# Vertex-Based Topological Indices of Double and Strong Double Graph of Dutch Windmill Graph

Authors: Muhammad Asad Ali; Muhammad Shoaib Sardar; Imran Siddique; Dalal Alrowaili
Journal: Journal of Chemistry
 (2021)

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## Abstract

A measurement of the molecular topology of graphs is known as a topological index, and several physical and chemical properties such as heat formation, boiling point, vaporization, enthalpy, and entropy are used to characterize them. Graph theory is useful in evaluating the relationship between various topological indices of some graphs derived by applying certain graph operations. Graph operations play an important role in many applications of graph theory because many big graphs can be obtained from small graphs. Here, we discuss two graph operations, i.e., double graph and strong double graph. In this article, we will compute the topological indices such as geometric arithmetic indexGA, atom bond connectivity index ABC, forgotten index F, inverse sum indeg index ISI, general inverse sum indeg index ISIα,β, first multiplicative-Zagreb index PM1 and second multiplicative-Zagreb index PM2, fifth geometric arithmetic index GA5, fourth atom bond connectivity index ABC4 of double graph, and strong double graph of Dutch Windmill graph D3p.


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## Body

## 1. Introduction and Preliminaries

For undetermined notations and terminologies, we recommend Robin J. Wilson book [1].Assume thatG is a simple graph that has no multiple edges and loops. VG and EG are the vertex and edge sets of graph G, respectively. The number of elements in VG and EG represents the order and size of graph G. Vertex degree is the number of edges joining to a vertex in a graph G. A vertex degree is indicated by drr∈VG and Sr=∑s∈NGrds, where NGr=s∈VG|rs∈EG. The following lemma is useful for computing the total number of edges in a graph G.Lemma 1.
IfG is a graph of size t, then(1)∑r∈VGdegr=2t.This is also called the handshake lemma and was observed by Lenford Euler in 1736. This observation is often called the first theorem of graph theory [2].The chemical graph theory connects graph theory and chemistry to solve organic chemistry problems. Structured-property (QSPR) and structured-activity (QSAR) relationships are among the most important topics in this field. QSPR/QSAR research relies heavily on topological indices. These topological indices analyse the structure of any finite graph and are based on mathematical equations. Several different kinds of topological indices exist, i.e., degree-based topological indices [3–5], distance-based topological indices [6], and counting-related topological indices [7, 8]. The topological index concept comes from the work of Wiener, who introduced the Wiener index, and thus, topological indexing history begins.The Wiener index is defined in [9] as follows:(2)WG=12∑r,sdr,s,where r,s is the order pair of vertices in G and dr,s is the distance of vertex r-s in G.The geometric-arithmetic indexGA [10] of graph G is defined as(3)GAG=∑rs∈EG2drdsdr+ds.The atomic bond connectivity indexABC of graph G is defined [11] as(4)ABCG=∑rs∈EGdr+ds−2drds.The forgotten indexF is defined [12] as(5)FG=∑rs∈EGdr2+ds2.The inverse sum indeg indexISI is defined [13] as(6)ISIG=∑rs∈EG11/dr+1/ds.The general inverse sum indeg indexISIα,β is defined [14] as(7)ISIα,βG=∑rs∈EGdrdsαdr+dsβ,where α and β are some real numbers.The first multiplicative-Zagreb (PM1) and second multiplicative-Zagreb index index PM2 is defined [15] as(8)PM1G=∏rs∈EGdr2,(9)PM2G=∏rs∈EGdr.ds.The first multiplicative-Zagreb indexPM1 can also be written in the sum of the edges [16] of G:(10)PM1G=∏rs∈EGdr+ds.TheGA5G index is defined [17] as(11)GA5G=∑rs∈EG2Sr×SsSr+Ss,where Sr is summation of degrees of all neighbor of vertex r, and the same for Ss.TheABC4G index is defined [18] as(12)ABC4G=∑rs∈EGSr+Ss−2Sr×Ss.We suggest that readers read the following articles for more detailed information on topological indices and molecular graphs [19–22].Definition 1.
A Dutch Windmill graph [23] is the graph which is formed by taking the p copies of the cycle graph by taking one mutual vertex. It is denoted by Dnp, where p≥1andn≥3. The order of Dutch Windmill graph is n−1p+1 and size pn. Dutch Windmill graph Dnp is depicted in Figure 1.Figure 1
Dutch windmill graphDnp.Definition 2.
The double graph of graphG is represented by DG. Assume two copies of a graph, and join each vertex in one copy to its neighbor in the other copy in order to produce the double graph of the graph [24]. For example, the double graph of Dutch Windmill graph D32 is depicted in Figure 2.Figure 2
(a). Dutch windmill graphD32. (b). Double graph of Dutch windmill graph D32.
(a)(b)Definition 3.
The strong double graphSDG of the graph G is attained by taking two graphs and joining the closed neighbourhoods of each vertex in one graph to the adjacent vertex in the other graph [25]. For example, strong double graph of graph G is depicted in Figure 3. A new type of equienergetic and L-equienergetic graph has been found by using strong double graphs.
The following is the structure of this paper. Sections2 and 3 will analyse some degree-based topological indices of double graphs and strong double graphs of Dutch Windmill graphs, respectively. In Section 4, we provide concluding remarks for the entire paper.Figure 3
Strong double graphSDG of graph G.


