Exploring structural variations and topological descriptors of square-hexagonal kink chains of type T 2,2 in engineering applications

Abstract

Square-hexagonal kink chains, also known as hexagonal kink chains or simply kink chains, are structural configurations used in engineering for various purposes, including material science, nano-technology and mechanics. The square hexagonal kink chain likely describes a chain-like structure where the individual units have square and hexagon-like shapes and are arranged with symmetry. Additionally, there are likely points in the chain where the linear arrangement deviates or kinks, possibly due to some structural irregularity. A nonterminal hexagon is regarded as a kink if it contains two neighbouring vertices of degree $2$ , while a nonterminal square is considered a kink if and only if it has a vertex of degree $2$ . The kinks from these two types of arrangements are called kinks of type $T_1$ and $T_2$ , respectively. Our objective is to discover three possible arrangements of kinks of type $T_2$ further, depending on the process of how different polygons are attached at different places, holding the condition to make a kink at each step. In this work, we have calculated forgotten, geometric-arithmetic, sum-connectivity and atom bond-connectivity indices for the kink chains. A comparison between these topological descriptor is given by computing their numerical values for each kink chain.

Introduction

A graph $G$ is a collection of vertices and edges, denoted as an ordered pair $G=(V(G),E(G))$ . The number of vertices and edges in a graph is called its order and size, respectively. The number of edges incident to a vertex $u\in V(G)$ is its degree, represented as $d_u$ or $\mathrm{deg}(u)$ . A graph $G$ is said to be planar if it can be drawn on a flat surface (like a piece of paper) in such a way that no two edges intersect, except possibly at their endpoints. The inner dual of a planar graph is obtained by replacing the faces of the original graph with vertices and connecting two vertices if their corresponding faces share an edge. For basic definitions related to graph theory. ¹

A geometric figure obtained by connecting regular squares and/or hexagons is known as a Square-hexagonal system. If the inner dual of a Square-hexagonal system is a path graph, it is specifically termed a Square-hexagonal chain. By inner dual (ID) of a Square-hexagonal system, we mean a graph whose vertices represent the polygons of the considered Square-hexagonal system. There is an edge between two vertices of the inner dual if and only if the corresponding polygons share a side. The term ‘Square-hexagonal chain’ refers to a finite graph obtained by concatenating $n$ cells (each being either a square or a hexagon) in such a way that any two adjacent cells have exactly one edge in common. Additionally, each cell is adjacent to exactly two other cells, except for the first and last cells (end cells), which are adjacent to exactly one other cell each. It should be noted that different Square-hexagonal chains may be obtained depending on the polygon type and the way the polygons are concatenated. If all the polygons are squares, then it is defined as a polyomino chain. ³ The graphs of linear and zig-zag polyomino chains are shown in Figure 1.

Figure 1.

Polyomino chains: (a) linear polyomino chain and (b) zig-zag polyomino chain.

If all the polygons are hexagons, then it is known as hexagonal chain. Figure 2 displays linear and zig-zag hexagonal chains.

Figure 2.

Hexagonal chains: (a) linear hexagonal chain and (b) zig-zag hexagonal chain.

Additionally, if hexagonal chain has alternate concatenations of squares and hexagons, then it is known as a phenylene chain, as shown in Figure 3.

Figure 3.

Square-hexagonal and phenylene chains: (a) linear square-hexagonal chain L_n and (b) phenylene chain H_n.

Square-hexagonal kinks

To obtain the main results, we need to introduce some fundamental terminology related to square-hexagonal kink chains. A polygon in a square-hexagonal chain is termed a terminal polygon if it is adjacent to one additional polygon, and a nonterminal polygon if it is adjacent to two additional polygons. If the centre of a nonterminal polygon is not collinear with the centres of the two adjacent polygons, the polygon is referred to as a kink. There are two types of Square-Hexagonal Kinks: $T_1$ and $T_2$ . ³ In type $T_1$ , a hexagon occurs as a kink, while in type $T_2$ , a square occurs as a kink. A nonterminal hexagon is a kink if and only if it contains two consecutive vertices of degree two. Conversely, a nonterminal square is a kink if and only if it has a vertex of degree two. If a square-hexagonal chain has no kinks, it is said to be linear. If every nonterminal polygon has a kink, the chain is said to be zig-zag. The following forms of kinks in square-hexagonal chains will be considered.

• Kinks of type $T_1$ : A square-hexagonal kink chain is of type $T_1$ if the non-terminal hexagon has exactly two vertices of degree two (see Figure 4).

• Kinks of type $T_{2,1}$ : A square-hexagonal kink chain is of type $T_{2,1}$ if non-terminal square with a vertex of degree two is adjacent to two squares (see Figure 5(a)).

• Kinks of type $T_{2,2}$ : A square-hexagonal kink chain is of type $T_{2,2}$ if non-terminal square with a vertex of degree two is adjacent to a square and a hexagon (see Figure 5(b)).

• Kinks of type $T_{2,3}$ : A square-hexagonal kink chain is of type $T_{2,3}$ if non-terminal square with a vertex of degree two is adjacent to two hexagons (see Figure 5(c)).

Figure 4.

Square-hexagonal kinks of type $T_1$ .

Figure 5.

Square-hexagonal kinks of type $T_2$ : (a) T_2,1, (b) T_2,2 and (c) T_2,3.

A segment is a square-hexagonal chain’s longest maximal linear chain that includes kinks and/or terminal polygons at its ends. The length $L(S)$ of a segment is defined as number of polygons including in it. Let $E(S)$ be the set of all edges of segment $S$ .

Topological descriptors

Topological descriptors/Molecular descriptors are quantitative measures that capture the structural characteristics and connectivity patterns of chemical or biological entities, such as molecules or proteins, without considering the spatial arrangement of atoms. These descriptors play a crucial role in computational chemistry, cheminformatics and bioinformatics for the analysis and prediction of various properties and activities. The numerical values assigned to a molecule based on its molecular graph topology are also referred to as topological indices. It provides information about the branching, connectivity and cycles within the molecular structure.

