# Topological Indices of Families of Bistar and Corona Product of Graphs

Authors: A. Khalid; N. Kausar; M. Munir; M. Gulistan; M. M. Al-Shamiri; T. Lamoudan
Journal: Journal of Mathematics
 (2022)

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## Abstract

Topological indices are graph invariants that are used to correlate the physicochemical properties of a chemical compound with its (molecular) graph. In this study, we study certain degree-based topological indices such as Randić index, Zagreb indices, multiplicative Zagreb indices, Narumi–Katayama index, atom-bond connectivity index, augmented Zagreb index, geometric-arithmetic index, harmonic index, and sum-connectivity index for the bistar graphs and the corona productKmoKn′, where Kn′ represents the complement of complete graph Kn.


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## Body

## 1. Introduction

Graph theory techniques [1–4] have applications in chemistry, biology, physics, and computer science. Topological indices are graph invariants that are used to study graph topology. Along with computer networks, graph theory is a powerful tool in other research areas such as coding theory, database management systems, circuit design, secret-sharing schemes, and theoretical chemistry. Cheminformatics is the combination of technology, graph theory, and chemistry. It links organic substances structure and physiochemical properties by using some useful graph invariants and their associated molecular graph. A molecular graph contains sets of points and lines which shows the atoms and covalent bond in the molecule. These points and lines are known as vertices and edges in graph theory, respectively [5]. Theoretical study of underlying chemical structure using useful graph invariants is an appealing area of research in mathematical chemistry due to its practical applications in QSAR/QSPR investigation [6]. Topological indices are used to estimate the physicochemical properties of the chemical compounds. A topological index can be considered as a function that maps a graph to a nonnegative real number [7].Some degree-based topological indices are Randić index, Zagreb indices, Narumi–Katayama and multiplicative Zagreb indices, atom-bond connectivity index, augmented Zagreb index, geometric-arithmetic index, harmonic index, and sum-connectivity index [8].Throughout this work, we use standard notationsG=V,E for graph, VG set of vertices, EG set of edges, and dwG degree of vertex w in G.For any simple graphG=V,E, w∼x denotes the adjacent vertices w and x in graph G.Definition 1.
Randić index is the most popular topological index among all topological indices and is defined as [9](1)RG=∑w∼x1dwGdxG.Definition 2.
The first Zagreb index of a graphG is defined as(2)M1G=∑w∈VGdwG2.
The second Zagreb index of a graphG is defined as(3)M2G=∑w∼xdwGdxG.
The first and second Zagreb index are known as Zagreb indices [10].Definition 3.
The first multiplicative Zagreb index of a graphG is defined as(4)∏1G=∏w∈VGdwG2,with product going overall vertices of a graph G.
The second multiplicative Zagreb index of a graphG is defined as(5)∏2G=∏w∼xdwGdxG.
The modified first multiplicative Zagreb index of a graphG is defined as(6)∏1∗G=∏w∼xdwG+dxG.
The first, second, and modified first multiplicative Zagreb indices are known as Narumi–Katayama indices [8, 11].Definition 4.
The atom-bond connectivity index of a graphG is defined as [12](7)ABCG=∑w∼xdwG+dxG−2dwGdxG.Definition 5.
The augmented Zagreb index of a graphG is defined as [13](8)AZIG=∑w∼xdwGdxGdwG+dxG−23.Definition 6.
The geometric-arithmetic index of a graphG is defined as [14](9)GAG=∑w∼x2dwGdxGdwG+dxG.Definition 7.
The harmonic index of a graphG is defined as [15, 16](10)HG=∑w∼x2dwG+dxG.Definition 8.
The sum-connectivity index of a graphG is defined as [17, 18](11)SCIG=∑w∼x1dwG+dxG.
Randić index was introduced by Milan Randić in 1975 [19]. After that, mathematicians did not pay attention to this index for nearly two decades. However, Pal, Erdos, and Bela Bollobas soon worked on this index and discovered the useful mathematics hidden within it. They published their first paper on this index in 1998. When the mathematical communities realized the value of the Randić index, they started to do research studies and soon a flood of publications on this descriptor were started. Randić also wrote two articles on this descriptor [8].
