# The Degree-Based Topological Indices for Two Special Families of Graphs of Diameter Three

Authors: Aqsa Khalid; Nasreen Kausar; Mohammad Munir; Hassen Aydi; Sajida Kousar; Yaé Ulrich Gaba
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this research article, we determine some vertex degree-based topological indices or descriptors of two families of graphs, i.e.,G=C4Kn and G=C4Kn+v1v3, where C4Kn is a graph obtained by identifying one of the vertices of Kn with one vertex of C4. Similarly, a graph formed by joining one of the vertices of Kn with one vertex of C4+v1v3 is known as the C4Kn+v1v3 graph.


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## Body

## 1. Introduction

Around the center of a century ago, theoretical experts found some interesting relationships between different properties of organic substances and those of molecular structure by analyzing a few invariants of the underlying molecular graph. These graph invariants are useful for molecular objects and are named as topological indices or topological descriptors. These are numerical parameters of graphs which can be used to describe the physiochemical properties like boiling point, molecular weight, density and refractive index of a molecule [1, 2]. The three main categories of topological indices are distance-based topological indices, degree-based indices and spectrum-based topological indices. Among these, degree-based topological indices are of great significance [1, 3, 4]. Some vertex degree-based topological indices are Randic or connectivity index, first and second Zagreb indices, Narumi Katayama and multiplicative Zagreb indices, atom bond connectivity index, augmented Zagreb index, geometric arithmetic index, harmonic index, and sum connectivity index [5]. The topological descriptors are beneficial tools to chemists that are provided by graph theory. The use of these theoretical invariants in QSPR/QSPR modeling has been a popular work in the past years [4, 6].The first most widely used topological index in chemistry is the Wiener index. Initially, it was presented by the American Chemist Harold Wiener in 1947 and is denoted byWG. After its success, many other topological indices have been introduced that are based on Wiener’s work [3, 7].Milan Randic introduced the Randic index in 1975. It was the most popular index among all topological indices [2, 8]. The mathematical form of this descriptor is
(1)RG=∑μ~ν1dμGdνG,with a summation going with overall pairs of adjacent vertices of the graph G.Later, mathematicians did not pay any attention to this index for almost two decades. However, Erdos together with Bollobas worked on this index and discovered the mathematical hidden in it. In 1998, they published their first paper on this index. When the mathematical community began to understand the importance of this descriptor, they started to do researches on it. There have been hundreds of articles and several books dedicated to this structural descriptor [9]. The chemists [1] found out some expressions during the study of molecular structure dependency of total pi-electron energy which includes the terms of the form
(2)M1G=∑μdμG2,M2G=∑μ~νdμGdνG.Immediately, it was identified that these two terms provide quantitative measures of the branching of the molecular carbon atom skeleton. In chemical theory,M1 and M2 are called first and second Zegreb indices. These two indices have been introduced by Gutman and Trinajestic in 1972. Initially, the first Zagreb index was also known as the Gutman index. But, Balaban et al. did not want to introduce these indices by the names of the discovers. After 1982, Balaban et al. included these two terms in the topological indices and named them as Zagreb indices. In 2012, multiplicative versions of multiplicative Zagreb indices have been proposed by Todeshine et al. and are known as Narum Katayama indices. They were the first topological index defined by the product of some graph theoretical quantities. In 1984, Narumi and Katayama determined the term ∏μdμG2 and named it as the first multiplicative Zagreb index. It is denoted by ∏1G. The second multiplicative Zagreb index and modified first multiplicative Zagreb index are defined, respectively, by