The first and second Zagreb indices ⁶ are one of the oldest and most studied topological indices. These topological indices were applied to branching problem in 1975 and exhibit a potential applicability for deriving multilinear regression models. The first and second Zagreb indices are defined as

$$M_1(G)=\sum _{\mathrm{uv}\in E(G)}(d_u+d_v)$$

$$M_2(G)=\sum _{\mathrm{uv}\in E(G)}(d_u\times d_v).$$

Atom-bond connectivity $(\mathrm{ABC})$ index is another degree based topological index introduced by Estrada et al. ⁷ which is denoted as.

$$\mathrm{ABC}(G)=\sum _{\mathrm{uv}\in E(G)}\sqrt{\frac{d_u+d_v-2}{d_u{d}_v}}.$$

The ABC index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes.^7,8

Recently the well-known connectivity topological index is geometric-arithmatic $(\mathrm{GA})$ which was introduced by Vukičević et al. ⁹ For a graph $G$ , the $\mathrm{GA}$ index is denoted and defined as

$$\mathrm{GA}(G)=\sum _{\mathrm{uv}\in E(G)}\frac{2\sqrt{d_u{d}_v}}{d_u+d_v}.$$

It has been demonstrated, on the example of octane isomers, that $\mathrm{GA}$ index is well-correlated with a variety of physico-chemical properties.

In Furtula and Gutman, ⁴ some fundamental characteristics of forgotten topological index are defined and demonstrated how it can greatly improve the first Zagreb index’s physico-chemical applicability. The forgotten topological index is given as

$$F(G)=\sum _{\mathrm{uv}\in E(G)}(d_v{)}^3=\sum _{\mathrm{uv}\in E(G)}[(d_u{)}^2+(d_v{)}^2].$$

The so-called ‘sum-connectivity index’ is a recent invention by Zhou and Trinajstic ⁵ and it’s defined as

$$\mathrm{SCI}(G)=\sum _{\mathrm{uv}\in E(G)}\frac{1}{\sqrt{d_u+d_v}}.$$

The geometric-arthemetic index of benzenoid systems and phenylenes was calculated by Xiao et al. ² Also a simple relation was established between the geometric-arithmetic of a phenylene and the corresponding hexagonal squeeze. Yang et al. ³ computed the exact expression for sum-connectivity index of polyomino chains. Imran and Akhter ¹⁵ calculated the general sum-connectivity index for polyomino, square, hexagonal and triangular cactus chains. Furthermore, in relation to the general sum-connectivity index, the extremal chains in the square, hexagonal and polyomino cactus chains are determined. Sigarreta et al. ¹⁷ studied the degree-based topological indices in a random polyomino chain. The key purpose of this manuscript was to obtain the asymptotic distribution, expected value and variance for the degree-based topological indices in a random polyomino chain by using a martingale approach. Consequently, computed the degree-based topological indices in a polyomino chain, hence some known results from the existing literature about polyomino chains are obtained as corollaries. ¹⁶ Ke et al. compared the excepted values of atom-bond connectivity and geometric-arthemetic indices in random spiro chains and established simple explicit formulae for the expected values of atom-bond connectivity and geometric-arthemetic indices in random polyphenyl chains which are graphs of a class of unbranched polycyclic aromatic hydrocarbons and presented the average values of atom-bond connectivity and geometric-arthemetic indices with respect to the set of all polyphenyl chains with n hexagons. Based on these formulae, a comparisons between the expected values of atom-bond connectivity and geometric-arthemetic indices in random polyphenyl chains was also made. For more results on topological indices see Refs.^10–14

In the study, Hayat ¹⁹ presented a computational method that depends on a computational method to calculate all the distance related indices of chemical graphs (including those related to eccentricity and valency distance). After that, a thorough statistical analysis is conducted using the top five distance descriptors, and the data suggests non-linear regression models as the most suitable option. Applications of the approach are shown in the relationship between linear polyacenes’ heat capacity and entropy. In, Hayat et al. ²⁰ investigated the usefulness of a particular class of graphical indices constructed based on eigenvalues, commonly known as valency-spectral graphical indices, for the structure-property modelling of thermodynamic characteristics of polycyclic aromatic hydrocarbons (PAHs). In Hayat et al. ²¹ the class of benzenoid hydrocarbons (BHs)are considered and it is observed that how temperature indices can predict the thermodynamic properties of benzenoid hydrocarbons. Our research is motivated from Ref. ¹⁸ in which Alraqad et al. represented the square hexagonal kinks and chains. The motivation to study the square hexagonal kink family stems from their significant impact on the properties of various materials and their potential applications, also by their ability to influence and enhance material properties across a range of engineering and technological applications. These kinks are crucial for optimizing mechanical, electronic and energy storage properties in advanced materials, such as nanomaterials and 2D materials, thereby driving innovation in nanotechnology, electronics and energy storage systems.

In this paper, we have introduced three type of kink chains. An exact expression for the atom bond connectivity index, sum connectivity index, geometric arithmetic index and forgotten index of these kink chains have been obtained. Furthermore, a graphical and numerical comparison between their computed values is given at the end. The comparison may be helpful to characterize the kink chains with maximum and minimum values among the considered topological indices.

The detailed analysis of these structural and topological features provides essential insights into how these kink chains can influence material properties, which is a critical first step in translating theoretical results into practical engineering solutions. The study lays the groundwork for future research that can directly apply these findings to specific engineering challenges, such as material design and optimization in advanced technologies.