The work on Randić index encouraged theoretical chemists and mathematicians to discover other topological indices that depend on vertex degree.
During the study of total pi electron energy on molecular structure, some expressions which included the terms of the form(12)M1G=∑wdwG2,M2G=∑w∼xdwGdvG,were occurred [8]. In the chemical theory, M1 and M2 are called first Zagreb index and second Zagreb index. These two indices were discovered by Trinajestic and Gutman in 1972 [8, 10]. At first, the first Zagreb index was also known as the Gutman index, but Balaban et al. did not want to introduce these descriptors by the names of the discoverers. So, they included M1 and M2 in the topological indices after 1982 and named them Zagreb group indices. However, soon, the name Zagreb group indices was changed to Zagreb indices. The second Zagreb index did not get any attention of mathematicians or mathematical chemists and that is why not a single property of M2 was introduced in [8]. Narumi–Katayama [8] introduced the first product descriptor in 1984 and named it simple topological index. This index is defined as(13)NKG=∏w∈VGdwG.
This product index was later renamed the Narumi–Katayama index by Tomovic and Gutman. In 2010, Todeshine et al. proposed multiplicative versions of multiplicative Zagreb indices. These two graph invariants are known as the first and second multiplicative Zagreb indices, after Gutman. Eliasi et al. and Iranmanesh and Gutman proposed a multiplicative version of the first Zagreb index. The authors termed it as modified multiplicative Zagreb index, respectively [20].
The first geometric-arithmetic index also known as geometric-arithmetic index is the modified version of the Randić index. This index was proposed by Vukicevic and Furtula in 2009 [6, 8].
The atom-bond connectivity index is another topological index that is similar to the Randić index. This topological descriptor was introduced by Estrada et al. in 1998 [6, 13]. A recent study of this index has caught the attention of some researchers. Furtula et al. [8] introduced the augmented Zagreb index in 2009, a modified version of the atom-bond connectivity index. This descriptor has higher predictive power than the atom-bond connectivity index. Tosovic and Gutman investigated in 2013 that AZI produced nice effects in the formation of heat and also yielded the best results in boiling factors of octane isomers. Furtula et al. investigated the structure sensitivity of twelve topological indices using trees and found that the AZI has the best structure sensitivity [13].
In 1980, Siemion Fajtlowiez introduced another topological index. In 2012, Zhang worked on this index and named it as harmonic index [8]. In sharp construct to other topological indices, no chemical applications of this index were found, but in mathematical chemistry, such research studies are expected very much [8]. In the last few years, this index has attracted the great attention of theoreticians [15, 16].
Bo Zhou and Nenad Trinajstic discovered that the termdwdx in the Randić index can be replaced by dw+dx and named it sum-connectivity index [8]. This index got the attention of both applied and pure researchers. Some recent work on this index can be found in [21].
There are a number of studies of various topological indices of graphs establishing formulas for computing the indices and also providing upper and lower bounds on the values of such indices. In this study, certain vertex degree-based topological indices are studied. We have determined and computed the closed formulas of these indices for two special families of graphs of diameter three. A graph formed by joining the centers of two-star graphs of orderm and n, i.e., K1,m and K1,n by an edge, is called the bistar graph and is denoted by Bn;m. The corona product KmoKn′ of two graphs is defined as the graph obtained by taking one copy of Km of order m and n copies of Kn′ and then joining each vertex of the ith copy of Kn′ to the ith vertex of Km. These graphs are undirected having no loops and multiple edges. These results are novel and significant contributions in graph theory and network science, and they provide a good basis to understand the topology of these graphs and networks (Figures 1 and 2).Figure 1
A representation of bistar graphBn;m of order m and n.Figure 2
A representation of corona productK3°K4′ of two graphs.


