Front. Phys.Frontiers in PhysicsFront. Phys.2296-424XFrontiers Media S.A.153844310.3389/fphy.2025.1538443PhysicsOriginal ResearchAnalysis of expected value of connectivity indices of random 2T2 kink chainsLing et al.10.3389/fphy.2025.1538443LingXuemin1RazzaqueAsima23*KhalilMaham4KanwalSalma4*NoorSaima23NazirRobina231General Education Department, Anhui Xinhua University, Hefei, China2Department of Basic Science, King Faisal University, Al Hofuf, Saudi Arabia3Department of Mathematics, College of Science, King Faisal University, Al Hofuf, Saudi Arabia4Department of Mathematics, Lahore College for Women University, Lahore, Pakistan
Edited by:Camillo La Mesa, Sapienza University of Rome, Italy
Reviewed by:Francisco Welington Lima, Federal University of Piauí, Brazil
Sarfraz Ahmad, COMSATS University Islamabad, Pakistan
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
In this work, we calculated the product connectivity index of three structures for the T2-type twists and formulated expressions of expected values of the forgotten index, atom bond connectivity index, sum connectivity index, product connectivity index and geometric arithmetic index of these random structures in the (ή−2) stage. In addition, we calculated the expected values numerically, graphically and analytically and found the topological indices that have the maximum and minimum value.
A square-hexagonal system is connected geometric shape and a unique geometric arrangement formed by connecting equal sized squares and/or hexagons together. In this system, the lattice points form a repeating pattern that resembles a combination of squares and hexagons. In contrast, a square-hexagonal chain is a one-dimensional linear sequence where square and hexagonal units alternate or connect in a line, creating a chain-like arrangement. This configuration is useful in fields like combinatorics and polymer chemistry, where alternating shapes along a single axis influence the overall behavior and properties of the chain. Different square-hexagoal chains are formed depending on the way how polygons are concatenated.
A [1, 2] polyomino chain is a sequence of connected squares, where each square shares at least one side with the next in the sequence. A hexagonal chain is a one-dimensional, linear sequence of connected hexagonal units, where each hexagon is linked to the next in a chain-like formation. Unlike a full hexagonal lattice, which extends in two dimensions, a hexagonal chain progresses in a single direction, creating a structure that resembles a series of hexagonal “tiles” arranged side by side. In graph theory, a graph ψ is defined as an ordered pair ψ=(Vψ,Eψ), where Vψ (also called vertices or nodes) is a set elements that represent pints in a graph and Eψ (called edges or arcs) is a set of pairs of vertices. Each pair (u,v) represents a connection (edge) between two vertices u and v.
Topological indices are the quantitative values that represent the structural characteristic of a graph which are used to relate the structure with its physical and chemical properties. These indices are usually known as descriptors of graphical structures for precisely this reason. These are the conclusions reached by applying logic and mathematics to a symbolic representation of a graph in an usable number. Some topological descriptors which we have used in our paper are described below.
The [3] geometric-arithmetic index is defined as;GAψ=∑uv∈Eψ2du×dvdu+dv
The sum-connectivity and product-connectivity indices are defined respectively as;SCψ=∑uv∈Eψ1du+dvPCψ=∑uv∈Eψ1du×dv
Different relations between these two indices are established in [4].
In article [5] the forgotten index is given as;Fψ=∑uv∈Eψdu3or it can also be defined as;Fψ=∑uv∈Eψdu2+dv2
The [6] atom bond connectivity index is defined as;ABCψ=∑uv∈Eψdu+dv−2du×dv
In mathematical chemistry the application of chemical graph theory to the study of complex structures, using different chemical invariants, has revolutionary effects. The total count of nodes or vertices in a graph is known as its order, represented by |Vψ|. Whereas the total count of edges between the nodes in a graph is called its size which is represented by |Eψ|. In a molecular graph atoms are represented by u∈Vψ, while uivj∈E(ψ) represents the bonds connecting the corresponding atoms. The degree of a vertex, indicated by deg(u) or du, refers to the count of edges connected to that vertex. Further notations of graph are discussed in [7].
Phenylene chains are part of a broader class of conjugated organic polymers, formed by joining squares and hexagons alternatively. Raza et al. in [8] compared numerically as well as graphically, the random phenylene chains among the expected values of atom bond connectivity and geometric-arithmetic indices. Wei et al. [9] formulated the expected values of various topological indices for random phenylene chains using auxiliary graphical structures, with a particular focus on the Merrifield-Simmons index.