Graphic structures of $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$

Observe that there are three possible kink chains of $T_{2,2}$ obtained depending on the way how different square and/or hexagons are attached at different places to form kink chains. Let $k$ be the number of kinks in a chain. In these chains, a square containing the vertex of degree $2$ becomes kink at each step. Holding the condition to form kink, two squares are attached with terminal hexagon to form kink, while a hexagon is attached with terminal square to form kink. We refer these possible arrangements as kink chains of type $T_{2,2}$ and indicate as $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ . These are defined as

• Kink chain $T_{2,2}^1$ :

A kink chain of type $T_{2,2}$ having no two adjacent vertices of degree $2$ in hexagon except at terminal polygons. It is represented in Figure 6.

Figure 6.

Kink chain $T_{2,2}^1$ .

These chains are descendent when at each step a square is added below the preceding square, whereas ascendant when at each step a square is attached above the preceding square.

• Kink chain $T_{2,2}^2$ :

A kink chain $T_{2,2}$ having only two adjacent vertices of degree $2$ in hexagon except at terminal polygons. It is shown in Figure 7.

• Kink chain $T_{2,2}^3$ :

Figure 7.

Kink chain $T_{2,2}^2$ .

A Kink chain $T_{2,2}$ having three adjacent vertices of degree $2$ in hexagon except at terminal polygons. It is shown in Figure 8.

Figure 8.

Kink chain $T_{2,2}^3$ .

In other words, there is no $(2,2)$ -degree edge, only one $(2,2)$ -degree edge and two adjacent $(2,2)$ -degree edges are in kink chains $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ respectively. It holds for all hexagons except at terminal polygons.

As, in each kink chain, every non-terminal square is kink so each kink chain is a zig-zag chain.

Odd and even number of kinks formed

Let $k$ represents the number of squares which are kink in a chain. It is to be noted that in each kink chain when two squares are attached, as a result even numbered kink chains are formed. On the other hand, when a hexagon is attached at terminal, odd numbered kink chains are obtained. Thus to sum up, when terminal(closing) polygon is a square then odd numbered kink chains and when terminal polygon is a hexagon, even numbered kink chains are obtained. It holds for $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ . For the sake of generality, we divide our all results into two cases as;

• Case-I: When $k=2n-1$ ; $n\in N$

• Case-II: When $k=2n$ ; $n\in N$ .

Order and size of $T_{2,2}^1,T_{2,2}^2$ and $T_{2,2}^3$

Now we define order and size of $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ . Let $\stackrel{⌣}{p}$ and $\stackrel{⌣}{q}$ be the order and size of kink chains respectively. It is to be noted that order of each kink chain remains same in both cases and is given as $\stackrel{⌣}{p}=10+4k$ . The size $\stackrel{⌣}{q}$ varies for $k=2n-1$ and $k=2n$ . For $k=2n-1$ ; $n\in N$ , the size of each kink chain is given as $\stackrel{⌣}{q}=\frac{11k+13}{2}$ . while for $k=2n$ , the size of each kink chain is given as $\stackrel{⌣}{q}=\frac{11k+12}{2}$ .

Vertex and corresponding edge partitions of $T_{2,2}^1,T_{2,2}^2$ and $T_{2,2}^3$

Let ${\stackrel{⌣}{V}}_i=\{w\in \stackrel{⌣}{V}$ $(G)|d_w=i\}$ be the subclass of vertex set of $T_{2,2}^1,T_{2,2}^2$ and $T_{2,2}^3$ . There are only vertices of degree $2$ , $3$ and $4$ in each kink chain. Table 1 represents the cardinalities of vertices for $k=2n-1$ and $k=2n$ ; $n\in N$ . It is interesting to note that cardinality of vertices remains same in both cases.

Table 1.

Vertex partitions of $T_{2,2}^1,T_{2,2}^2$ and $T_{2,2}^3$ .

Vertex partition	For $k=2n-1$	For $k=2n$
$|V_i|$	$T_{2,2}^1,T_{2,2}^2$ and $T_{2,2}^3$	$T_{2,2}^1,T_{2,2}^2$ and $T_{2,2}^3$
$|V_2|$	$2k+5$	$2k+6$
$|V_3|$	$k+1$	$k$
$|V_4|$	$k$	$k$

Let $E_{\mathrm{ij}}=\{e=\mathrm{wx};d_w=i,d_x=j\}$ be the subclass of edge sets of $T_{2,2}^1,T_{2,2}^2$ and $T_{2,2}^3$ for corresponding edge partitions then |Ě_ij| denotes the edge partitions for each kink chain. Edge partitions also depends on the number of kinks $k$ in each kink chain. Observe that there are only $(2,2)$ , $(2,3)$ , $(2,4)$ , $(3,4)$ and $(4,4)$ -type of edges in each kink chain $T_{2,2}^1,T_{2,2}^2$ and $T_{2,2}^3$ . So, we have the cardinalities of type $|E_{22}|$ , $|E_{23}|$ , $|E_{24}|$ , $|E_{34}|$ and $|E_{44}|$ respectively. Table 2 represents the cardinalities of these edges of each kink chain accordingly.

Table 2.

Edge partitions of $T_{2,2}^1,T_{2,2}^2$ and $T_{2,2}^3$ .

Edge partition	For $k=2n-1$			For $k=2n$
$|E_{\mathrm{ij}}|$	$T_{2,2}^1$	$T_{2,2}^2$	$T_{2,2}^3$	$T_{2,2}^1$	$T_{2,2}^2$	$T_{2,2}^3$
$|E_{22}|$	$4$	$\frac{k+7}{2}$	$k+3$	$6$	$\frac{k+10}{2}$	$k+4$
$|E_{23}|$	$2(k+1$ )	$\frac{3k+5}{2}$	$4$	$2k$	$\frac{3k+2}{2}$	$4$
$|E_{24}|$	$2k$	$\frac{3k+1}{2}$	$3k-1$	$2k$	$\frac{3k+2}{2}$	$3k-2$
$|E_{34}|$	$k+1$	$\frac{3k+1}{2}$	$2$	$k$	$\frac{3k-2}{2}$	$2$
$|E_{44}|$	$\frac{k-1}{2}$	$\frac{k-1}{2}$	$\frac{3(k-1)}{2}$	$\frac{k}{2}$	$\frac{k}{2}$	$\frac{3k-4}{2}$

Let $T_{2,2}^p$ denotes the kink chain for kinks of type $T_{2,2}$ and p varies from $1$ to $3$ .