## 2. Topological Indices of Families of Bistar Graphs, i.e.,G=Bn;m

By looking at the earlier results for computing the topological indices for families of bistar graphs, here we introduce new degree-based topological indices to compute their values for these families of graphs.Theorem 1.
The sum-connectivity index of families of bistar graphsBn;m is(14)nn+2+1n+m+2+mm+2.Proof.
To find the sum-connectivity index ofBn;m, firstly, we select a vertex u on Bn;m of degree n+1. There are n vertices u1,u2,…,un of degree 1 which are adjacent to u. For the vertices u and ui, where i=1,2,3,…,n, the sum is obtained as(15)∑u∼ui1duG+duiG=n1+n+1=nn+2,i=1,2,3,…,n.
Since the degree ofu is n+1, the other vertex which is adjacent to u is v of degree m+1. For the vertices u and v, we have(16)1duG+dvG=1n+1+m+1=1n+m+2.
Now, we select a vertexv of degree m+1 on Bn;m; there are m vertices v1,v2,v3,…,vm of degree 1 which are adjacent to v. For the vertices v and vj, the sum is obtained as(17)∑v∼vj1dvG+dvjG=m1+m+1=mm+2,j=1,2,3,…,m.
Adding above equations, we haveSCIG = sum-connectivity index of Bn;m:(18)nn+2+1n+m+2+mm+2.Theorem 2.
The Zagreb indices of the families of bistar graphsBn;m are(19)M1G=n+12+m+12+n+m,M2G=nn+1+n+1m+1+mm+1.Proof.
To find the first Zagreb indexM1G of Bn;m, we select a vertex u on Bn;m of degree n+1. For the vertex u, we have(20)degu2=n+12.
Since there aren vertices u1,u2,u3,…,un of Bn;m of degree 1 which are adjacent to u, thus, we have(21)∑uidegui2=n,i=1,2,3,…,n.
As the degree ofu is n+1, the other vertex which is adjacent to u is v of degree m+1. For the vertex v, we have(22)degv2=n+12.
Since the degree ofv is m+1, the other vertices which are adjacent to v are m in numbers of degree 1:(23)∑vjdegvj2=m,j=1,2,3,…,m.
Adding equations (20) to (23), we have M1G = first Zagreb index of Bn;m:(24)M1G=n+12+m+12+n+m.
Similarly,M2G = second Zagreb index of Bn;m is(25)M2G=n.n+1+n+1m+1+mm+1.Theorem 3.
The atom-bond connectivity index and augmented Zagreb index of families of bistar graphsBn;m are(26)nnn+1+n+mn+1m+1+mmm+1,nn+1n3+n+1m+1n+m3+mm+1m3.The proof of this Theorem is the same as Theorem1.Theorem 4.
The geometric-arithmetic index and harmonic index of families of bistar graphsBn;m are(27)2nn+1n+2+2n+1m+1n+m+2+2mm+1m+2,2nn+2+2n+m+2+2mm+2.The proof of this theorem is the same as Theorem1.Theorem 5.
The Narumi–Katayama and multiplicative Zagreb indices of families of bistar graphsBn;m are(28)∏1G=n+12m+12,∏2G=n+1n+1m+1m+1,∏1∗G=n+2nn+m+2m+2m.Proof.
To find the first multiplicative Zagreb index ofBn;m, we select a vertex u on Bn;m of degree n+1. For the vertex u, we have(29)degu2=n+12.
Since there aren vertices u1,u2,u3,…,un of Bn;m of degree 1, thus, we have(30)∏uidegui2=degu12degu22,…,degun2=1.
Similarly, for the vertexv, we have(31)degv2=m+12.
Also, for the verticesv1,v2,v3,…,vm of Bn;m of degree 1, we have(32)∏vjdegvj2=degv12degv22,…,degvm2=1.
Multiplying equations (29) to (32), we have ∏1G = first multiplicative Zagreb index of families of bistar graphs Bn;m:(33)∏1G=n+12m+12.
Similarly, second multiplicative Zagreb index and modified first multiplicative Zagreb index of families of bistar graphs are(34)∏2G=n+1n+1m+1m+1,∏1∗G=n+2nn+m+2m+2m.Example 1.
Topological indices of bistar graphB5;6 are shown in Table 1.Table 1
Topological indices ofG = B5;6.
SCIG4.288RG4.463GAG8.465ABCG10.631M1G96∏1G1764