Several researchers [10–12] have provided exact expressions and comparative analyses of topological indices for chemical graph structures, including random phenylene and polyphenyl chains. In [2], Sigarreta et al. computed the Sombor, Forgotten, Zagreb, Atom-Bond Connectivity, Randic, and Geometric-Arithmetic indices for polyomino chains, deriving precise expressions for their expected values and variances in random polyomino structures.
The first Zagreb connection index, a widely studied topological descriptor, has been extensively used to investigate the structural properties and chemical stability of molecular networks. Recent studies [14] have explored its expected value in random cyclooctatetraene chains, random polyphenyl chains, and random chain networks, providing valuable insights into the probabilistic behavior of these indices in complex molecular structures. This research contributes significantly to the understanding of topological indices in stochastic chemical graph theory, particularly in the context of polymers and organic chain networks. Distance-based graphical indices have proven highly effective in predicting the thermodynamic properties of benzenoid hydrocarbons [15], highlighting their practical applications in computational chemistry and materials science.
Similarly, eigenvalue-based graphical indices have demonstrated significant predictive capabilities in modeling and determining the thermodynamic properties of polycyclic aromatic hydrocarbons [16]. This method has been particularly useful in studying polyacenes, providing valuable insights into their structural and thermal behavior.
Furthermore, temperature-based topological indices have been utilized in structure-property modeling to predict the thermodynamic properties of benzenoid hydrocarbons [17], contributing to a deeper understanding of their thermal characteristics. Depending on the attachment of polygons and graphical representation, there are two types of square-hexagonal kinks, kinks of type T1 and T2. In type T1 hexagon occurs as a kink holding the criteria to have two adjacent vertices of degree two. While in type T2, a square is said to be a kink if contains a vertex of degree 2. Kinks of type T2 are divided into three types [1] based on the possibilities of connecting polygons (square and hexagon) at different places of a square. These three types, T12,T22 and T32, are shown in Figure 1
Kinks of type T12,T22 and T32.
By considering ή being the kink, we have further discovered three types of kink chains of type T22, named as [18] T212, T222 and T232. Also computed Forgotten, atom-bond connectivity, Sum-connectivity and Product-connectivity indices of these chains and found out the maximizing and minimizing index using graphical representation. For the sake of generality, we expressed our results into odd and even numbered kink chains. In graph theory, the concept of expected value is not inherently a property of a graph, but it is often used in the context of random graphs or randomized processes on graphs.
In [19], we calculated 1st and 2nd Gourava, 1st and 2nd Revan, Redefind 1st and 2nd Zagreb and Hyper-Zagreb indices of these three structures of kink chains of type T22 and found out maximizing and minimizing index. In addition to that, we also got the expected valued values for these descriptors at (ή−1)th stage and made comparison among them.
We are motivated to consider square and hexagonal kink chains in our work due to their superior mechanical and structural properties, which are highly relevant in material science and engineering applications due to their unique geometric and mechanical properties. They enhance stress distribution and energy absorption, making them ideal for impact-resistant materials in aerospace, automotive, and structural engineering. A comprehensive examination of these structural and topological characteristics offers valuable understanding of how kink chains impact material properties, forming a crucial foundation for bridging theoretical outcomes with practical engineering applications. This research establishes a basis for future studies to implement these insights in addressing specific engineering challenges, including the design and optimization of materials for advanced technological applications.
2 Methodology
In this study, we employed a comprehensive methodology to analyze the connectivity indices of random T2 kink chains. We began by graphically representing the kink structures as graphs, where vertices denote atoms and edges represent bonds. We calculated several topological indices including the Product Connectivity Index (PC), Sum Connectivity Index (SC), Atom Bond Connectivity Index (ABC), and Forgotten Index (F) using established mathematical definitions. The expected values of these indices were derived at various stages of the kink evolution, employing relationships from previous literature in mathematical chemistry. Numerical analysis were performed to identify the maximizing and minimizing values of these indices, complemented by graphical visualizations to portray their behavior throughout the kinking process. Finally, a comparative analysis with existing studies allowed us to validate our results and deepen our understanding of the structural implications of these connectivity indices in molecular networks.
3 Main results
In this article, we will expand our work and calculate PC index of three possible arrangements of kinks of type T22 and the expected value of PC, SC, F, ABC and GA indices of three random structures for the kink of type T22. We will numerically, graphically as well as analytically find out the maximizing and minimizing expectation at (ή−2)th stage.
Let ή represents the kink. Tables 1, 2 represents the vertex and edge partitions of each kink chain accordingly [18].