Theorem 5.1. Let $n\in N$ , then the forgotten topological index of kink chain $T_{2,2}^p$ is given as;

$$F(T_{2,2}^p)=\{\begin{array}{c}107k+67\mathrm{if}p=1,2;\mathrm{for}k=2n-1\hfill \\ \multicolumn{1}{c}{116k+58\mathrm{if}p=3}\end{array}$$

$$F(T_{2,2}^p)=\{\begin{array}{c}107k+48\mathrm{if}p=1,2;\mathrm{for}k=2n\hfill \\ \multicolumn{1}{c}{116k+30\mathrm{if}p=3}\end{array}$$

Proof. Let $k=2n-1$ . Using the edge partition given in Table 2 and the definition of forgotten topological index, we get

$$\begin{array}{c}\multicolumn{1}{c}{F(T_{2,2}^1)=(4)(8)+(2(k+1))(13)+(2k)(20)+(k+1)(25)+\left(\frac{k-1}{2}\right)(32)}\\ \multicolumn{1}{c}{=107k+67}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{F(T_{2,2}^2)=\left(\frac{k+7}{2}\right)(8)+\left(\frac{3k+5}{2}\right)(13)+\left(\frac{3k+1}{2}\right)(20)+\left(\frac{3k+1}{2}\right)(25)+\left(\frac{k-1}{2}\right)(32)}\\ \multicolumn{1}{c}{=107k+67}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{F(T_{2,2}^3)=(k+3)(8)+(4)(13)+(3k-1)(20)+(2)(25)+\left(\frac{3(k-1)}{2}\right)(32)}\\ \multicolumn{1}{c}{=116k+58}\end{array}$$

Let $k=2n$ . Using the edge partition given in Table 2 and the definition of forgotten topological index, we get

$$\begin{array}{c}\multicolumn{1}{c}{F(T_{2,2}^1)=(6)(8)+(2k)(13)+(2k)(20)+(k)(25)+\left(\frac{k}{2}\right)(32)}\\ \multicolumn{1}{c}{=107k+48}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{F(T_{2,2}^2)=\left(\frac{k+10}{2}\right)(8)+\left(\frac{3k+2}{2}\right)(13)+\left(\frac{3k+2}{2}\right)(20)+\left(\frac{3k-2}{2}\right)(25)+\left(\frac{k}{2}\right)(32)}\\ \multicolumn{1}{c}{=107k+48}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{F(T_{2,2}^3)=(k+4)(8)+(4)(13)+(3k-2)(20)+(2)(25)+\left(\frac{3k-4}{2}\right)(32)}\\ \multicolumn{1}{c}{=116k+30}\end{array}$$

Theorem 5.2. Let $n\in N$ , then the geometric arthemetic topological index of kink chain $T_{2,2}^p$ is given as;

$$\mathrm{GA}(T_{2,2}^p)=\{\begin{array}{c}\multicolumn{1}{c}{5.334953196k+6.449335113\mathrm{if}p=1}\\ \multicolumn{1}{c}{5.368522386k+6.415765923\mathrm{if}p=2;\mathrm{for}k=2n-1}\\ \multicolumn{1}{c}{5.328427125k+6.455861184\mathrm{if}p=3}\end{array}$$

$$\mathrm{GA}(T_{2,2}^p)=\{\begin{array}{c}\multicolumn{1}{c}{5.334953196k+6\mathrm{if}p=1}\\ \multicolumn{1}{c}{5.368522386k+5.93286162\mathrm{if}p=2;\mathrm{for}k=2n}\\ \multicolumn{1}{c}{5.328427125k+6.013052143\mathrm{if}p=3}\end{array}$$

Proof. Let $k=2n-1$ . Using the edge partition given in Table 2 and the definition of geometric arithmetic topological index, we get

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{GA}(T_{2,2}^1)=(4)\left(2\frac{\sqrt{2*2}}{2+2}\right)+(2(k+1))\left(2\frac{\sqrt{2*3}}{2+3}\right)+(2k)\left(2\frac{\sqrt{2*4}}{2+4}\right)+(k+1)\left(2\frac{\sqrt{3*4}}{3+4}\right)}\\ \multicolumn{1}{c}{+\left(\frac{k-1}{2}\right)\left(2\frac{\sqrt{4*4}}{4+4}\right)}\\ \multicolumn{1}{c}{=4(\frac{\sqrt{6}}{5}+\frac{\sqrt{2}}{3}+\frac{\sqrt{3}}{7}+\frac{1}{8})k+4+4\frac{\sqrt{6}}{5}+2\frac{\sqrt{12}}{7}-\frac{1}{2}=5.334953196k+6.449335113}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{GA}(T_{2,2}^2)=\left(\frac{k+7}{2}\right)\left(2\frac{\sqrt{2*2}}{2+2}\right)+\left(\frac{3k+5}{2}\right)\left(2\frac{\sqrt{2*3}}{2+3}\right)+\left(\frac{3k+1}{2}\right)\left(2\frac{\sqrt{2*4}}{2+4}\right)+\left(\frac{3k+1}{2}\right)\left(2\frac{\sqrt{3*4}}{3+4}\right)}\\ \multicolumn{1}{c}{+\left(\frac{k-1}{2}\right)\left(2\frac{\sqrt{4*4}}{4+4}\right)}\\ \multicolumn{1}{c}{=(1+3\frac{\sqrt{6}}{5}+\sqrt{2}+6\frac{\sqrt{3}}{7})k+\frac{7}{2}+\sqrt{6}+\frac{\sqrt{2}}{3}+2\frac{\sqrt{3}}{7}-\frac{1}{2}=5.368522386k+6.415765923}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{GA}(T_{2,2}^3)=(k+3)\left(2\frac{\sqrt{2*2}}{2+2}\right)+(4)\left(2\frac{\sqrt{2*3}}{2+3}\right)+(3k-1)\left(2\frac{\sqrt{2*4}}{2+4}\right)+(2)\left(2\frac{\sqrt{3*4}}{3+4}\right)}\\ \multicolumn{1}{c}{+\left(\frac{3(k-1)}{2}\right)\left(2\frac{\sqrt{4*4}}{4+4}\right)}\\ \multicolumn{1}{c}{=(1+6\frac{\sqrt{2}}{3}+\frac{3}{2})k+3+8\frac{\sqrt{6}}{5}-2\frac{\sqrt{2}}{3}+8\frac{\sqrt{3}}{7}-\frac{3}{2}=5.328427125k+6.455861184}\end{array}$$