(3)∏2G=∏μ~νdμGdνG,∏1∗G=∏μ~νdμG+dνG,with a product going with overall adjacent vertices of a graph G [10, 11]. The first, second, and modified multiplicative Zagreb indices are known as Narumi Katayama multiplicative Zagreb indices. The geometric arithmetic index is the modified form of the Randic index. This topological descriptor was introduced by Vukicevic and Furtula in 2009. The mathematical form [12] of this index is
(4)GAG=∑μ~νdμGdνGdμG+dνG/2.Another vertex degree-based topological index that keeps the spirit of Randic index is the atom bond connectivity index. This topological descriptor was introduced by Estrada et al. in 1998. This index has turned out to be an important index for the stability of Alkanes. The mathematical form [8, 13] of this index is
(5)ABCG=∑μ~νdμG+dνG−2dμGdνG.Furtula et al. introduced the modified form of atom bond connectivity index in 2009 and named it as the augmented Zagreb index. The mathematical form [8, 14] of this descriptor is
(6)AZIG=∑μ~νdμGdνGdμG+dνG−23.In the investigation of heat creation in heptanes and octanes, the prediction capability of the augmented Zagreb index outperforms that of the atom bond connectivity index. Ali et al. investigated the correlation abilities of 20 vertex degree-based topological descriptors for the case of normal heats of formation and ordinary boiling factor of octane isomers, finding that the AZI produces the best results [15].In 1980, Fajtlowiez introduced another topological index. In 2012, Zhang worked on this index and named it as the harmonic index. This index is defined as(7)HG=∑μ~ν2dμG+dνG.No chemical applications of this index have been found so far, but in the last few years, this index has attracted the great attention of theoreticians [16, 17]. Another vertex degree-based topological index was proposed by Zhou and Trinajstic and was named as the sum connectivity index. They observed in the definition of Randic index that the term dμ×dν can be replaced by dμ+dν which is defined by ([2, 18, 19]).
(8)SCIG=∑μ~ν1dμG+dνG.Many topological indices are bond-additive, i.e., they can be presented as a sum of edge contributions and have the following form:(9)∑μν∈EGfgμ,gν,where gμ are usually degrees or the sum of distances from μ to all other vertices of G. Inspired by the most successful indices of this type, such as the Randic index, the second Zagreb index and others, a whole family of Adriatic indices [20] was defined. A particularly intriguing subclass of these descriptors is made up of 148 discrete Adriatic indices. They were examined on the testing sets provided by the International Academy of Mathematical Chemistry, and it had been shown that they have good predictive properties in many cases and it was found that they have good predictive properties in many cases. One of these useful discrete Adriatic indices is the symmetric division deg (SDD) index, which is defined as
(10)SDDG=∑μν∈EGdμdν+dνdμ,where EG is the edge set of a molecular graph G (a graph in which vertices correspond to atoms of a (hydrogen-suppressed) molecule and edges correspond to bonds between the atoms) and dμ and dν denote the degrees of the vertices μν∈VG, respectively [21].In this paper, we compute the degree-based topological indices, namely, Randic or connectivity 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 for two special families of graphs of diameter three. These graphs are undirected having no loops and multiple edges. In Section2, the topological indices of families of graphs C4Kn are determined, while in Section 3, the topological indices of families of graphs C4Kn+v1v3 are computed.