Edge partitions of T212, T222 and T232; n∈N.
|Eij|
For ή=2n−1
For ή=2n
T212
T222
T232
T212
T222
T232
|E22|
4
ή+72
ή+3
6
ή+102
ή+4
|E23|
2(ή+1)
3ή+52
4
2ή
3ή+22
4
|E24|
2ή
3ή+12
3ή−1
2ή
3ή+22
3ή−2
|E34|
ή+1
3ή+12
2
ή
3ή−22
2
|E44|
ή−12
ή−12
3(ή−1)2
ή2
ή2
3ή−42
Vertex partitions of T12, T222 and T232.
|Vi|
For ή=2n−1
For ή=2n
T212, T222 and T232
T212, T222 and T232
|V2|
2ή+5
2ή+6
|V3|
ή+1
ή
|V4|
ή
ή
Letn∈N, then the product connectivity topological index of kink chainT2,2pis given as;PCT2,2p=1.937278497ή+2.980171716ifp=11.950715223ή+2.966734989ifp=2;forή=2n−11.935660172ή+2.98179004ifp=3PCT2,2p=1.937278497ή+3ifp=11.950715223ή+2.973126546ifp=2;forή=2n1.935660172ή+3.00323665ifp=3
Proof. Let ή=2n−1. Using the edge partition given in Table 1 and the definition of product connectivity topological index, we getPCT2,21=412+2ή+116+2ή122+ή+1123+ή−1214=1.937278497ή+2.980171716PCT2,22=ή+7212+3ή+5216+3ή+12122+3ή+12123+ή−1214=1.950715223ή+2.966734989PCT2,23=ή+312+416+3ή−1122+2123+3ή−1214=1.935660172ή+2.98179004
Let ή=2n. Using the edge partition given in Table 1 and the definition of product connectivity topological index, we getPCT2,21=612+2ή16+2ή122+ή123+ή214=1.937278497ή+3PCT2,22=ή+10212+3ή+2216+3ή+22122+3ή−22123+ή214=1.950715223ή+2.973126546PCT2,23=ή+412+416+3ή−2122+2123+3ή−4214=1.935660172ή+3.00323665
Observe that there is no edge between two adjacent vertices of degree 2, only one edge between two adjacent vertices of degree 2, and two edges between two adjacent vertices of degree two in three random (the second and third arrangements are same) structures of kink of type T22 respectively, except at terminal polygons, so we have only three possible arrangements of type T22, holding the conditions to make kink at each step. The possible arrangement for ή=1 is same as shown in Figure 1b and for ή=2 is shown in Figure 2. For ή≥2, the possible arrangements for attaching terminal polygons are shown in Figure 3 in three different ways, resulting three types [18]T212, T222 and T232. Let γ́́ be the probability of attaching terminal polygons in the first or second kind of arrangement, then 1−2γ́́ be the probability of attaching the terminal polygon in the third type of arrangement.
Kinks of type T12,T22and T32.
Possible arrangements of kink chains of type ⊤22.
Consider kink chain T22 with ή number of kinks and probability γ́́ is represented by (T2γ́́2)ή́. Now we compute expected values of product-connectivity, Forgotten, atom-bond connectivity, product-connectivity and sum-connectivity indices of possible square-hexagonal kink chains (T2γ́2)ή. Let |Eij|=βij denotes the number of edges for (T2γ́2)ή with end vertices of degree i and j accordingly. There are only β22, β23, β24, β34 and β44-type of edges in (T2γ́2)ή. From Equations 1–5, topological descriptors can be expressed asFT2γ́2ή=8β22T2γ2ή+13β23T2γ2ή+20β24T2γ2ή+25β34T2γ2ή+32β44T2γ2ήGAT2γ2ή=β22T2γ2ή+265β23T2γ2ή+223β24T2γ2ή+437β34T2γ2ή+β44T2γ2ήSCT2γ2ή=12β22T2γ2ή+15β23T2γ2ή+16β24T2γ2ή+17β34T2γ2ή+122β44T2γ2ήABCT2γ2ή=24β22T2γ2ή+36β23T2γ2ή+14β24T2γ2ή+512β34T2γ2ή+616β44T2γ2ήPCT2γ2ή=12β22T2γ2ή+16β23T2γ2ή+122β24T2γ2ή+123β34T2γ2ή+14β44T2γ2ή
As (T2γ́2)ή is a possible kink chain, it proceeds that F(T2γ́2)ή, GA(T2γ́2)ή, SC(T2γ́2)ή, ABC(T2γ́2)ή and PC(T2γ́2)ή are possible variables. Let us denote by EήF=E[F(T2γ́2)ή], EήGA=E[GA(T2γ́2)ή], EήABC=E[ABC(T2γ́2)ή], EήSC=E[SC(T2γ́2)ή] and EήPC=E[PC(T2γ́2)ή] the expected values of these indices respectively.