Let $k=2n$ . Using the edge partition given in Table 2 and the definition of geometric arithmetic topological index, we get

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{GA}(T_{2,2}^1)=(6)\left(2\frac{\sqrt{2*2}}{2+2}\right)+(2k)\left(2\frac{\sqrt{2*3}}{2+3}\right)+(2k)\left(2\frac{\sqrt{2*4}}{2+4}\right)+(k)\left(2\frac{\sqrt{3*4}}{3+4}\right)+\left(\frac{k}{2}\right)\left(2\frac{\sqrt{4*4}}{4+4}\right)}\\ \multicolumn{1}{c}{=4(\frac{\sqrt{6}}{5}+\frac{\sqrt{2}}{3}+\frac{\sqrt{3}}{7}+\frac{1}{8})k+6=5.334953196k+6}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{GA}(T_{2,2}^2)=\left(\frac{k+10}{2}\right)\left(2\frac{\sqrt{2*2}}{2+2}\right)+\left(\frac{3k+2}{2}\right)\left(2\frac{\sqrt{2*3}}{2+3}\right)+\left(\frac{3k+2}{2}\right)\left(2\frac{\sqrt{2*4}}{2+4}\right)+\left(\frac{3k-2}{2}\right)\left(2\frac{\sqrt{3*4}}{3+4}\right)}\\ \multicolumn{1}{c}{+\left(\frac{k}{2}\right)\left(2\frac{\sqrt{4*4}}{4+4}\right)}\\ \multicolumn{1}{c}{=(1+3\frac{\sqrt{6}}{5}+\sqrt{2}+6\frac{\sqrt{3}}{7})k+5+2\frac{\sqrt{6}}{5}+2\frac{\sqrt{2}}{3}-4\frac{\sqrt{3}}{7}=5.368522386k+5.93286162}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{GA}(T_{2,2}^3)=(k+4)\left(2\frac{\sqrt{2*2}}{2+2}\right)+(4)\left(2\frac{\sqrt{2*3}}{2+3}\right)+(3k-2)\left(2\frac{\sqrt{2*4}}{2+4}\right)+(2)\left(2\frac{\sqrt{3*4}}{3+4}\right)+\left(\frac{3k-4}{2}\right)\left(2\frac{\sqrt{4*4}}{4+4}\right)}\\ \multicolumn{1}{c}{=(1+6\frac{\sqrt{2}}{3}+\frac{3}{2})k+4+8\frac{\sqrt{6}}{5}-4\frac{\sqrt{2}}{3}+8\frac{\sqrt{3}}{7}-2=5.328427125k+6.013052143}\end{array}$$

Theorem 5.3. Let $n\in N$ , then the sum connectivity topological index of kink chain $T_{2,2}^p$ is given as

$$\mathrm{SC}(T_{2,2}^p)=\{\begin{array}{c}\multicolumn{1}{c}{2.26566494k+3.095614969\mathrm{if}p=1}\\ \multicolumn{1}{c}{2.276916234k+3.084363675\mathrm{if}p=2;\mathrm{for}k=2n-1}\\ \multicolumn{1}{c}{2.255074957k+3.106204952\mathrm{if}p=3}\end{array}$$

$$\mathrm{SC}(T_{2,2}^p)=\{\begin{array}{c}\multicolumn{1}{c}{2.26566494k+3\mathrm{if}p=1}\\ \multicolumn{1}{c}{2.276916234k+2.977497413\mathrm{if}p=2;\mathrm{for}k=2n}\\ \multicolumn{1}{c}{2.255074957k+3.021179966\mathrm{if}p=3}\end{array}$$

Proof. Let $k=2n-1$ . Using the edge partition given in Table 2 and the definition of sum connectivity topological index, we get