## 2. Degree-Based Topological Indices of Double Graph of Dutch Windmill GraphD3p

We will compute the vertex-based indices of the double graph of the Dutch Windmill graphD3p in this section.Theorem 4.
LetDD3p be the double graph of Dutch Windmill graph D3p. Then, the geometric arithmetic index of DD3p is(13)GADD3p=4p1+4p1+p.Proof.
The total number of vertices and edges in the double graph of Dutch Windmill graph is22p+1 and 12p, respectively. In DD3p, we have 4p vertices of degree 4 and 2 vertices of degree 8p. We spilt the edges of DD3p into those of the type Edr,ds in which r, s are edges. DD3p contains edges of type E4,4 and E4,4p, and Table 1 presents the edges of these types.
By applying equation (3) and Table 1, we acquire the desired results, i.e.,(14)GAG=∑rs∈EG2drdsdr+ds,GADD3p=E4,4∑rs∈EDD3p2drdsdr+ds+E4,4p∑rs∈EDD3p2drdsdr+ds,GADD3p=4p2444+4+8p244p4+4p,GADD3p=4p1+4p1+p.Table 1
Partitioning of edges.
Edr,dsE4,4E4,4pNumber of edges4p8pTheorem 5.
LetDD3p be the double graph of Dutch Windmill graph D3p. Then, the ABC index of DD3p is (15)ABCDD3p=p6+22p1+2p.Proof.
By applying equation (4) and Table 1, we acquire the desired results, i.e.,(16)ABCG=∑rs∈EGdr+ds−2drds,ABCDD3p=E4,4∑rs∈EDD3pdr+ds−2drds+E4,4p∑rs∈EDD3pdr+ds−2drds,=4p4+4−216+8p4+4p−216p,=4p616+8p2+4p16p,ABCDD3p=p6+22p1+2p.Theorem: 6.
LetDD3p be the double graph of Dutch Windmill graph D3p. Then, the forgotten index of DD3p is(17)FDD3p=128p+1+p2.Proof.
By applying equation (5) and Table 1, we acquire the desired results, i.e.,(18)FG=∑rs∈EGdr2+ds2FDD3p=E4,4∑rs∈EDD3pdr2+ds2+E4,4p∑rs∈EDD3pdr2+ds2=4p42+42+8p42+4p2FDD3p=128p+1+p2.Theorem 7.
LetDD3p be the double graph of Dutch Windmill graph D3p. Then, the inverse sum indeg index of DD3p is (19)ISIDD3p=8p1+4p1+p.Proof.
By applying equation (6) and Table 1, we acquire the desired results, i.e.,(20)ISIG=∑rs∈EG11/dr+1/ds=∑rs∈EGdrdsdr+ds,ISIDD3p=E4,4∑rs∈EDD3pdrdsdr+ds+E4,4p∑rs∈EDD3pdrdsdr+ds,ISIDD3p=4p444+4+8p44p4+4p,ISIDD3p=8p1+4p1+p.Theorem 8.