To compute the expected values for above mentioned indices we will compute the change in edge partitions of (T2γ́2)ή for three possible constructions of kink chains as shown in Figure 3. It is to be noted that if ή is odd in possible arrangements (T212, T222 and T232) of kink chain then at (ή−1)th step, even numbered kink chains are formed and at (ή−2)th step, odd numbered kink chains are formed. Similarly, if ή is even in possible arrangements of kink chain then at (ή−1)th step, odd numbered kink chains are obtained and at (ή−2)th step, even numbered kink chains are obtained again. We will formulate our expressions at (ή−2)th stage.
The three possible constructions at (ή−2)th stage are as follows:1.(T22)ή−2→(T212)ή2.(T22)ή−2→(T222)ή3.(T22)ή−2→(T232)ή
It is interested to note that change in edge partitions of (T2γ́2)k remains same for ή=2n−1 and ή=2n when we take (ή−2)th stage. Change in edge partitions of (T2γ́2)ή at (ή−2)th stage for ή=2n−1 and ή=2n is shown in Table 3
Change in edge partitions of (T2γ́2)k at (ή−2)th step for ή=2n−1 and ή=2n; n∈n.
Type p
(T22)ή−2→(T2p2)ή
(βij)ή=(βij)ή−2± no. of edges
(β22)ή
(β23)ή
(β24)ή
(β34)ή
(β44)ή
1
(T22)ή−2→(T212)ή
(β22)ή−2
(β23)ή−2+4
(β24)ή−2+4
(β34)ή−2+2
(β44)ή−2+1
2
(T22)ή−2→(T222)ή
(β22)ή−2+1
(β23)ή−2+3
(β24)ή−2+3
(β34)ή−2+3
(β44)ή−2+1
3
(T22)ή−2→(T232)ή
(β22)ή−2+2
(β23)ή−2
(β24)ή−2+6
(β34)ή−2
(β44)ή−2+3
For square-hexagonal kink chainT22withήnumber of kinks, we have
(a) Forή=2n−1;n∈NEήF=ή−18γ́+116+18γ́+58
(b) Forή=2n;n∈NEήF=ή−18γ́+116+36γ́+30
Proof. Let ή≥2, then there are three possibilities. Using Table 3 and Equation 3, we get
1. If (T22)ή−2→(T212)ή, thenFT212ή=8β22T22ή−2+13β23T22ή−2+4+20β24T22ή−2+4+25β34T22ή−2+2+32β44T22ή−2+1=FT22ή−2+2142. If (T22)ή−2→(T222)ή, thenFT222ή=8β22T22ή−2+1+13β23T22ή−2+3+20β24T22ή−2+3+25β34T22ή−2+3+32β44T22ή−2+1=FT22ή−2+2143. If (T22)ή−2→(T232)ή, thenFT232ή=8β22T22ή−2+2+13β23T22ή−2+20β24T22ή−2+6+25β34T22ή−2+32β44T22ή−2+3=FT22ή−2+232
Thus, we haveEήF=γ́FT212ή+γ́FT222ή+1−2γ́FT232ή
Using Equations 11–13, we get the following relationEήF=γ́FT22ή−2+214+γ́FT22ή−2+214+1−2γ́FT22ή−2+232=FT22ή−2−36γ́+232
Applying operator E on both sides and ∴E(EήF)=EήFEήF=Eή−2F−36γ́+232• Let ή=2n−1
For n=1⇒ή=1, E1=174, which is indeed true. Using recursive relation upto ή−1 trermsEήF=Eή−ή−1F+ή−12−36γ́+232=174+ή−1−18γ́+116=ή−18γ́+116+18γ́+58which completes the result.
• Let ή=2n
For n=1⇒ή=2, E2=262, which is indeed true. Using recursive relation upto ή−2 termsEήF=Eή−ή−2F+ή−22−36γ́+232=262+ή−2−18γ́+116=ή−18γ́+116+36γ́+30which completes the proof.
