# Omega, Sadhana, Theta, and PI Polynomials of Double Benzonoid Chain

Authors: Fozia Bashir Farooq; Saima Parveen; Nadeem Ul Hassan Awan; Rakotondrajao Fanja
Journal: Journal of Mathematics
 (2022)

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## Abstract

Counting polynomials are closely related to certain features of chemical graphs and provide an elegant means of expressing graph topological invariants. The current paper aims to calculate four polynomials for double benzenoid chains, Sadhana, omega, theta, and Padmakar–Ivan (PI). The edge-cut method is used to derive analytical closed expressions for these polynomials.


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## Body

## 1. Introduction

In chemistry, graph theory has a wide range of applications, particularly in the mathematical modeling of chemical structures [1–3]. Atoms are referred to as vertices in chemical graph theory, while the bonds that connect them are referred to as edges. Graph H (V, E) is a connected, finite, simple chemical graph, where vertex and edge sets are debited by V and E, respectively. Pólya [4] coined the concept of a counting polynomial in the field of chemistry in 1936. However, chemists did not pay much attention to the subject for several decades, even though characteristic polynomials are computed from molecular orbitals of unsaturated hydrocarbons [5]. Polynomial counting is a common method for expressing molecular invariants in a chemical graph in polynomial form. Chemical graph features such as equidistant edges, independent sets, matching sets, and chromatic numbers influence these polynomials. Characteristic polynomials, Hosoya polynomial, rotational polynomial, Wiener polynomial, and sextet polynomial are some well-known polynomials. Polynomials can be used to produce a variety of significant topological indices, either directly or thenceforth integrals or derivatives. A topological index is a numeric quantity that can quantify many chemical properties in organic chemistry and is invariant under the graph’s automorphism and derived according to specific rules. Topological indices were first used in chemistry in 1947; Harary Wiener introduced a renowned topological descriptor called the Wiener index [6], which is given as(1)WH=12∑p∈VG∑q∈VGdp,q,where d(p, q) is the shortest distance between vertices p and q.For a molecular graphHV,E,the edges e=xy and f=wz of H are called codistant, denoted by eCof if they satisfy the following topologically parallel relation: dx,z=dy,w=dx,w+1=dy,z+1 and. dx,w=dy,z.Co is symmetric and reflexive. It is not transitive in general. SetCe=f∈EG:eCof. The given relation “Co” is transitive on Ce, and Ce is said to be orthogonal cut oc in the graph. It is named a quasiorthogonal cut qoc if the relation Co is not transitive.By quasiorthogonal cuts, omega, theta, PI polynomials, and Sadhana are well-defined. Diudea [7] defined omega polynomial ΩH,x as calculating “quasiorthogonal cut” strips, qoc strips in H as(2)ΩH,t=∑kmH,ktk,where m(H, k) represents the number of qoc of length k.Diudea et al. [8] were the first to introduce the theta polynomial θH,x in 2008. It is defined as counting several edges equidistant for each edge f of the graph and given as(3)θH,t=∑kmH,kktk.The Sadhana polynomial denoted bysdH,t was introduced by Ashrafi et al. [9]. It is given as follows:(4)sdH,t=∑kmH,kte−k.After Khadikar et al. [10] presented the Padmakar–Ivan (PI) index, Ashrafi et al. [11] proposed the PI polynomial. The polynomial, denoted by PI(H, t), counts no equidistant edges in G and is given as follows:(5)PIH,t=∑kmH,kkte−k.Counting polynomials have piqued the interest of analysts and researchers working in the field of chemical graph theory, with some recent work included in [12–16]. In addition, various topological features of hexagonal chains have been considered in [17–20]. This paper calculated four counting polynomials of double benzenoid chains: Sadhana, omega, theta, and Padmakar–Ivan (PI).