## 3. Topological Indices of Families of Corona Product of Graphs, i.e.,G=Km°Kn′

Based on previous results for computing topological indices for families of corona product of graphs, we present new degree-based topological indices to compute their values for these families of graphs.Theorem 6.
The Randić index and sum-connectivity index of families of corona product of graphsKm°Kn′ are(35)RG=mnm+n−1+mm−12m+n−1,SCIG=mnm+n+mm−122m+n−1.Proof.
To find the Randić index of the family of graphsKm°Kn′ obtained by taking the corona product of complete graph Km of order m and the complement of Kn of order n, firstly, we select a vertex v1 on Km°Kn′ as v1 is adjacent to m−1 vertices v2,v3,v4,…,vm and n vertices u1,u2,u3,…,un. Therefore, the degree of v1 is m+n−1.
For the verticesv1 and ui, where i=1,2,3,…,n, the sum is obtained as(36)∑v1∼ui1degv1GdeguiG=nm+n−1.
As the graph is symmetric, the same result is obtained for the remainingm−1 vertices v2,v3,…,vm. So, combining all the vertices v1,v2,v3,…,vm equation (36) becomes(37)∑v1∼ui1degv1GdeguiG+∑v2∼wi1degv2GdegwiG+⋯+∑vm∼xi1degvmGdegxiG=mnm+n−1mtimes.
Sincev1 is also adjacent to m−1 vertices v2,v3,v4,…,vm of degree m+n−1, for the vertices v1 and vj, where j=2,3,4,…,m, the sum is obtained as(38)∑v1∼vj1dv1GdvjG=m−1m+n−1.
SinceKm is symmetric, the same result is obtained for remaining m−1 vertices v2,v3,…,vm. Combining the result for all m vertices v1,v2,v3,…,vm, we have(39)=m−1m+n−1+m−2m+n−1+m−3m+n−1+⋯+m−m−1m+n−1=1m+n−1m−1+m−2+m−3+⋯+m−m−1=1m+n−1m+m+m+,…,+mm−1times−1+2+3+,…,+m−1=1m+n−1mm−1−m−1m−1+12=1m+n−1mm−1−mm−12=1m+n−1mm−12.
Adding equations (37) and (39), we have RG = Randić index of Km°Kn′:(40)RG=mnm+n−1+mm−12m+n−1.
Similarly,SCIG = sum-connectivity index of Km°Kn′ is(41)SCIG=mnm+n+mm−122m+n−1.Theorem 7.
The atom-bond connectivity index and augmented Zagreb index of families of corona product of graphsKm°Kn′ are(42)ABCG=mnm+n−2m+n−1+mm−12m+n−12m+2n−4,AZIG=mnm+n−1m+n−23+mm−1m+n−1622m+2n−43.The proof of this theorem is the same as Theorem6.Theorem 8.
The geometric-arithmetic index and harmonic index of families of corona product of graphsKm°Kn′ are(43)GAG=2mnm+n−1m+n+mm−1m+n−12m+2n−2,HG=2mnm+n+mm−12m+2n−2.The proof of this theorem is the same as Theorem6.Theorem 9.
The Zagreb indices of families of corona product of graphsKm°Kn′ are(44)M1G=mn+mm+n−12,M2G=mnm+n−1+mm−1m+n−122.Proof.
For the families of corona product of graphs there aremn vertices of degree one and m vertices v1,v2,v3,…,vm of degree m+n−1, respectively. For the vertices u1,u2,u3,…,un, we have(45)∑uidegui2=degu12+degu22+⋯+degun2,i=1,2,3,…,n=12+12,…,+12ntimes=n.
Since there aremn vertices of degree 1, combining the above result for all mn vertices, thus, the above equation becomes(46)=n+n+n+⋯+nmtimes=mn.
Also, for the vertexv1, we have(47)degv12=m+n−12.
Since the graph is symmetric, the same result is obtained for remainingm−1 vertices v2,v3,…,vm:(48)∑vjdegvj2=degv12+degv22+⋯+degvm2,j=1,2,3,…,m=m+n−12+m+n−12+⋯+m+n−12mtimes=mm+n−12.