For square-hexagonal kink chainT22withήnumber of kinks, we haveEήGA=ή0.04662133γ́+5.328427125−0.04662133γ́+6.455861185
Proof. Let ή≥2, then there are three possibilities. Using Table 3 and Equation 7, we get
1. If (T22)ή−2→(T212)ή, thenGAT212ή=β22T22ή−2+265β23T22ή−2+4+223β24T22ή−2+4+437β34T22ή−2+2+β44T22ή−2+1=GAT22ή−2+10.669906392. If (T22)ή−2→(T222)ή, thenGAT222ή=β22T22ή−2+1+265β23T22ή−2+3+223β24T22ή−2+3+437β34T22ή−2+3+β44T22ή−2+1=GAT22ή−2+10.737044773. If (T22)ή−2→(T232)ή, thenGAT232ή=β22T22ή−2+2+265β23T22ή−2+223β24T22ή−2+6+437β34T22ή−2+β44T22ή−2+3=GAT22ή−2+10.65685425
Thus, we haveEήGA=γ́GAT212ή+γ́GAT222ή+1−2γ́GAT232ή
Using Equations 14–16, we get the following relationEήGA=γ́GAT22ή−2+10.66990639+γ́GAT22ή−2+10.73704477+1−2γ́GAT22ή−2+10.65685425EήGA=GAT22ή−2+0.09324266γ́+10.65685425
Applying operator E on both sides and ∴E(EήGA)=EήGAEήGA=Eή−2GA+0.09324266γ́+10.65685425• Let ή=2n−1
For n=1⇒ή=1, E1=11.78428831, which is indeed true. Using recursive relation upto ή−1 trermsEήGA=Eή−ή−1GA+ή−120.09324266γ́+10.65685425=11.78428831+ή−10.04662133γ́+5.328427125=ή0.04662133γ́+5.328427125−0.04662133γ́+6.455861185which completes the result.
• Let ή=2n
For n=1⇒ή=2, E2=16.66990639, which is indeed true. Using recursive relation upto ή−2 termsEήGA=Eή−ή−2GA+ή−220.09324266γ́+10.65685425=16.66990639+ή−20.04662133γ́+5.328427125=ή0.04662133γ́+5.328427125−0.09324266γ́+6.01305214which completes the proof.
For square-hexagonal kink chainT22withήnumber of kinks, we haveEήSC=ή0.032431259γ́+2.255074958−0.032431259γ́+3.106204951
Proof. Let ή≥2, then there are three possibilities. Using Table 3 and Equation 8, we get
1. If (T22)ή−2→(T212)ή, thenSCT212ή=12β22T22ή−2+15β23T22ή−2+4+16β24T22ή−2+4+17β34T22ή−2+2+122β44T22ή−2+1=SCT22ή−2+4.531329882. If (T22)ή−2→(T222)ή, thenSCT222ή=12β22T22ή−2+1+15β23T22ή−2+3+16β24T22ή−2+3+17β34T22ή−2+3+122β44T22ή−2+1=SCT22ή−2+4.5538324683. If (T22)ή−2→(T232)ή, then=12β22T22ή−2+2+15β23T22ή−2+16β24T22ή−2+6+17β34T22ή−2+122β44T22ή−2+3SCT232ή=SCT22ή−2+4.510149915
Thus, we haveEήSC=γ́SCT212ή+γ́SCT222ή+1−2γ́SCT232ή
Using Equations 17–19, we get the following relationEήSC=γ́SCT22ή−2+4.53132988+γ́SCT22ή−2+4.553832468+1−2γ́SCT22ή−2+4.510149915EήSC=SCT22ή−2+0.064862518γ́+4.510149915
Applying operator E on both sides and ∴E(EήSC)=EήSCEήSC=Eή−2SC+0.064862518γ́+4.510149915• Let ή=2n−1; n∈N
For n=1⇒ή=1, E1=5.361279909, which is indeed true. Using recursive relation upto ή−1 trermsEήSC=Eή−ή−1SC+ή−120.064862518γ́+4.510149915=5.361279909+ή−10.032431259γ́+2.255074958=ή0.032431259γ́+2.255074958−0.032431259γ́+3.106204951which completes the result.
• Let =́2n; n∈N
For n=1⇒ή=2, E2=7.53132988, which is indeed true. Using recursive relation upto ή−2 termsEήSC=Eή−ή−2SC+ή−220.064862518γ́+4.510149915=7.53132988+ή−20.032431259γ́+2.255074958=ή0.032431259γ́+2.255074958−0.064862518γ́+3.021179965which completes the proof.