## 2. Results and Discussion

### 2.1. Counting Polynomials and a Double Benzenoid Hexagonal Chain

Benzenoid hydrocarbons play a significant role in organic chemistry. This hexagonal system has congruent interior regions and contains no cut vertex. The system is named pericondensed if at least one vertex belongs to three hexagons; otherwise, the hexagonal system is said to be catacondensed. The chain is an arrangement of hexagons in which no two hexagons are adjacent to each other. We obtainn-tuple hexagonal chain which is formed when condensed identical hexagonal chains combine [9, 21].A double hexagonal chain can be obtained whenn = 2 [22, 23]. A pericondensed hexagonal system is formed with the aid of a new fused naphthalene, which is a double hexagonal (benzenoid) chain.Let us have a look at the naphthalene structure with horizontal internal edges. There are two types of naphthalene fusion in this regard, as shown in Figure1.(6)α−type:v=1,w=2,x=3,y=4,β−type:u=2,v=3,w=4,x=5.Figure 1
(a) Naphthalene fusionβ type in the double hexagonal benzenoid chain. (b) A double hexagonal benzenoid chain Hη1,η2,η3,η4,η5,η6.
(a)(b)At every stage, fusion type is obtained byη, where η∈α,β. A double (benzenoid) chain is denoted by Hη1,η2,…,ηn where η1,η2,…,ηn∈α,β are fusion types, respectively. It is seen that Hη1,η2,…,ηn contains 2n + 2 regular hexagons and n+1 naphthalene, see Figure 1. The number of codistant edges of the double benzenoid chain Hα,β,α,β,…,α,β is given in Table 1.Table 1
Number of codistant edges of the double benzenoid hexagonal chainHα,β,α,β,…,α,β (using Figure 2).
Types of qocsTypes of edgesNo. of codistant edgesNo. of qocsMs122425n − 2Ss23n + 3The details of double hexagonal (benzenoid) chains can be seen in [21, 24–27]. The edge-cut procedure proposed by Klavzar [28] will be used to compute the counting polynomials.Figure 2
The elementary cuts of the double benzenoid hexagonal chainHα,β,α,β,α,β.Theorem 1.
The omega polynomial of the double hexagonalHα,β,α,β,…,β is given by(7)ΩHα,β,α,β,…,β,t=2t2+2t4+n+2t3+n−2t5.Proof.
Consider a graphH of a double hexagonal chain Hα,β,α,β,…,β, with 8n+11 number of edges. The omega polynomial of graph H is ΩH,t=∑kmH,ktk.
The elementary cuts of the double benzenoid hexagonal chain are described in Figure2.
Using Table1 for the number of qocs and the number of codistant edges, we get the desired result.(8)ΩHα,β,α,β,…,β,t=2t2+2t4+n+3t3+n−2t5.Theorem 2.
The Sadhana polynomial of the double hexagonalHα,β,α,β,…,α,β is given by(9)sdHα,β,α,β,…,β,t=t8n+62t3+n+3t2+2t+n−2.Proof.
Consider a graphH of a double hexagonal chain Hα,β,α,β,…,β. The Sadhana polynomial of graph H is sdH,t=∑kmH,kte−k, where e=E=8n+11.
Using Table1 for the number of qocs and the number of codistant edges, we get the desired result.(10)SdHα,β,α,β,…,β,t=n−2t8n+11−5+n+3t8n+11−3+2t8n+11−4+2t8n+11−2.
On simplifying it becomes(11)SdHα,β,α,β,…,β,t=t8n+62t3+n+3t2+2t+n−2.Theorem 3.
The theta polynomial of the double hexagonalHα,β,α,β,…,α,β is given by(12)θHα,β,α,β,…,β,t=4t2+8t4+3n+3t3+5n−2t5.Proof.
Consider a graphH of a double hexagonal chain Hα,β,α,β,…,β. The theta polynomial of graph H is(13)θH,t=∑kmH,kktk.
Using Table1 for several qocs and several codistant edges, we get the desired result.(14)θHα,β,α,β,…,β,t=4t2+8t4+3n+3t3+5n−2t5.Theorem 4.
The PI polynomial of the double hexagonalHα,β,α,β,…,α,β is given by(15)PIHα,β,α,β,…,β,t=t8n+64t3+3n+3t2+8t+5n−10.Proof.
Consider a graphH of a double hexagonal chain Hα,β,α,β,…,β,α. The PI polynomial of graph H with e=E=8n+3 is PIH,t=∑kmH,kkte−k.
Using Table1 for the number of qocs and the number of codistant edges, we get(16)PIHα,β,α,β,…,β,α,t=5n−2t8n+11−5+3n+3t8n+11−3+8t8n+11−4+4t8n+11−2.
On simplifying, we get(17)PIHα,β,α,β,…,β,α,t=t8n+64t3+3n+3t2+8t+5n−10.


