑i=1n∑p,q=1mdzr,xi|F3H1δzr,yp+12∑r=1k∑i=1n∑p,q=1mdyp,yq|H2δzr,yp+12∑r=1k∑i=1n∑p,q=1mdzr,xi|F3H1δxi+12∑r=1k∑i=1n∑p,q=1mdzr,xi|F3H1δyq+12∑r=1k∑i=1n∑p,q=1mdyp,yq|H2δxi+12∑r=1k∑i=1n∑p,q=1mdyp,yq|H2δyq=m22∑r=1k∑i=1ndzr,xi|F3H1δzr,yp+n4k−tWH2+m22∑r=1k∑i=1ndzr,xi|F3H1δxi+lm∑r=1k∑i=1ndzr,xi|F3H1+2k2WH2+kn2DDH2.
The distance between old and new vertices is twice theB1, i.e.,(25)B=m2∑r=1k∑i=1ndzr,xi|F3H1δzr,yp+2n4k−tWH2+m2∑i=1n∑r=1kdzr,xi|F3H1δxi+2lm∑r=1k∑i=1ndzr,xi|F3H1+4k2WH2+knDDH2.Case 9.
Between two new vertices:(26)C=12∑zr,yp,zs,yq∈H1+F3H2dzr,yp,zs,yq(δ(zr,yp+δzs,yq|H1+F3H2.
Also,C=C1+C2+C3, where(27)C1=12∑zr,yp,zs,yq∈H1+F3H2dzr,yp,zs,yqδzr,yp+δzs,yq|H1+F3H2:p≠q,r=s,C2=12∑zr,yp,zs,yq∈H1+F3H2dzr,yp,zs,yqδzr,yp+δzs,yq|H1+F3H2:p=q,r≠s,C3=12∑zr,yp,zs,yq∈H1+F3H2dzr,yp,zs,yqδzr,yp+δzs,yq|H1+F3H2:p≠q,r≠s,where p,q=1 to m and r,s=1 to k.(28)C1=12∑p,q=1,p≠qm∑r,s=1,r=sk2+dyp,yq|H2δzr,yp+δzs,yq|H1+F3H2,C1=∑p,q=1,p≠qm∑r,s=1,r=skδzr,yp+δzs,yq|H1+F3H2+12∑p,q=1,p≠qm∑r,s=1,r=skdyp,yq|H2δzr,yp+δzs,yq=24k−tm2−m+24k−tWH2,C2=12∑p,q=1,p=qm∑r,s=1,r≠skdzr,zs|F3H1δzr,yp+δzs,yq=m2∑r,s=1,r≠skdzr,zs|F3H1(δzr,yp+δzs,yq,C3=12∑p,q=1,p≠qm∑r,s=1,r≠sk1+dzr,zs|F3H1+dyp,yq|H2δzr,yp+δzs,yq=12∑p,q=1,p≠qm∑r,s=1,r≠skδzr,yp+δzs,yq+12∑r,s=1,r≠sk∑p,q=1,p≠qmdzr,zs|F3H1δzr,yp+12∑p,q=1,p≠qm∑r,s=1,r≠skdzr,zs|F3H1δzs,yq+12∑p,q=1,p≠qm∑r,s=1,r≠skdyp,yq|H2δzr,yp+12∑p,q=1,p≠qm∑r,s=1,r≠skdyp,yq|H2δzs,yq)=k−14k−tm2−m+mm−12∑r,s=1,r≠skdzr,zs|F3H1δzr,yp+δzs,yq+2k−14k−tWH2,C=24k−tm2−m+24k−tWH2+m2∑r,s=1,r≠skdzr,zs|F3H1δzr,yp+δzs,yq+m2−m4k−tk−1+mm−12∑r,s=1,r≠skdzr,zs|F3H1δzr,yp+δzs,yq+24k−tk−1WH2,C=k+14k−tm2−m+2k4k−tWH2+m22∑r,s=1,r≠skdzr,zs|F3H1δzr,vp+δzs,yq,DDH1+F3H2=A+B+C,DDH1+F3H2=m22∑i,j=1ndxi,xj|F3H1δxi+dδj+n2DDH2+4knWH2+2lm∑i,j=1ndxi,xj|F3H1+m2∑r=1k∑i=1ndzr,xi|F3H1δzr,yp+2n4k−tWH2+m2∑r=1k∑i=1ndzr,xi|F3H1δxi+2lm∑r=1k∑i=1ndzr,xi|F3H1+4k2WH2+knDDH2+4k−tk+1m2−m+2k4k−tWH2+m22∑r,s=1,r≠skdzr,zs|F3H1δzr,yp+δzs,yqDDH1+F3H2=m212∑i,j=1ndxi,xj|F3H1δxi+δxj+∑r=1k∑i=1ndzr,xi|F3H1δzr,yp+∑r=1k∑i=1ndzr,xi|F3H1δxi+12∑r,s=1,r≠skδzr,zs|F3H1δzr,yp+δzs,yq+n2DDH2+knDDH2+WH24kn+2n4k−t+2k4k−t+4k2+k+1m2−m4k−t+2lm∑i,j=1ndxi,xj|F3H1+2lm∑r=1k∑i=1ndzr,xi|F3H1,DDH1+F3H2=m2DDF3H1+nn+kDDH2+26k−tk+nWH2+k+1m2−m4k−t+2lm∑i,j=1ndxi,xj|F3H1+2lm∑r=1k∑i=1ndzr,xi|F3H1.Theorem 4.