LetDD3p be the double graph of Dutch Windmill graph D3p. Then, the general inverse sum indeg index ISIα,β of DD3p is (21)ISIα,βDD3p=4p16α8β+8p16pα41+pβ.Proof.
By applying equation (7) and Table 1, we acquire the desired results, i.e.,(22)ISIα,βG=∑rs∈EGdudvαdu+dvβ=E4,4∑rs∈EDD3pdrdsαdr+dsβ+E4,4p∑rs∈EDD3pdrdsαdr+dsβ,ISIα,βDD2p=4p44α4+4β+8p44pα4+4pβ,ISIα,βDD2p=4p16α8β+8p16pα41+pβ.Theorem 9.
LetDD3p be the double graph of Dutch Windmill graph D3p. Then, the first multiplicative-Zagreb index of DD3p is(23)PM1DD3p=1024p21+p.Proof.
By applying equation (9) and Table 1, we acquire the desired results, i.e.,(24)PM1G=∏rs∈EGdr+ds,PM1DD3p=E4,4∏rs∈EDD3pdr+ds×E4,4p∏rs∈EDD3pdr+ds,PM1DD3p=4p4+4×8p4+4p,PM1DD3p=1024p21+p.Theorem 10.
LetDD3p be the double graph of Dutch Windmill graph D3p. Then, the second multiplicative-Zagreb index of DD3p is(25)PM2DD3p=8192p3.Proof.
By applying equation (9) and Table 1, we acquire the desired results, i.e.,(26)PM2G=∏rs∈EGdr.ds,PM2DD3p=E4,4∏rs∈EDD3pdr.ds×E4,4p∏rs∈EDD3pdr.ds,PM2DD3p=4p4×4×8p4×4p,PM2DD3p=8192p3.Theorem 11.
LetDD3p be the double graph of Dutch Windmill graph D3p. Then, GA5G index of DD3p is(27)GA5DD3p=4pp+11p+1+4p23p+1.Proof.
We spilt the edges ofDD3p into those of the type ESr,Ss in which “rs” is an edge. DD3p contains edge of the type E8p+1,8p+1 and E8p+1,16p, and Table 2 presents the edges of these types.
By applying equation (11) and Table 2, we acquire the desired results, i.e.,(28)GA5G=∑rs∈EG2Sr×SsSr+Ss=E8p+1,8p+1∑rs∈EDD3p2Sr×SsSr+Ss+E8p+1,16p∑rs∈EDD3p2Sr×SsSr+Ss,GA5DD3p=4p16p+128p+1+8p88pp+183p+1GA5DD3p=4pp+11p+1+4p23p+1.Table 2
Partitioning of edges.
ESu,SvE8p+1,8p+1E8p+1,16pNumber of edges4p8pTheorem 12.
LetDD3p be the double graph of Dutch Windmill graph D3p. Then, the fourth atom bond connectivity index of DD3p is (29)ABC4DD3p=p2p+116p−14+p34p−1pp+1.Proof.
By applying equation (12) and Table 2, we acquire the desired results, i.e.,(30)ABC4G=∑rs∈EGSr+Ss−2Sr×Ss=E8p+1,8p+1∑rs∈EDD3pSr+Ss−2Sr×Ss+E8p+1,16p∑rs∈EDD3pSr+Ss−2Sr×Ss,ABC4DD3p=4p16p−148p+12+8p64p−1128pp+1,ABC4DD3p=p2p+116p−14+p34p−1pp+1.