### 2.2. Double Benzenoid Hexagonal Linear Chain and Counting Polynomials

Hη1,η2,…,ηn is said to be a double linear hexagonal (benzenoid) chain, L2×n, if for all i,ηi=ηi+1 in Hη1,η2,…,ηn, see Figure 3. L2×n has 8n+11 edges. The number of codistant edges of the double benzenoid chain L2×n is given in Table 2.Figure 3
(a) Naphthaleneα type fusion in the double hexagonal benzenoid chain. (b) A double linear hexagonal benzenoid chain L2×5.
(a)(b)Table 2
Number of codistant edges of the double linear benzenoid chainL2×n.
Types of qocsTypes of edgesNo. of codistant edgesNo. of qocsMs1n + 22Ns232n + 1Ns222Theorem 5.
The omega polynomial of the double hexagonalL2×n is given by(18)ΩL2×n,t=2tn+2+2n+1t3+2t2.Proof.
Consider the double hexagonal chainL2×n and omega polynomial of the graph H as(19)ΩH,t=∑kmH,ktk.
The elementary cuts of the double benzenoid linear chain are described in Figure4.
Using the definition of omega polynomial, and putting the values ofm(H, k) and k from Table 2, we get the desired result that(20)ΩL2×n,t=2tn+2+2n+1t3+2t2.Figure 4
The elementary cuts of the double benzenoid linear chainL2×5.Theorem 6.
The Sadhana polynomial of the double linear hexagonalL2×n is given as follows:(21)SdL2×n,t=t7n+82tn+1+2n+1tn+2t.Proof.
SincesdH,t=∑kmH,kte−k, e = 8n + 11.
Again using Table2, and putting the values of m(H, k) and k in the definition of Sadhana polynomial, we get(22)SdL2×n,t=2t8n+11−n−2+2n+1t8n+11−3+2t8n+11−2.
On simplifying, we get(23)SdL2×n,t=t7n+82tn+1+2n+1tn+2t.Theorem 7.
The theta polynomial of the double linear hexagonalL2×n is given by(24)θL2×n,t=2n+4tn+2+6n+3t3+4t2.Proof.
We know thatθH,t=∑kmH,kktk.
Putting values ofmH,k and k from Table 1, we get(25)θL2×n,t=2n+2tn+2+32n+1t3+4t2,which simplifies to(26)θL2×n,t=2n+4tn+2+6n+3t3+4t2.Theorem 8.
The PI polynomial of the double linear hexagonalL2×n is given by(27)PIL2×n,t=t7n+84tn+1+6n+3tn+4n+2t.Proof.
By definition, PI polynomial is given ase=E=8n+11 and(28)PIL2×n,t=∑kmH,kkte−k.
Again employing Table2 for values of k and mH,k, we get the following polynomial:(29)PIL2×n,t=2n+2t8n+11−n−2+32n+1t8n+11−3+4t8n+11−2,which leads the result.(30)PIL2×n,t=t7n+84tn+1+6n+3tn+2n+4t.