For square-hexagonal kink chainT22withήnumber of kinks, we have
Proof. Let ή≥2, then there are three possibilities. Using Table 3 and Equation 9, we get1. If (T22)ή−2→(T212)ή, thenABCT212ή=24β22T22ή−2+36β23T22ή−2+4+14β24T22ή−2+4+512β34T22ή−2+2+616β44T22ή−2+1=ABCT22ή−2+2.6804716442. If (T22)ή−2→(T222)ή, thenABCT222ή=24β22T22ή−2+1+36β23T22ή−2+3+14β24T22ή−2+3+512β34T22ή−2+3+616β44T22ή−2+1=ABCT22ή−2+2.6816888983. If (T22)ή−2→(T232)ή, thenABCT232ή=24β22T22ή−2+2+36β23T22ή−2+14β24T22ή−2+6+512β34T22ή−2+616β44T22ή−2+3ABCT232ή=ABCT22ή−2+2.666386108
Thus, we haveEήABC=γ́ABCT212ή+γ́ABCT222ή+1−2γ́ABCT232ή
Using Equations 20–22, we get the following relationEήABC=γ́ABCT22ή−2+2.680471644+γ́ABCT22ή−2+2.681688898+1−2γ́ABCT22ή−2+2.666386108EήABC=ABCT22ή−2+0.029388326γ́+2.666386108
Applying operator E on both sides and ∴E(EήABC)=EήABCEήABC=Eή−2ABC+0.029388326γ́+2.666386108
For ή=2n−1 For n=1⇒ή=1, E1=3.441592097, which is indeed true. Using recursive relation upto ή−1 trermsEήABC=Eή−ή−1ABC+ή−120.029388326γ́+2.666386108=3.441592097+ή−10.014694163γ́+1.333193054=0.014694163γ́+1.333193054ή−0.014694163γ́+2.108399043which completes the result.
For ή=2n For n=1⇒ή=2, E2=4.801791987, which is indeed true. Using recursive relation upto ή−2 termsEήABC=Eή−ή−2ABC+ή−220.029388326γ́+2.666386108=4.801791987+ή−20.014694163γ́+1.333193054=0.014694163γ́+1.333193054ή−0.029388326γ́+2.135405879which completes the proof.
For square-hexagonal kink chainT22withήnumber of kinks, we have
Proof. Let ή≥2, then there are three possibilities. Using Table 3 and Equation 10, we get
1. If (T22)ή−2→(T212)ή, thenPCT212ή=12β22T22ή−2+16β23T22ή−2+4+122β24T22ή−2+4+123β34T22ή−2+2+14β44T22ή−2+1=PCT22ή−2+3.8745569932. If (T22)ή−2→(T222)ή, thenPCT222ή=12β22T22n−2+1+16β23T22n−2+3+122β24T22n−2+3+123β34T22n−2+3+14β44T22n−2+1=PCT22ή−2+3.9014304473. If (T22)ή−2→(T232)ή, thenPCT232ή=12β22T22ή−2+2+16β23T22ή−2+122β24T22ή−2+6+123β34T22ή−2+14β44T22ή−2+3PCT232ή=PCT22ή−2+3.871320344
Thus, we haveEήPC=γ́PCT212ή+γ́PCT222ή+1−2γ́PCT232ή
Using Equations 23–25, we get the following relationEήPC=γ́PCT22ή−2+3.874556993+γ́PCT22ή−2+3.901430447+1−2γ́PCT22ή−2+3.871320344EήPC=PCT22ή−2+0.033346752γ́+3.871320344
Applying operator E on both sides and ∴E(EήPC)=EήPCEήPC=Eή−2PC+0.033346752γ́+3.871320344
For ή=2n−1 For n=1⇒ή=1, E1=4.917450212, which is indeed true. Using recursive relation upto ή−1 trermsEήPC=Eή−ή−1PC+ή−120.033346752γ́+3.871320344=4.917450212+ή−10.016673376γ́+1.935660172=0.016673376γ́+1.935660172ή−0.016673376γ́+2.98179004which completes the result.
For ή=2n For n=1⇒ή=2, E2=6.874556993, which is indeed true. Using recursive relation upto ή−2 termsEήPC=Eή−ή−2PC+ή−220.033346752γ́+3.871320344=6.874556993+ή−20.016673376γ́+1.935660172=0.016673376γ́+1.935660172ή−0.033346752γ́+3.003236649which completes the proof.
From Figure 3 it is easy to that three possible kink chains can be obtained from (T2γ́2)ή by taking the value of γ́=13.
In this section we analytically prove that forgotten index attains the greatest expected values at (ή−2)th stage, for any value of γ́ and ή and for ή=2n−1 and ή=2n, while atom-bond connectivity attains minimum expectations.