IfH1+F4H2 is the F4-sum graph of two connected and simple graphs H1 and H2, then(29)DDH1+F4H2=m2DDF4H1+nn+kDDH2+28k−tn+kWH2+k+14k−tm2−m+2lm∑i,j=1ndxi,xj|F4H1+2lm∑r=1k∑i=1ndzr,xi|F4H1,where t is the number of pendant vertices in graph H1.Proof.Case 10.
Between two old vertices,A can be calculated similarly as in Theorem 2 Case 4:(30)A=m2∑i,j=1ndxi,xj|F4H1dxi+dxj+n2DDH2+2lm∑i,j=1ndxi,xj|F4H1+8knWH2.Case 11.
Between old and new vertices,(31)B1=12∑(zr,yp),(xi,yq)∈H1+F4H2dzr,xp,xi,yq(δzr,yp+δxi,yq|H1+F4H2,where p,q=1 to m, i=1 to nr=1 to k and δxi,yq=2δxi+δyj(32)=12∑r=1k∑i=1n∑p,q=1mdzr,xi|F4H1+d(yp,yq|H2δzr,yp+2δxi+δyq=12∑i=1n∑p,q=1m∑r=1kdzr,xi|F4H1δzr,yp+12∑i=1n∑p,q=1m∑r=1kd(yp,yq|H2δzr,yp+∑i=1n∑p,q=1m∑r=1kdzr,xi|F4H1δxi+12∑i=1n∑p,q=1m∑r=1kdzr,xi|F4H1δyq+∑i=1n∑p,q=1m∑r=1kdyp,yq|H2δxi+12∑i=1n∑p,q=1m∑r=1kdyp,yq|H2δyq=m22∑i=1n∑r=1kdzr,xi|F4H1δzr,yp+n4k−tWH2+m2∑i=1n∑r=1kdzr,xi|F4H1δxi+lm∑i=1n∑r=1kdzr,xi|F4H1+4k2WH2+kn2DDH2.
The distance between old and new vertices is twice theB1, i.e.,(33)B=m2∑i=1n∑r=1kdzr,xi|F4H1δzr,yp+2n4k−tWH2+2m2∑i=1n∑r=1kdzr,xi|F4H1δxi+2lm∑i=1n∑r=1kdzr,xi|F4H1+8k2WH2+knDDH2.Case 12.
Between two new vertices,C can be calculated similarly as in Theorem 3 Case 9:(34)C=k+14k−tm2−m+2k4k−tWH2+m22∑r,s=1,r≠skdzr,zs|F4H1δzr,yp+δzs,yqDDH1+F4H2=A+B+CDDH1+F4H2=m2∑i,j=1ndxi,xj|F4H1δxi+δxj+n2DDH2+2lm∑i,j=1ndxi,xj|F4H1+8knWH2+m2∑i=1n∑r=1kdzr,xi|F4H1δzr,yp+2n4k−tWH2+2m2∑i=1n∑r=1kdzr,xi|F4H1δui+2lm∑i=1n∑r=1kdzr,xi|F4H1+8k2WH2)+knDDH2+4k−tk+1m2−m+2k4k−tWH2+m22∑r,s=1,r≠skdzr,zs|F4H1δzr,yp+δzs,yqDDH1+F4H2=m2∑i,j=1ndxi,xj|F4H1δxi+δxj+∑i=1n∑r=1kδzr,yp+2∑i=1n∑r=1kdzr,xi|F4H1δxi+12∑r,s=1,r≠skdzr,zs|F4H1δzr,yp+δzs,yq+n2DDH2+knDDH2+WH28kn+2n4k−t+8k2+2k4k−t+k+1m2−m4k−t+2lm∑i,j=1ndxi,xj|F4H1+2lm∑i=1n∑r=1kdzr,xi|F4H1DDH1+F4H2=m2DD(F4H1)+nn+kDDH2+28k−tk+nWH2+k+1m2−m4k−t+2lm∑i,j=1ndxi,xj|F4H1+2lm∑i=1n∑r=1kdzr,xi|F4H1.