## 3. Degree-Based Topological Indices of Strong Double Graphs of Dutch Windmill GraphD3p

We will compute the vertex-based indices of the strong double graph of the Dutch Windmill graphD3p in this section. The strong double graph of D32 is depicted in Figure 4.Figure 4
(a) Dutch Windmill graphD32. (b). Strong double graph of Dutch Windmill graph D3p.
(a)(b)Theorem 13.
LetSDD3p be the strong double graph of Dutch Windmill graph D3p. Then, the geometric arithmetic index of SDD3p is (31)GASDD3p=1+6p+8p20p+52p+3.Proof.
The total number of vertices and edges inSDD3p are 22p+1 and 14p+1, respectively. In SDD3p, we have 4p vertices of degree 5 and 2 vertices of degree 4p+1. We spilt the edges of SDD3p into those of the type Edr,ds in which “rs” is an edge. SDD3p contains edges of the type E5,5, E5,4p+1, and E4p+1,4p+1, and Table 3 presents the edges of these types.
By applying Equation (3) and Table 3, we acquire the desired results, i.e.,(32)GAG=∑rs∈EG2drdsdr+ds=E5,5∑rs∈EDD3p2drdsdr+ds+E5,4p+1∑rs∈EDD3p2drdsdr+ds+E4p+1,4p+1∑rs∈EDD3p2drdsdr+ds=6p255+8p254p+15+4p+1+124p+14p+14p+1+4p+1,GASDD3p=1+6p+8p20p+52p+3.Table 3
Partitioning of edges.
Edr,dsE5,5E5,4p+1E4p+1,4p+1Number of edges6p8p1Theorem 14.
LetSDD3p be the strong double graph of Dutch Windmill graph D3p. Then, ABC index for(33)SDD3pisABCSDD3p=12p25+16pp+120p+5+22p4p+1.Proof.
By applying equation (4) and Table 3, we acquire the desired results, i.e.,(34)ABCG=∑rs∈EGdr+ds−2drds,ABCSDD3p=E5,5∑rs∈EDD3pdr+ds−2drds+E5,4p+1∑rs∈EDD3pdr+ds−2drds+E4p+1,4p+1∑rs∈EDD3pdr+ds−2drds,ABCSDD3p=6p825+8p5+4p+1−254p+1+8p+2−24p+12,ABCSDD3p=12p25+16pp+120p+5+22p4p+1.Theorem 15.
LetSDD3p be the strong double graph of Dutch Windmill graph D3p. Then, the forgotten index of SDD3p is(35)FSDD3p=128p3+96p2+524p+2.Proof.
By applying equation (5) and Table 3, we acquire the desired results, i.e.,(36)FG=∑rs∈EGdr2+ds2FSDD3p=E5,5∑rs∈EDD3pdr2+ds2+E5,4p+1∑rs∈EDD3pdr2+ds2+E4p+1,4p+1∑rs∈EDD3pdr2+ds2=6p50+8p25+16p2+8p+1+116p2+8p+1+16p2+8p+1,FSDD3p=128p3+96p2+524p+2.Theorem 16.
LetSDD3p be the strong double graph of Dutch Windmill graph D3p. Then, the inverse sum indeg index of SDD3p is (37)ISISDD3p=15p+4p20p+52p+3+4p+1224p+1.Proof.
By applying equation (6) and Table 3, we acquire the desired results, i.e.,(38)ISIG=∑rs∈EG11/dr+1/ds=∑rs∈EGdrdsdr+ds,ISISDD3p=E5,5∑rs∈EDD3pdrdsdr+ds+E5,4p+1∑rs∈EDD3pdrdsdr+ds+E4p+1,4p+1∑rs∈EDD3pdrdsdr+ds,ISISDD3p=6p2510+8p54p+15+4p+1+4p+14p+18p+2,ISISDD3p=15p+4p20p+52p+3+4p+1224p+1.Theorem 17.
LetSDD3p be the strong double graph of Dutch Windmill graph D3p. Then, general inverse sum indeg index ISIα,β of SDD3p is(39)ISIα,βSDD3p=6p25α10β+8p20p+5α22p+3β+4p+12α24p+1β.Proof.
By applying equation (7) and Table 3, we acquire the desired results, i.e.,(40)ISIα,βG=∑rs∈EGdrdsαdr+dsβ=E5,5∑rs∈EGdrdsαdr+dsβ+E5,4p+1∑rs∈EGdrdsαdr+dsβ+E4p+1,4p+1∑rs∈EGdrdsαdr+dsβ=6p25α10β+8p54p+1α4p+6β+4p+12α4p+1+4p+1βISIα,βSDD3p=6p25α10β+8p20p+5α22p+3β+4p+12α24p+1β.Theorem 18.
LetSDD3p be the strong double graph of Dutch Windmill graph D3p. Then, the first multiplicative-Zagreb index of SDD3p is(41)PM1SDD3p=15360p4+26880p3+5760p2.Proof.