Forή=2n−1andή=2n;n∈N, we haveEFT2γ́2ή>EGAT2γ́2ή
Proof. • For ή=2n−1EFT2γ́2ή−EGAT2γ́2ή=ή−18γ́+116+18γ́+58−ή0.04662133γ́+5.328427125−0.04662133γ́+6.455861185=18.04662133γ́1−ή+110.6715729ή+51.54413881>0which holds for γ́=13 and for all ή∈N, so we haveEFT2γ́2ή>EGAT2γ́2ή• For ή=2nEFT2γ́2ή−EGAT2γ́2ή=ή−18γ́+116+36γ́+30−ή0.04662133γ́+5.328427125−0.09324266γ́+6.01305214=18.04662133γ́2−ή+110.6715783ή+23.98694786>0which holds for γ́=13 and for all ή∈N, so we haveEFT2γ́2ή>EGAT2γ́2ή
Forή=2n−1andή=2n;ή∈N, we haveEGAT2γ́2ή>ESCT2γ́2ή
Proof. • For ή=2n−1EGAT2γ́2ή−ESCT2γ́2ή=ή0.04662133γ́+5.328427125−0.04662133γ́+6.455861185−ή0.032431259γ́+2.255074958−0.032431259γ́+3.106204951=0.014190071γ́ή−1+3.073352167ή+3.349656234>0which holds for γ́=13 and for all ή∈N, so we haveEGAT2γ́2ή>ESCT2γ́2ή• For ή=2nEGAT2γ́2ή−ESCT2γ́2ή=ή0.04662133γ́+5.328427125−0.09324266γ́+6.01305214−ή0.032431259γ́+2.255074958−0.064862518γ́+3.021179965=0.014190071γ́ή−2+3.073352167ή+2.991872175)>0which holds for γ́=13 and for all ή∈N, so we haveEGAT2γ́2ή>ESCT2γ́2ή
Forή=2n−1andή=2n;ή∈N, we haveESCT2γ́2ή>EPCT2γ́2ή
Proof. • For ή=2n−1ESCT2γ́2ή−EPCT2γ́2ή=ή0.032431259γ́+2.255074958−0.032431259γ́+3.106204951−0.016673376γ́+1.935660172ή−0.016673376γ́+2.98179004=0.015757883γ́ή−1+0.319414786ή+0.124414911)>0which holds for γ́=13 and for all ή∈N, so we haveESCT2γ́2ή>EPCT2γ́2ή• For ή=2nESCT2γ́2ή−EPCT2γ́2ή=ή0.032431259γ́+2.255074958−0.064862518γ́+3.021179965−0.016673376γ́+1.935660172ή−0.033346752γ́+3.003236649=0.015757883γ́ή−2+0.319414786ή+0.017943316>0which holds for γ́=13 and for all ή∈N, so we haveESCT2γ́2ή>EPCT2γ́2ή
Forή=2n−1andή=2n;ή∈N, we haveEPCT2γ́2ή>EABCT2γ́2ή
Proof. • For ή=2n−1EPCT2γ́2ή−EABCT2γ́2ή=ή0.016673376γ́+1.935660172−0.016673376γ́+2.98179004−0.014694163γ́+1.333193054ή−0.014694163γ́+2.108399043=0.001979213γ́ή−1+0.602467118ή+0.873390997>0which holds for γ́=13 and for all ή∈N, so we haveEPCT2γ́2ή>EABCT2γ́2ή• For ή=2nEPCT2γ́2ή−EABCT2γ́2ή=0.016673376γ́+1.935660172ή−0.033346752γ́+3.003236649−0.014694163γ́+1.333193054ή−0.029388326γ́+2.135405879=0.001979213γ́ή−2+0.602467118ή+0.86783077>0which holds for γ́=13 and for all ή∈N, so we haveEPCT2γ́2ή>EABCT2γ́2ή
From these analytical expressions, we conclude that that the expected value of Forgotten index for γ́=13 is greatest among all other computed topological indices at (ή−2)th stage.
The numerical values as well as graphical representation in Figures 4–7 meet with the results of analytical comparison. So, we conclude that the expected value of F index reaches maximum value for both odd and even numbered kinked chains at (ή−2)th stage. And the expected value of ABC index for both odd and even numbered kinked chains attains minimum value at (ή−2)th stage, and for γ́=13.
Expected values of PC, ABC and SC at (ή−2)th stage for ή=2n−1.
Expected values of PC, ABC and SC at (ή−2)th stage for ή=2n.
Expected values of F and GA at (ή−2)th stage for ή=2n−1.
Expected values of F and GA at (ή−2)th stage for ή=2n.