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{SC}(T_{2,2}^1)=(4)\left(\frac{1}{\sqrt{2+2}}\right)+(2(k+1))\left(\frac{1}{\sqrt{2+3}}\right)+(2k)\left(\frac{1}{\sqrt{2+4}}\right)+(k+1)\left(\frac{1}{\sqrt{3+4}}\right)}\\ \multicolumn{1}{c}{+\left(\frac{k-1}{2}\right)\left(\frac{1}{\sqrt{4+4}}\right)}\\ \multicolumn{1}{c}{=(\frac{2}{\sqrt{5}}+\frac{2}{\sqrt{6}}+\frac{1}{\sqrt{7}}+\frac{1}{4\sqrt{2}})k+2+\frac{2}{\sqrt{5}}+\frac{1}{\sqrt{7}}-\frac{1}{4\sqrt{2}}=2.26566494k+3.095614969}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{SC}(T_{2,2}^2)=\left(\frac{k+7}{2}\right)\left(\frac{1}{\sqrt{2+2}}\right)+\left(\frac{3k+5}{2}\right)\left(\frac{1}{\sqrt{2+3}}\right)+\left(\frac{3k+1}{2}\right)\left(\frac{1}{\sqrt{2+4}}\right)+\left(\frac{3k+1}{2}\right)\left(\frac{1}{\sqrt{3+4}}\right)}\\ \multicolumn{1}{c}{+\left(\frac{k-1}{2}\right)\left(\frac{1}{\sqrt{4+4}}\right)}\\ \multicolumn{1}{c}{=\frac{1}{4}(1+\frac{6}{\sqrt{5}}+\frac{6}{\sqrt{6}}+\frac{6}{\sqrt{7}}+\frac{1}{\sqrt{2}})k+\frac{7}{4}+\frac{5}{2\sqrt{5}}+\frac{1}{2\sqrt{6}}+\frac{1}{2\sqrt{7}}-\frac{1}{4\sqrt{2}}=2.276916234k+3.084363675}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{SC}(T_{2,2}^3)=(k+3)\left(\frac{1}{\sqrt{2+2}}\right)+(4)\left(\frac{1}{\sqrt{2+3}}\right)+(3k-1)\left(\frac{1}{\sqrt{2+4}}\right)+(2)\left(\frac{1}{\sqrt{3+4}}\right)}\\ \multicolumn{1}{c}{+\left(\frac{3(k-1)}{2}\right)\left(\frac{1}{\sqrt{4+4}}\right)}\\ \multicolumn{1}{c}{=(\frac{1}{2}+\frac{3}{\sqrt{6}}+\frac{3}{4\sqrt{2}})k+\frac{3}{2}+\frac{4}{\sqrt{5}}-\frac{1}{\sqrt{6}}+\frac{2}{\sqrt{7}}-\frac{3}{4\sqrt{3}}=2.255074957k+3.106204952}\end{array}$$

Let $k=2n$ . Using the edge partition given in Table 2 and the definition of sum connectivity topological index, we get

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{SC}(T_{2,2}^1)=(6)\left(\frac{1}{\sqrt{2+2}}\right)+(2k)\left(\frac{1}{\sqrt{2+3}}\right)+(2k)\left(\frac{1}{\sqrt{2+4}}\right)+(k)\left(\frac{1}{\sqrt{3+4}}\right)+\left(\frac{k}{2}\right)\left(\frac{1}{\sqrt{4+4}}\right)}\\ \multicolumn{1}{c}{=(3+\frac{2}{\sqrt{5}}+\frac{2}{\sqrt{6}}+\frac{1}{\sqrt{7}}+\frac{1}{4\sqrt{2}})k+3=2.6566494k+3}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{SC}(T_{2,2}^2)=\left(\frac{k+10}{2}\right)\left(\frac{1}{\sqrt{2+2}}\right)+\left(\frac{3k+2}{2}\right)\left(\frac{1}{\sqrt{2+3}}\right)+\left(\frac{3k+2}{2}\right)\left(\frac{1}{\sqrt{2+4}}\right)+\left(\frac{3k-2}{2}\right)\left(\frac{1}{\sqrt{3+4}}\right)}\\ \multicolumn{1}{c}{+\left(\frac{k}{2}\right)\left(\frac{1}{\sqrt{4+4}}\right)}\\ \multicolumn{1}{c}{=\frac{1}{4}(1+\frac{6}{\sqrt{5}}+\frac{6}{\sqrt{6}}+\frac{6}{\sqrt{7}}+\frac{1}{\sqrt{2}})k+\frac{10}{4}+\frac{1}{\sqrt{5}}+\frac{1}{\sqrt{6}}-\frac{1}{\sqrt{7}}=2.276916234k+2.977497413}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{SC}(T_{2,2}^3)=(k+4)\left(\frac{1}{\sqrt{2+2}}\right)+(4)\left(\frac{1}{\sqrt{2+3}}\right)+(3k-2)\left(\frac{1}{\sqrt{2+4}}\right)+(2)\left(\frac{1}{\sqrt{3+4}}\right)+\left(\frac{3k-4}{2}\right)\left(\frac{1}{\sqrt{4+4}}\right)}\\ \multicolumn{1}{c}{=(\frac{1}{2}+\frac{3}{\sqrt{6}}+\frac{3}{4\sqrt{2}})k+2+\frac{4}{\sqrt{5}}-\frac{2}{\sqrt{6}}+\frac{2}{\sqrt{7}}-\frac{1}{\sqrt{2}}=2.255074957k+3.021179966}\end{array}$$

Theorem 5.4. Let $n\in N$ , then the atom bond connectivity topological index of kink chain $T_{2,2}^p$ is given as;

$$\mathrm{ABC}(T_{2,2}^p)=\{\begin{array}{c}\multicolumn{1}{c}{1.340235822k+2.101356275\mathrm{if}p=1}\\ \multicolumn{1}{c}{1.340844449k+2.100747648\mathrm{if}p=2;\mathrm{for}k=2n-1}\\ \multicolumn{1}{c}{1.333193054k+2.108399043\mathrm{if}p=3}\end{array}$$

$$\mathrm{ABC}(T_{2,2}^p)=\{\begin{array}{c}\multicolumn{1}{c}{1.340235822k+2.121320344\mathrm{if}p=1}\\ \multicolumn{1}{c}{1.340844449k+2.120103089\mathrm{if}p=2;\mathrm{for}k=2n}\\ \multicolumn{1}{c}{1.333193054k+2.135405879\mathrm{if}p=3}\end{array}$$

Proof. Let $k=2n-1$ . Using the edge partition given in Table 2 and the definition of atom bond connectivity topological index, we get