By applying equation (10) and Table 3, we acquire the desired results, i.e.,(42)PM1G=∏rs∈EGdr+ds,PM1SDD3p=E5,5∏rs∈EDD3pdr+ds×E5,4p+1∏rs∈EDD3pdr+ds×E4p+1,4p+1∏rs∈EDD3pdr+ds,PM1SDD3p=6p10×16p2p+3×8p+2,PM1SDD3p=15360p4+26880p3+5760p2.Theorem 19.
LetSDD3p be the strong double graph of Dutch Windmill graph D3p. Then, the second multiplicative-Zagreb index of SDD3p is (43)PM2SDD3p=4p+1224000p3+6000p2.Proof.
By applying equation (9) and Table 3, we acquire the desired results, i.e.,(44)PM2G=∏rs∈EGdr.ds,PM2SDD3p=E5,5∏rs∈EDD3pdr.ds×E5,4p+1∏rs∈EDD3pdr.ds×E4p+1,4p+1∏rs∈EDD3pdr.ds,PM2DD3p=6p25×40p4p+1×4p+12,PM2SDD3p=4p+1224000p3+6000p2.Theorem 20.
LetSDD3p be the strong double graph of Dutch Windmill graph D3p. Then, the GA5 index of SDD3p is(45)GA5SDD3p=2p48p+27+48p24p+52+1716p+9.Proof.
We spilt the edges ofSDD3p into those of the type ESr,Ss in which “rs” is an edge. SDD3p contains edges of the type E8p+17,8+17, E8p+17,24p+1, and E24p+1,24p+1, and Table 4 presents the edges of these types.
By applying equation (11) and Table 4, we acquire the desired results, i.e.,(46)GA5G=∑rs∈EG2Sr×SsSr+Ss,GA5SDD3p=E8p+17,8p+17∑rs∈EDD3p2Sr×SsSr+Ss+E8p+17,24p+1∑rs∈EDD3p2Sr×SsSr+Ss+E24p+1,24p+1∑rs∈EDD3p2Sr×SsSr+Ss,=6p28p+17216p+34+8p28p+1724p+132p+18+1224p+1248p+2GA5SDD3p=2p48p+27+48p24p+52+1716p+9.Table 4
Partitioning of edges.
ESr,SsE8p+17,8p+17E8p+17,24p+1E24p+1,24p+1Number of edges6p8p1Theorem 21.
LetSDD3p be the strong double graph of Dutch Windmill graph D3p. Then, the fourth atom bond connectivity index of SDD3p is (47)ABC4SDD3p=12p4p+88p+17+32p2p+18p24p+52+17+43p24p+1.Proof.
By applying equation (11) and Table 4, we acquire the desired results, i.e.,(48)ABC4G=∑rs∈EGSr+Ss−2Sr×Ss=E8p+17,8p+17∑rs∈EDD3pSr+Ss−2Sr×Ss+E8p+17,24p+1∑rs∈EDD3pSr+Ss−2Sr×Ss+E24p+1,24p+1∑rs∈EDD3pSr+Ss−2Sr×Ss,ABC4SDD3p=6p16p+328p+172+8p32p+168p+1724p+1+48p24p+12,ABC4SDD3p=12p4p+88p+17+32p2p+18p24p+52+17+43p24p+1.


## 4. Conclusion

The topological indices provide key information about a molecule's chemical structure and chemical activity, for example, in order to derive quantitative structure-activity relationships (QSARs). These models are derived by applying statistical measures of the molecular structure or properties with descriptors representative of the biological activity (including undesirable side effects) of chemicals (drugs, toxicants, and environmental pollutants). Drug discovery, lead optimization, and toxicity prediction are just a few of the areas in which QSAR is being used. Our purpose in this article was to construct two new graphs from the Dutch Windmill graph using two graph operations, namely, double graph and strong double graph. After that, we calculated some vertex-based topological indices of double graph and strong double graph of the Dutch Windmill graph. Many graph theory applications rely on graph operations. We recommend the readers to compute the topological indices for double and strong double graphs of some other classes of graphs or networks.



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