## 2.1. Counting Polynomials and a Double Benzenoid Hexagonal Chain

Benzenoid hydrocarbons play a significant role in organic chemistry. This hexagonal system has congruent interior regions and contains no cut vertex. The system is named pericondensed if at least one vertex belongs to three hexagons; otherwise, the hexagonal system is said to be catacondensed. The chain is an arrangement of hexagons in which no two hexagons are adjacent to each other. We obtainn-tuple hexagonal chain which is formed when condensed identical hexagonal chains combine [9, 21].A double hexagonal chain can be obtained whenn = 2 [22, 23]. A pericondensed hexagonal system is formed with the aid of a new fused naphthalene, which is a double hexagonal (benzenoid) chain.Let us have a look at the naphthalene structure with horizontal internal edges. There are two types of naphthalene fusion in this regard, as shown in Figure1.(6)α−type:v=1,w=2,x=3,y=4,β−type:u=2,v=3,w=4,x=5.Figure 1
(a) Naphthalene fusionβ type in the double hexagonal benzenoid chain. (b) A double hexagonal benzenoid chain Hη1,η2,η3,η4,η5,η6.
(a)(b)At every stage, fusion type is obtained byη, where η∈α,β. A double (benzenoid) chain is denoted by Hη1,η2,…,ηn where η1,η2,…,ηn∈α,β are fusion types, respectively. It is seen that Hη1,η2,…,ηn contains 2n + 2 regular hexagons and n+1 naphthalene, see Figure 1. The number of codistant edges of the double benzenoid chain Hα,β,α,β,…,α,β is given in Table 1.Table 1
Number of codistant edges of the double benzenoid hexagonal chainHα,β,α,β,…,α,β (using Figure 2).
Types of qocsTypes of edgesNo. of codistant edgesNo. of qocsMs122425n − 2Ss23n + 3The details of double hexagonal (benzenoid) chains can be seen in [21, 24–27]. The edge-cut procedure proposed by Klavzar [28] will be used to compute the counting polynomials.Figure 2
The elementary cuts of the double benzenoid hexagonal chainHα,β,α,β,α,β.Theorem 1.
The omega polynomial of the double hexagonalHα,β,α,β,…,β is given by(7)ΩHα,β,α,β,…,β,t=2t2+2t4+n+2t3+n−2t5.Proof.
Consider a graphH of a double hexagonal chain Hα,β,α,β,…,β, with 8n+11 number of edges. The omega polynomial of graph H is ΩH,t=∑kmH,ktk.
The elementary cuts of the double benzenoid hexagonal chain are described in Figure2.
Using Table1 for the number of qocs and the number of codistant edges, we get the desired result.(8)ΩHα,β,α,β,…,β,t=2t2+2t4+n+3t3+n−2t5.Theorem 2.
The Sadhana polynomial of the double hexagonalHα,β,α,β,…,α,β is given by(9)sdHα,β,α,β,…,β,t=t8n+62t3+n+3t2+2t+n−2.Proof.
Consider a graphH of a double hexagonal chain Hα,β,α,β,…,β. The Sadhana polynomial of graph H is sdH,t=∑kmH,kte−k, where e=E=8n+11.
Using Table1 for the number of qocs and the number of codistant edges, we get the desired result.(10)SdHα,β,α,β,…,β,t=n−2t8n+11−5+n+3t8n+11−3+2t8n+11−4+2t8n+11−2.
On simplifying it becomes(11)SdHα,β,α,β,…,β,t=t8n+62t3+n+3t2+2t+n−2.Theorem 3.
The theta polynomial of the double hexagonalHα,β,α,β,…,α,β is given by(12)θHα,β,α,β,…,β,t=4t2+8t4+3n+3t3+5n−2t5.Proof.
Consider a graphH of a double hexagonal chain Hα,β,α,β,…,β. The theta polynomial of graph H is(13)θH,t=∑kmH,kktk.
Using Table1 for several qocs and several codistant edges, we get the desired result.(14)θHα,β,α,β,…,β,t=4t2+8t4+3n+3t3+5n−2t5.Theorem 4.
The PI polynomial of the double hexagonalHα,β,α,β,…,α,β is given by(15)PIHα,β,α,β,…,β,t=t8n+64t3+3n+3t2+8t+5n−10.Proof.
Consider a graphH of a double hexagonal chain Hα,β,α,β,…,β,α. The PI polynomial of graph H with e=E=8n+3 is PIH,t=∑kmH,kkte−k.
Using Table1 for the number of qocs and the number of codistant edges, we get(16)PIHα,β,α,β,…,β,α,t=5n−2t8n+11−5+3n+3t8n+11−3+8t8n+11−4+4t8n+11−2.
On simplifying, we get(17)PIHα,β,α,β,…,β,α,t=t8n+64t3+3n+3t2+8t+5n−10.