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{ABC}(T_{2,2}^1)=(4)\left(\frac{\sqrt{2+2-2}}{2*2}\right)+(2(k+1))\left(\frac{\sqrt{2+3-2}}{2*3}\right)+(2k)\left(\frac{\sqrt{2+4-2}}{2*4}\right)+(k+1)\left(\frac{\sqrt{3+4-2}}{3*4}\right)}\\ \multicolumn{1}{c}{+\left(\frac{k-1}{2}\right)\left(\frac{\sqrt{4+4-2}}{4*4}\right)}\\ \multicolumn{1}{c}{=(\frac{\sqrt{3}}{3}+\frac{1}{2}+\frac{\sqrt{5}}{12}+\frac{\sqrt{6}}{32})k+\sqrt{2}+\frac{\sqrt{3}}{3}+\frac{\sqrt{5}}{12}-\frac{\sqrt{6}}{32}=1.340235822k+2.101356275}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{ABC}(T_{2,2}^2)=\left(\frac{k+7}{2}\right)\left(\frac{\sqrt{2+2-2}}{2*2}\right)+\left(\frac{3k+5}{2}\right)\left(\frac{\sqrt{2+3-2}}{2*3}\right)+\left(\frac{3k+1}{2}\right)\left(\frac{\sqrt{2+4-2}}{2*4}\right)}\\ \multicolumn{1}{c}{+\left(\frac{3k+1}{2}\right)\left(\frac{\sqrt{3+4-2}}{3*4}\right)+\left(\frac{k-1}{2}\right)\left(\frac{\sqrt{4+4-2}}{4*4}\right)}\\ \multicolumn{1}{c}{=\frac{1}{4}(\frac{\sqrt{2}}{2}+\sqrt{3}+\frac{3}{2}+\frac{3\sqrt{5}}{6}+\frac{\sqrt{6}}{8})k+\frac{7\sqrt{2}}{8}+\frac{5\sqrt{3}}{12}+\frac{1}{8}+\frac{\sqrt{5}}{24}-\frac{\sqrt{6}}{32}=1.340844449k+2.100747648}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{ABC}(T_{2,2}^3)=(k+3)\left(\frac{\sqrt{2+2-2}}{2*2}\right)+(4)\left(\frac{\sqrt{2+3-2}}{2*3}\right)+(3k-1)\left(\frac{\sqrt{2+4-2}}{2*4}\right)+(2)\left(\frac{\sqrt{3+4-2}}{3*4}\right)}\\ \multicolumn{1}{c}{+\left(\frac{3(k-1)}{2}\right)\left(\frac{\sqrt{4+4-2}}{4*4}\right)}\\ \multicolumn{1}{c}{=\frac{1}{4}(\sqrt{2}+3+\frac{3\sqrt{3}}{8})k+\frac{3\sqrt{2}}{4}+\frac{2\sqrt{3}}{3}-\frac{1}{4}+\frac{\sqrt{5}}{6}-\frac{3\sqrt{6}}{32}=1.333193054k+2.108399043}\end{array}$$

Let $k=2n$ . Using the edge partition given in Table 2 and the definition of atom bond connectivity topological index, we get

$$\begin{array}{l}ABC(T_{2,2}^1)=(6)\left(\frac{\sqrt{2+2-2}}{2\ast 2}\right)+(2k)\left(\frac{\sqrt{2+3-2}}{2\ast 3}\right)+(2k)\left(\frac{\sqrt{2+4-2}}{2\ast 4}\right)+(k)\left(\frac{\sqrt{3+4-2}}{3\ast 4}\right)+\left(\frac{k}{2}\right)\left(\frac{\sqrt{4+4-2}}{4\ast 4}\right)\\ =\left(\frac{\sqrt{3}}{3}+\frac{1}{2}+\frac{\sqrt{5}}{12}+\frac{\sqrt{6}}{32}\right)k+\frac{3\sqrt{2}}{2}=1.340235822k+2.121320344\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{ABC}(T_{2,2}^2)=\left(\frac{k+10}{2}\right)\left(\frac{\sqrt{2+2-2}}{2*2}\right)+\left(\frac{3k+2}{2}\right)\left(\frac{\sqrt{2+3-2}}{2*3}\right)+\left(\frac{3k+2}{2}\right)\left(\frac{\sqrt{2+4-2}}{2*4}\right)}\\ \multicolumn{1}{c}{+\left(\frac{3k-2}{2}\right)\left(\frac{\sqrt{3+4-2}}{3*4}\right)+\left(\frac{k}{2}\right)\left(\frac{\sqrt{4+4-2}}{4*4}\right)}\\ \multicolumn{1}{c}{=\frac{1}{4}(\frac{\sqrt{2}}{2}+\sqrt{3}+\frac{3}{2}+\frac{3\sqrt{5}}{6}+\frac{\sqrt{6}}{8})k+\frac{5\sqrt{2}}{4}+\frac{\sqrt{3}}{6}+\frac{1}{4}-\frac{\sqrt{5}}{12}=1.340844449k+2.120103089}\end{array}$$

$$\begin{array}{l}ABC(T_{2,2}^3)=(k+4)\left(\frac{\sqrt{2+2-2}}{2\ast 2}\right)+(4)\left(\frac{\sqrt{2+3-2}}{2\ast 3}\right)+(3k-2)\left(\frac{\sqrt{2+4-2}}{2\ast 4}\right)+(2)\left(\frac{\sqrt{3+4-2}}{3\ast 4}\right)\\ +\left(\frac{3k-4}{2}\right)\left(\frac{\sqrt{4+4-2}}{4\ast 4}\right)\\ =\frac{1}{4}\left(\sqrt{2}+3+\frac{3\sqrt{3}}{8}\right)k+\sqrt{2}+\frac{2\sqrt{3}}{3}-\frac{1}{2}+\frac{\sqrt{5}}{6}-\frac{\sqrt{6}}{8}=1.333193054k+2.135405879\end{array}$$

A comparison of topological descriptors of $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$

In this section, we give a comparison of forgotten, geometric-arthemetic, sum-connectivity and atom-bond connectivity descriptors for kink chains $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ .

Tables 3 and 5 show the numerical values of forgotten, geometric-arithmetic, sum-connectivity and atom bond connectivity descriptors of $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ for $k=2n-1$ . Tables 4 and 6 show the numerical values of forgotten, geometric-arithmetic, sum-connectivity and atom bond connectivity descriptors of $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ for $k=2n$ . We can see that the forgotten topological descriptor of $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ attains the maximum numerical value for both the cases, $k=2n-1$ and $k=2n$ , in comparison to the other topological descriptors. While the ABC descriptor of $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ has minimum numerical value in both the cases.