## 4. Application

The main results of Section3 are explained in this section. Consider path Pn of order n≥2, WPn=nn2−1/6 and DDPn=nn−12n−1/3. Now, before to find DDPn+FiPm, we construct Tables 1 and 2.Table 1
WI and DDI ofFiPn.
FWFiPnDDFiPnF12nn−12n−1/32n−12n−14n−3/3F2n−12n2+2n−3/34n+1n−12F32nn−1n+1/34n3−6n2+2n−2F4nn−14n+1/62/38n3−15n2+10n−6Table 2
Some particular distance sums ofFiPn.
F∑i,j=1ndxi,xj|FiPn∑r=1k∑i=1ndzr,xi|FiPnF12nn−1n+1/3nn−12n−1/3F2nn2−1/3nn2−1/3F3nn−1n+4/3nn−1n+1/3F4nn−1n+1/3nn−1n+1/3Now, using Definition2, Tables 1 and 2, we obtain(35)iDDPn+F1Pm=2m2n−12n−14n−33+nm2n−1m−12m−13+4mn−1m−1+4mn−12n−1m2−13+2mnm−1n−14n+13,iiDDPn+F2Pm=4m2n−12n+1+mn2n−1m−12m−13+2mn−12n−1m2−1+4mn−1m−1+4mnm−1n2−13,iiiDDPn+F3Pm=m24n3−6n2+2n−2+nm2n−1m−12m−13+2mnm−12n−3+2m2n−13n−4m2−13+2mnm−1n−12n+53,ivDDPn+F4Pm=2m28n2−15n2+10n−63+nm2n−1m−12m−13+2mnm−12n−3+2m2n−14n−5m2−13+4mnm−1n−1n+13.Consider cycleCn of n, then(36)WCn=n38if n is evennn2−18if n is odd,DDCn=n32if n is evennn2−12if n is odd..Now, before to findDDCn+FiPm, we construct Tables 3 and 4.Table 3
WI and DDIs ofFiCn.
FWFCnDDFCnF1n34n3F2nn2+2n−1/2nn+23n−1F3nn2+2n−1/2nn+23n−1F4n2n+1/24n2n+1Table 4
Some particular distance sums ofFCn.
F∑i,j=1ndxi,xj|FCn∑r=1k∑i=1ndzr,xi|FCnF1n3/2ifnis evennn2−1/2ifnis oddn3/2ifnis evennn2+1/2ifnis oddF2n3/4ifnis evennn2+1/4ifnis oddn2n+2/4ifnis evennn2+1/4ifnis oddF3nn2+4n−4/4ifnis evennn2+4n−5/4ifnis oddn2n+2/4ifnis evennn+12/4ifnis oddF4n3/4ifnis evennn2−12/4ifnis oddn2n+2/4ifnis evennn2+12/4ifnis oddNow, using Definition2, Tables 2 and 4, we obtain(37)iDDCn+F1Pm=2mnn23m−1+m−1n2m+1+2,iiDDCn+F2Pm=mnmn+23n−1+m−12n32m−1+n4m+n+5+4,iiiDDCn+F3Pm=mnmn+23n−1+m−12n32m−1+n4m+n+11+2,ivDDCn+F4Pm=mn4mnn+1+m−12n310m+7+n+1n+4.Now, we construct Table5 such that values of Columns 3, 4 and 5 are obtained using Theorems 1–4 of Section 3, and aforesaid results of the same section and the bounds obtained in [26], respectively. We note that the computed and exact values are same, and the values of the upper bounds exceed them.Table 5
Numerical comparison of DD index.
No.F-sum graphExact valueComputed valueBounded value1DDP3+F1P24704705622DDP3+F2P24864865703DDP3+F3P24864865384DDP3+F4P25505505945DDC3+F1P27447448526DDC3+F2P28288289247DDC3+F3P29249249968DDC3+F4P2106810681140


## 5. Conclusion

In the present study, it is finally concluded that we have improved the already existing bounds of degree distance indices of theF-sum graphs (F1-sum, F2-sum, F3-sum and F4-sum) ([26]) in the form of their exact values. In the section of Application, the obtained results are also illustrated with the help of the comparison among bounded, computed, and exact values for the particular F-sum graphs. However, the exact values or the improved bounded values of the other topological indices for the F-sum graphs are still to be determined.



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