5 Conclusion
In conclusion, at the (ή−2)th stage, the expected value of a certain topological descriptor for three kink chains with ή=2n−1 is equal to the average of the corresponding topological descriptor for ή=2n−1, and similarly, the expected value of the topological descriptor for ή=2n equals the average for ή=2n. This establishes the validity of the results. The values of the topological descriptors R1, R2, HZ, ReZ2, GO1, and GO2 can be calculated using the theorems proven above for γ́=13 at the (ή−2)th stage. Furthermore, at this stage, for both ή=2n−1 and ή=2n, the expected values of the descriptors ER1ή, ER2ή, EHZή, EReZ2ή, EGO1ή, and EGO2ή depend on both γ́ and the number of kinks ή. However, the expected value of ReZ1 depends solely on the number of kinks ή and is identical in both cases. Moreover, the expected values for the forgotten index attains maximum value, whereas those for the ABC index attains minimum value, for both the cases.
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.
The author(s) declare that financial support was received for the research and/or publication of this article. This work was supported by Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [KFU251155] and Anhui Province quality Engineering” online and offline mixed and Social practice course” (2020xsxxkc222) Scientific Research Program of Colleges and Universities in Anhui Province (2022AH052889, 2024AH050616).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The authors declare that no Generative AI was used in the creation of this manuscript.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
ReferencesAlraqadTASaberHAliAMazorodzeJP. On-bond incident degree indices of square-hexagonal chains. J Mathematics (2022) 2022(1):1864828. 10.1155/2022/1864828SigarretaSCSigarretaSMCruz-SuarezH. On degree-based topological indices of random polyomino chains. Math Biosciences Eng (2022) 19(9):8760–73. 10.3934/mbe.2022406XiaoLChenSGuoZChenQ. The geometric-arithmetic index of benzenoid systems and phenylenes. Int J Contemp Math Sci (2010) 5(45):2225–2230.ZhouBTrinajsticN. Relations between the product-and sum-connectivity indices. Croatica chemica acta (2012) 85(3):363–365.FurtulaBGutmanI. A forgotten topological index. J Math Chem (2015) 53(4):1184–90. 10.1007/s10910-015-0480-zGaoWHusinMNFarahaniMRImranM. On the edges version of atom-bond connectivity index of nanotubes. J Comput Theor Nanoscience (2016) 13(10):6733–40. 10.1166/jctn.2016.5621TrinajsticN. Chemical graph theory. Boca Raton: CRC Press (1992).RazaZ. The expected values of arithmetic bond connectivity and geometric indices in random phenylene chains. Heliyon (2020) 6(7):e04479. 10.1016/j.heliyon.2020.e04479WeiLBianHYuHDingJ. The expected values of Merrifield-Simmons index in random phenylene chains. Iran J Math (2020) 11(4):271–81.FangXYouLLiuH. The expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs. Int J Quan Chem. (2021) 121(17):e26740. 10.1002/qua.26740KeXWeiSHuangJ. The atom-bond connectivity and geometric-arithmetic indices in random polyphenyl chains. Polycyclic Aromatic Compounds (2021) 41(9):1873–82. 10.1080/10406638.2019.1703763RazaZ. The expected values of some indices in random phenylene chains. The Eur Phys J Plus (2021) 136(1):91–15. 10.1140/epjp/s13360-021-01082-yRazaZAkhterSShangY. Expected value of first Zagreb connection index in random cyclooctatetraene chain, random polyphenyls chain, and random chain network. Front Chem (2023) 10:1067874. 10.3389/fchem.2022.1067874HayatS. Distance-based graphical indices for predicting thermodynamic properties of benzenoid hydrocarbons with applications. Comput Mater Sci (2023) 230:112492. 10.1016/j.commatsci.2023.112492HayatSMahadiHAlanaziSJWangS. Predictive potential of eigenvalues-based graphical indices for determining thermodynamic properties of polycyclic aromatic hydrocarbons with applications to polyacenes. Comput Mater Sci (2024) 238:112944. 10.1016/j.commatsci.2024.112944HayatSKhanAAliKLiuJB. Structure-property modeling for thermodynamic properties of benzenoid hydrocarbons by temperature-based topological indices. Ain Shams Eng J (2024) 15(3):102586. 10.1016/j.asej.2023.102586ChunsongBKhalilMRazzaqueAKanwalSNazirRNoorS. Exploring structural variations and topological descriptors of square-hexagonal kink chains of type T2,2 in engineering applications. Adv Mech Eng (2024) 16(10):16878132241290418. 10.1177/16878132241290418ChenRRazzaqueAKhalilMKanwalSNoorSNazirR. Expected value of topological descriptors for possible kink chains of type 2T2. Front Chem (2024) 12:1517892. 10.3389/fchem.2024.1517892Garcia-DomenechRGalvezJde Julian-OrtizJVPoglianiL. Some new trends in chemical graph theory. Chem Rev (2008) 108(3):1127–69. 10.1021/cr0780006