Table 3.

Numerical values of forgotten and geometric-arthemetic topoloical descriptors of $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ for $k=2n-1$ .

Number of kinks	Forgotten index			Geometric-arthemetic index
$k$	$T_{2,2}^1$	$T_{2,2}^2$	$T_{2,2}^3$	$T_{2,2}^1$	$T_{2,2}^2$	$T_{2,2}^3$
1	174	174	174	11.78428831	11.78428831	11.78428831
3	388	348	406	22.4541947	22.52133308	22.44114256
5	602	602	638	33.12410109	33.25837785	33.09799681
7	816	816	870	43.79400749	43.99542263	43.75485106
9	1030	1030	1102	54.46391388	54.7324674	54.41170531
11	1244	1244	1334	65.13382027	65.46951217	65.06855956
13	1458	1458	1566	75.80372666	76.20655694	75.72541381
15	1672	1672	1798	86.47363305	86.94360171	86.38226806
17	1886	1886	2030	97.14353945	97.68064648	97.03912231
19	2100	2100	2262	107.8134458	108.4176913	107.6959766

Table 4.

Numerical values of forgotten and geometric-arthemetic topoloical descriptors of $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ for $k=2n$ .

Number of kinks	Forgotten index			Geometric-arthemetic index
$k$	$T_{2,2}^1$	$T_{2,2}^2$	$T_{2,2}^3$	$T_{2,2}^1$	$T_{2,2}^2$	$T_{2,2}^3$
2	262	262	262	16.66990639	16.66990639	16.66990639
4	476	476	494	27.33981278	27.40695116	27.32676064
6	690	690	726	38.00971918	38.14399594	37.98361489
8	904	904	958	48.67962557	48.88104071	48.64046914
10	1118	1118	1190	59.34953196	59.61808548	59.29732339
12	1332	1332	1424	70.01943835	70.35513025	69.95417764
14	1546	1546	1654	80.68934474	81.09217502	80.61103189
16	1760	1760	1886	91.35925114	91.8292198	91.26788614
18	1974	1974	2118	102.0291575	102.5662646	101.9247404
20	2188	2188	2350	112.6990639	113.3033093	112.5815946

Table 5.

Numerical values of sum-connectivity and atom bond connectivity topoloical descriptors of $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ for $k=2n-1$ .

Number of kinks	Sum-connectivity index			Atom bond connectivity index
$k$	$T_{2\mathbf{,}2}^1$	$T_{2\mathbf{,}2}^2$	$T_{2\mathbf{,}2}^3$	$T_{2\mathbf{,}2}^1$	$T_{2\mathbf{,}2}^2$	$T_{2\mathbf{,}2}^3$
1	5.361279909	5.361279909	5.361279909	3.441592079	3.441592079	3.441592097
3	9.892609789	9.915112377	9.871429823	6.122063723	6.123280995	6.107978205
5	14.42393967	14.46894484	14.38157974	8.802535367	8.804969893	8.774364313
7	18.95526955	19.02277731	18.89172965	11.48300701	11.48665879	11.44075042
9	23.48659943	23.57660978	23.40187957	14.16347866	14.16834769	14.10713653
11	28.01792931	28.13044225	27.91202948	16.8439503	16.85003659	16.77352264
13	32.54925919	32.68427472	32.42217939	19.52442194	19.53172548	19.43990875
15	37.08058907	37.23810718	36.93232931	22.20489359	22.21341438	22.10629485
17	41.61191895	41.79193965	41.44247922	24.88536523	24.89510328	24.77268096
19	46.14324883	46.34577212	45.95262914	27.56583687	27.57679218	27.43906707

Table 6.

Numerical values of sum-connectivity and atom bond connectivity topoloical descriptors of $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ for $k=2n$ .

Number of kinks	Sum-connectivity index			Atom bond connectivity index
$k$	$T_{2,2}^1$	$T_{2,2}^2$	$T_{2,2}^3$	$T_{2,2}^1$	$T_{2,2}^2$	$T_{2,2}^3$
2	8.3132988	7.531329881	7.53132988	4.801791988	4.801791987	4.801791987
4	13.6265976	12.08516235	12.04147979	7.482263632	7.483480885	7.468178095
6	18.9398964	16.63899482	16.55162971	10.16273528	10.16516978	10.1345642
8	24.2531952	21.19282728	21.06177962	12.84320692	12.84685868	12.80095031
10	29.566494	25.74665975	25.57192954	15.52367856	15.52854758	15.46733642
12	34.8797928	30.30049222	30.08207945	18.20415021	18.21023648	18.13372253
14	40.1930916	34.85432469	34.59222936	20.88462185	20.89192537	20.80010864
16	45.5063904	39.40815716	39.10237928	23.5650935	23.57361427	23.46649474
18	50.8196892	43.96198963	43.61252919	26.24556514	26.25530317	26.13288085
20	56.132988	48.51582209	48.12267911	28.92603678	28.93699207	28.79926696

Conclusion

In this article, we represent graphical structures of kink chains of type $T_{2,2}$ , named as $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ . We computed their order, size and corresponding vertex and edge partitions in two cases, for $k=2n-1$ and $k=2n$ , (odd and even numbered kink chains respectively). We determined the explicit expression of forgotten index, sum connectivity index, atom bond connectivity index and geometric arithmetic index of these kink chains. Following that, we calculated the numerical values of attained topological descriptors for both the cases and observe that the numerical value of forgotten topological descriptor of $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ in both cases attains maximum value than other topological descriptors. And numerical value of atom-bond connectivity descriptor of $T_{2,2}^1$ , $T_{2,2}^2$ and $T_{2,2}^3$ in both cases attains minimum value than others.