## 2. Topological Indices of Families of GraphsC4Kn

In this section, we have determined and computed the closed formulas of degree-based topological indices for families of graphsC4Kn. The results are analyzed, and the general formulas are derived for these families of graphs.Theorem 1.
The Randic and sum connectivity indices of families of graphsC4Kn are
(11)RG=n−1n2−1+1n−1n2−3n+22+22n+2+1,SCIG=n−12n+2n+3+12n−2n2−3n+22+1.Proof.
To find the Randic or connectivity index ofC4Kn as shown in Figure 1, we first select a vertex u1 on C4Kn of degree n+1. There are n−1 vertices u2,u3,⋯,un of degree n−1, which are adjacent to u1. For the vertices u1 and ui, where i=2,3,4,⋯,n, the sum is obtained as
(12)∑u1~ui1du1GduiG=n−1n+1n−1i=2,3,,⋯,n=n−1n2−1.
Since the degree ofu1 is n+1, the other two vertices which are adjacent to u1, are v1 and v3 of degree two. For the vertices u1 and v1, we have
(13)1du1Gdv1G=12n+2.
Similarly, for the verticesu1 are v3, we have
(14)1du1Gdv3G=12n+2.
Now, we select a vertexu2 in C4Kn. Since the degree of u2 is n−1, the other n−2 vertices which are adjacent to u2, are u3,u4,⋯,un. For the vertices u2 and uj where j=3,4,5,⋯,n, the sum is obtained as
(15)∑u2~uj1du2GdujG=n−2n−1.
SinceKn is symmetric, the same result is obtained for the remaining n−2 vertices u3,u4,⋯,un. So combining the result for all n−1 vertices u2,u3,⋯,un, we have
(16)=n−2n−1+n−3n−1+n−4n−1+⋯+n−n−1n−1=1n−1n2−3n+22.
Now, we selectv1 on C4Kn. The degree of v1 is two, the other vertex which is adjacent to v1 is v2 of degree two. For the vertices v1 and v2, we have
(17)1dv1Gdv2G=12.
Similarly, we selectv2 on C4Kn. The degree of v2 is two, the other vertex which is adjacent to v2 is v3 of degree two. For vertices v2 and v3, we have
(18)1dv2Gdv3G=12.
Adding equations (12) to (18), we have RG = Randic or connectivity index of C4Kn is
(19)RG=n−1n2−1+1n−1n2−3n+22+22n+2+1.
Similarly,SCIG = sum connectivity index of C4Kn is
(20)SCIG=n−12n+2n+3+12n−2n2−3n+22+1.Theorem 2.
The atom bond connectivity index and augmented Zagreb index of families of graphsC4Kn are defined, respectively, by
(21)ABCG=n−12n+1+2n−4n−1n2−3n+22+22,AZIG=n−1n+123+n−162n−43n2−3n+22+32.
The proof is the same as that of Theorem 1.Theorem 3.
The geometric arithmetic index and harmonic index of families of graphsC4Kn are defined, respectively, by
(22)GAG=n−1n2−1n+42n+2n+3+n2−3n+22+2,HG=n−1n+4n+3+1n−1n2−3n+22+1.
The proof is similar to the proof of Theorem 1.Theorem 4.
The Zagreb indices of families of graphsC4Kn are
(23)M1G=n3−2n2+5n+12,M2G=n3−n2+3n+13+n2−3n+2n−122.Proof.
For the family of graphsC4Kn, there are n−1 vertices u2,u3,⋯,un of degree n−1, one vertex u1 of degree n+1, and three vertices v1, v2, and v3 of degree two, respectively. For the vertex u1, we have
(24)degu12=n+12.
For the verticesv1, v2 and v3, we have
(25)degv12+degv22+degv32=12.
For verticesu2,u3,⋯,un, we have
(26)degu22+degu32+⋯+degun2=n−13.
Adding equations (24) to (26), we have M1G = first Zagreb index of C4Kn is
(27)M1G=n3−2n2+5n+12.
Similarly,M2G = second Zagreb index of C4Kn is
(28)M2G=n3−n2+3n+13+n2−3n+2n−122.Theorem 5.
The Katayma and multiplicative Zagreb indices of families of graphsC4Kn are
(29)∏1G=64n+12n2−2n+1n−1,∏2G=16n2−1n−12n+22n−1n2−3n+2,∏1∗G=162nn−13+n22n−2n2−3n+2/2.Proof.
Since there aren−1 vertices u2,u3,⋯,un of degree n−1, we have one vertex of degree n+1 and three vertices of degree two in C4Kn graph. Thus, we have
(30)degu22degu32,⋯,degun2=n−12n−12,⋯,n−12n−1.times=n2−2n+1n−1.
For the vertexu1, we have
(31)degu12=n+12.
Similarly, for the verticesv1, v2, and v3, we have
(32)degv12degv22degv32=22.22.22=64.
Multiplying equations (30) to (32), ∏1G =first multiplicative Zagreb index of C4Kn is
(33)∏1G=64n+12n2−2n+1n−1.
Similarly, second multiplicative and modified first multiplicative Zagreb indices ofC4Kn are defined, respectively, by
(34)∏2G=16n2−1n−12n+22n−1n2−3n+2,∏1∗G=162nn−13+n22n−2n2−3n+2/2.Example 6.
Topological indices of graphC4K6 are shown in Table 1.Figure 1
G=C4K5 (graph obtained by identifying one of the vertices of K5 with one vertex of C4).Table 1
Topological indices ofG=C4K6.
∏1G3.0625×1010HG4.278RG4.380SCIG6.272ABCG11.158GAG18.593M1G186