Adding equations (46) and (48), we have M1G = first Zagreb index of families of corona product of graphs Km°Kn′:(49)M1G=mn+mm+n−12.
Similarly,M2G = second Zagreb index of families of corona product of graphs Km°Kn′:(50)M2G=mnm+n−1+mm−1m+n−122.Theorem 10.
The Narumi–Katayma and multiplicative Zagreb indices of families of corona product of graphsKm°Kn′ are(51)∏1G=m+n−12m,∏2G=m+n−1m2−m+mn,∏1∗G=m+nmn2m+2n−2m2−m/2.Proof.
For the families of corona product of graphs, there aremn vertices of degree one and m vertices v1,v2,v3,…,vm of degree m+n−1. For the vertices u1,u2,u3,…,un, we have(52)∏uidegui2=degu12degu22,…,degun2,i=1,2,3,…,n=1212,…,12ntimes=1n=1.
Since there aremn vertices of degree 1, so the same result is obtained for all mn vertices. Thus, the above equation becomes(53)1mn=1.
Also, for the vertexv1, we have(54)degv12=m+n−12.
Since the graph is symmetric the same result is obtained for remainingm−1 vertices v2,v3,…,vm. Thus equation (54) becomes(55)∏vjdegvj2=degv12degv22degv32,…,degvm2,j=1,2,3,…,m=m+n−12m+n−12m+n−12,…,m+n−12mtimes=m+n−12m.
Multiplying equations (53) to (55), we have ∏1G = first Zagreb index of families of corona product of graphs Km°Kn′(56)∏1G=1.m+n−12m=m+n−12m.
Similarly, second multiplicative Zagreb index and modified first multiplicative Zagreb index ofKm°Kn′ are(57)∏2G=m+n−1m2−m+mn,∏1∗G=m+nmn2m+2n−2m2−m/2.Example 2.
Topological indices of corona product of graphsK6°K3′ are shown in Table 2.Table 2
Topological indices ofG = K6°K3′.
∏1G6,8710000000HG5.875SCIG9.750RG8.239GAG26.314ABCG23.853M1G402


## 4. Concluding Remarks

There are several articles published on calculating topological indices for different families of graphs. Some have found applications, but others were devoted to the mathematical side to throw more light on the relationship between these concepts. This study introduces Randić index, Zagreb indices, Narumi–Katayama, and multiplicative Zagreb indices. We further discussed the atom-bond connectivity index, augmented Zagreb index, geometric-arithmetic index, harmonic index, and sum-connectivity index for two particular families of graphs for the first time. We have determined and computed the closed formulas of these families of graphs. We had checked that all vertex degree-based topological indices of families of bistar graphs and corona product of graphs remain the same for all values ofm and n. We have also determined the values of some topological indices for B5;6 and K6°K3′ graphs, as shown in Tables 1 and 2. This work will give new directions for considering and computing topological indices of several other families of graphs. In the future, we are interested in investigating and calculating some other topological indices of two more special families of graphs whose diameters are greater than three.



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