## 2.2. Double Benzenoid Hexagonal Linear Chain and Counting Polynomials

Hη1,η2,…,ηn is said to be a double linear hexagonal (benzenoid) chain, L2×n, if for all i,ηi=ηi+1 in Hη1,η2,…,ηn, see Figure 3. L2×n has 8n+11 edges. The number of codistant edges of the double benzenoid chain L2×n is given in Table 2.Figure 3
(a) Naphthaleneα type fusion in the double hexagonal benzenoid chain. (b) A double linear hexagonal benzenoid chain L2×5.
(a)(b)Table 2
Number of codistant edges of the double linear benzenoid chainL2×n.
Types of qocsTypes of edgesNo. of codistant edgesNo. of qocsMs1n + 22Ns232n + 1Ns222Theorem 5.
The omega polynomial of the double hexagonalL2×n is given by(18)ΩL2×n,t=2tn+2+2n+1t3+2t2.Proof.
Consider the double hexagonal chainL2×n and omega polynomial of the graph H as(19)ΩH,t=∑kmH,ktk.
The elementary cuts of the double benzenoid linear chain are described in Figure4.
Using the definition of omega polynomial, and putting the values ofm(H, k) and k from Table 2, we get the desired result that(20)ΩL2×n,t=2tn+2+2n+1t3+2t2.Figure 4
The elementary cuts of the double benzenoid linear chainL2×5.Theorem 6.
The Sadhana polynomial of the double linear hexagonalL2×n is given as follows:(21)SdL2×n,t=t7n+82tn+1+2n+1tn+2t.Proof.
SincesdH,t=∑kmH,kte−k, e = 8n + 11.
Again using Table2, and putting the values of m(H, k) and k in the definition of Sadhana polynomial, we get(22)SdL2×n,t=2t8n+11−n−2+2n+1t8n+11−3+2t8n+11−2.
On simplifying, we get(23)SdL2×n,t=t7n+82tn+1+2n+1tn+2t.Theorem 7.
The theta polynomial of the double linear hexagonalL2×n is given by(24)θL2×n,t=2n+4tn+2+6n+3t3+4t2.Proof.
We know thatθH,t=∑kmH,kktk.
Putting values ofmH,k and k from Table 1, we get(25)θL2×n,t=2n+2tn+2+32n+1t3+4t2,which simplifies to(26)θL2×n,t=2n+4tn+2+6n+3t3+4t2.Theorem 8.
The PI polynomial of the double linear hexagonalL2×n is given by(27)PIL2×n,t=t7n+84tn+1+6n+3tn+4n+2t.Proof.
By definition, PI polynomial is given ase=E=8n+11 and(28)PIL2×n,t=∑kmH,kkte−k.
Again employing Table2 for values of k and mH,k, we get the following polynomial:(29)PIL2×n,t=2n+2t8n+11−n−2+32n+1t8n+11−3+4t8n+11−2,which leads the result.(30)PIL2×n,t=t7n+84tn+1+6n+3tn+2n+4t.


## 3. Conclusion

Counting polynomials are a simple technique to encode topological indices of chemical graphs, which are quantifiers of various physiochemical aspects of compounds and are commonly employed in structure-activity correlations. The edge-cut method of Klavzar is used to compute distance-based counting polynomials of the double benzenoid chain, such as omega, Sadhana, theta and PI polynomials. These polynomials are well-known methods for matching the chemical graph with the physiological features of various double benzenoid chains.



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