## 3. Topological Indices of Families of GraphsC4Kn+v1v3

By looking at the earlier results for computing the topological indices for different families of graphs, here, we introduce new degree-based topological indices to compute their values for families of graphsC4Kn+v1v3.Theorem 7.
The Randic and sum connectivity indices of families of graphsC4Kn+v1v3 are defined, respectively, by
(35)RG=n−1n2−1+1n−1n2−3n+22+23n+3+26+13,SCIG=n−12n+2n+4+12n−2n2−3n+22+25+16.Proof.
To find the Randic or connectivity index ofC4Kn+v1v3 as presented in Figure 2, we first select a vertex u1 on C4Kn+v1v3 of degree n+1. There are n−1 vertices u2,u3,⋯,un of degree n−1, which are adjacent to u1. For the vertices u1 and ui where i=2,3,4,⋯,n, the sum
(36)∑u1~ui1du1GduiG,i=2,3,4,⋯,n,is obtained as
(37)=n−1n+1n−1=n−1n2−1.
Since the degree ofu1 is n+1, the other two vertices which are adjacent to u1, are v1 and v3 of degree 3. For the vertices u1 and v1, we have
(38)1du1Gdv1G=13n+3.
Similarly, for the verticesu1 are v3, we have
(39)1du1Gdv3G=13n+3.
Now, we select a vertexu2 on C4Kn. Since the degree of u2 is n−1, the other n−2 vertices which are adjacent to u2, are u3,u4,cdots,un. For the vertices u2 and uj where j=3,4,5,⋯,n, the sum is obtained as
(40)∑u2~uj1du2GdujG=n−2n−1.
SinceKn is symmetric, the same result is obtained for the remaining n−2 vertices u3,u4,⋯,un. By using equation (16), the above equation can be written as
(41)=1n−1n2−3n+22.
Now, we selectv1 on C4Kn. Since the degree of v1 is three, the other two vertices which are adjacent to v1, are v2 and v3. For vertices v1 and v2, we have
(42)1dv1Gdv2G=16.
Similarly, for the verticesv1 and v3, we have
(43)1dv1Gdv3G=13.
Now, we selectv2 on C4Kn. The degree of v2 is two, the other vertex which is adjacent to v2, is v3 of degree three. For vertices v2 and v3, we have
(44)1dv2Gdv3G=16.
Adding equations (37) to (44) RG = Randic or connectivity index of C4Kn+v1v3 is RG=n−1/n2−1+1/n−1n2−3n+2/2+2/2n+2+1.
Similarly,SCIG = sum connectivity index of C4Kn+v1v3 is
(45)SCIG=n−12n+2n+4+12n−2n2−3n+22+25+16.
Using the same arguments as in Theorem 6, we determine some other topological indices ofC4Kn+v1v3.Theorem 8.
The atom bond connectivity index and augmented Zagreb index of families of graphsC4Kn+v1v3are defined, respectively, by(46)ABCG=n−12n+1+2n−4n−1n2−3n+22+2n+23n+3+2+23,AZIG=n−1n+123+n−162n−43n2−3n+22+23n+3n+23+175364.Theorem 9.
The geometric arithmetic index and harmonic index of families of graphsC4Kn+v1v3 are defined, respectively, by
(47)GAG=n−1n2−1n+43n+3n+4+n2−3n+22+465+1HG=n−1n+4n+4+1n−1n2−3n+22+1715.Theorem 10.
The Zagreb indices of the families of graphsC4Kn+v1v3 are
(48)M1G=n3−2n2+5n+22,M2G=n3−n2+5n+28+n2−3n+2n−122.Proof.
For the families of graphsC4Kn+v1v3 there are n−1 vertices u2,u3..⋯un of degree n−1, one vertex u1 of degree n+1 and two vertices v1 and v3 of degree three and one vertex v2 of degree two, respectively.
(49)degu12=n+12.
For verticesv1 and v3, the first Zagreb index is
(50)degv12+degv32=32+32=18.
For verticesu2,u3..⋯un, we have
(51)degu22+degu32+⋯⋯⋯+degun2=n−12+n−12+.⋯⋯n−12n−1times=n−1n−12=n−13.
Similarly, for vertexv2 we have
(52)degv22=22=4.
Adding equations (49) to (52), we have M1G = first Zagreb index of C4Kn+v1v3(53)M1G=n+12+n−13+22=n3−2n2+5n+22.
Similarly,M2G = second Zagreb index of C4Kn+v1v3(54)M2G=n3−n2+5n+28+n2−3n+2n−122.Theorem 11.
The Katayma and multiplicative Zagreb indices of families of graphsC4Kn+v1v3 are
(55)∏1G=324n+12n2−2n+1n−1,∏2G=324n2−1n−13n+32n−1n2−3n+2,∏1∗G=1502nn−1n+422n−2n2−3n+2/2.Proof.
Since there aren−1 vertices u2,u3..⋯un of degree n−1, one vertex of degree n+1 and three vertices of degree two in C4Kn+v1v3 graph. Thus, we have
(56)degu22degu32..⋯⋯degun2=n−12n−12.⋯⋯n−12n−1times=n2−2n+1n−1.
For vertexu1, we have
(57)degu12=n+12.
Similarly, for verticesv1 and v3, we have
(58)degv12degv32=32.32=81.
Also, for vertexv2, we have
(59)degv22=22=4.
Multiplying equations (56) to (59), we have ∏1G = first multiplicative Zagreb index of C4Kn+v1v3(60)∏1G=324n+12n2−2n+1n−1.
Similarly, second multiplicative Zagreb index and modified first multiplicative Zagreb index ofC4Kn+v1v3 are
(61)∏2G=324n2−1n−13n+32n−1n2−3n+2∏1∗G=1502nn−1n+422n−2n2−3n+2/2.Example 12.
Topological indices of graphC4K6+v1v3 are shown in Table 2.Figure 2
G=C4K5+v1v3 (graph obtained by identifying one of the vertices of K5 with one vertex of C4+v1v3).Table 2
Topological indices ofG=C4K6+v1v3.
∏1G1.5503×1011HG4.366RG4.431SCIG6.541ABCG10.231GAG19.723M1G196


## 4. Concluding Remarks

This research work was done to understand the relationship between various concepts about topological indices and graphs. In this article, we have studied certain vertex degree-based topological indices and derived the closed formulas of these indices for two special families of graphs, i.e.,C4Kn and C4Kn+v1v3. It was easily checked that all vertex degree-based topological indices of families of these two graphs remain the same for all values of n and the topology of a graph is completely changed if we add an additional edge in it. In the future, we are interested in investigating and calculating some other topological indices of two more special families of graphs.



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