Graph theory is a mathematical discipline that studies graphs, which are abstract structures used to model and analyze relationships between objects. A Graph ζ = ( V ( ζ ) , E ( ζ ) ) is defined to be the collection of vertices (nodes) and edges (links or arcs) where V ( ζ ) and E ( ζ ) denotes set of vertices and edges respectively. The order of a graph, denoted as n = | V ( ζ ) | refers to the total number of vertices in the graph. The size of a graph, denoted as m = | E ( ζ ) | refers to the total number of edges in the graph. The degree of a vertex, denoted by d e g ( w ) or d w , is the number of edges connected to that vertex. The distance d ( w , x ) between two vertices w , x is the length of the shortest path joining them. For basic definitions related to graph theory, we refer (Trinajstic, 1992).

In chemical graph theory, the numerical values assigned to a molecular graph, known as topological indices or molecular descriptors, are often used to correlate with chemical structures and their properties. In other words, topological indices refer to graph invariants or descriptors that have significant chemical relevance. These indices are based on the graphical representation of a molecule and can encode chemical information such as atom types and bond multiplicities. Topological indices are valuable for predicting specific chemical and physical properties of the underlying molecular structure, combining logical and mathematical principles to translate a molecule’s symbolic representation into a useable numerical form. Chemical graph theory, which merges the fields of chemistry and graph theory, uses graphs to represent chemical structures, providing insights into the physical and chemical characteristics of molecules.

The first degree based topological descriptor was introduced by Milan Randic in 1975 in his paper (Randic, 1975) “On characterization of molecular branching.” This index is referred to as Randic index and is defined as R ζ = ∑ w x ∈ E ζ 1 d w + d x . The Randic index has been recognized as a valuable tool in drug design and has been widely used for this purpose in various studies (Randic, 1975).

The first and second Zagreb indices are the oldest degree based graph invariants introduced by Gutman and Trinajstic (Gutman and Trinajstic, 1972b) in 1972. They were later included among topological descriptors and are defined as M 1 ζ = ∑ w x ∈ E ζ d w + d x . M 2 ζ = ∑ w x ∈ E ζ d w × d x .

The first and second Zagreb indices were initially applied to branching problem (Gutman et al., 1975b). Later, they found applications in QSPR and QSAR studies (Balaban, 1979; Bonchev and Trinajsti, 2001; Devillers and Balaban, 1999).

The applicability of Zagreb indices motivated the researchers to define different variants of Zagreb indices. The Hyper Zagreb index was put forwarded by Shirdel et al. (Shirdel et al., 2013) in 2013 and is defined as H Z ζ = ∑ w x ∈ E ζ d w + d x 2 .

Another variant of Zagreb indices namely, first and second redefined Zagreb indices were introduced by Ranjini et al. (Ranjini et al., 2013) R e Z 1 ζ = ∑ w x ∈ E ζ d w + d x d w ⋅ d x . R e Z 2 ζ = ∑ w x ∈ E ζ d w ⋅ d x d w + d x .

Motivated by the definitions of first and second Zagreb indices and their chemical applicability, V. Kulli (Kulli, 2017a) introduces the first and second Gourava indices. These topological indices are defined as G O 1 ζ = ∑ w x ∈ E ζ d w + d x + d w ⋅ d x . G O 2 ζ = ∑ w x ∈ E ζ d w + d x d w ⋅ d x .

The first and second Revan descriptors were introduced by V. Kulli (Kulli, 2017b) and are defined as R 1 ζ = ∑ w x ∈ E ζ r w + r x R 2 ζ = ∑ w x ∈ E ζ r w ⋅ r x where r w is defined as r w = Δ ζ + δ ζ − d w , where Δ and δ denotes the maximum and minimum degree among the vertices of ζ .

For more details on the importance of topological indices and their applications see (Noreen and Mahmood, 2018; Wei and Shiu, 2019; Raza, 2020; Wei et al., 2020; Fang et al., 2021; Alraqad et al., 2022; Zhang X. et al., 2023; Zhang Guoping et al., 2023; Hui et al., 2023a; Hui et al., 2023b; Huang et al., 2023). For results related to mathematical properties of the topological indices, we refere (Zhou, 2004; Zhao et al., 2016; Gao et al., 2017; Kulli, 2017c; Kulli, 2017e; Zhang et al., 2024; Govardhan et al., 2024; Prabhu et al., 2024a; Prabhu et al., 2024b).

A square-hexagonal system, also known as a rectangular hexagonal system, is a connected geometric structure created by joining equal-sized squares and hexagons. This arrangement blends elements of square and hexagonal lattices, forming a distinctive repeating pattern that combines the characteristics of both shapes. The lattice points in this system create a regular and continuous design, where each polygon is linked to its neighbors. Two polygons are considered neighboring if they share a common edge, emphasizing the interconnected nature of this hybrid structure. This system is widely used in crystallography and materials science, particularly for analyzing the structures of materials with hexagonal crystal systems that exhibit square symmetry along specific crystallographic directions. It provides a geometric framework for understanding the arrangement of atoms, ions, or other structural components within such materials.

A square-hexagonal system is a two-dimensional lattice structure that combines square and hexagonal elements in a unified arrangement. In contrast, a square-hexagonal chain is a one-dimensional sequence where square and hexagonal configurations alternate along its length. While both concepts incorporate square and hexagonal features, they differ in their structure and intended applications. The structure of a square-hexagonal chain varies depending on the types of polygons used and how they are concatenated. A square-hexagonal chain composed of n polygons is denoted as R n . If all the polygons in R n are squares, it is referred to as a polyomino chain (Li et al., 2023). Similarly, if all the polygons are hexagons, R n is called a hexagonal chain (Alraqad et al., 2022). However, when squares and hexagons alternate in the chain, R n is specifically known as a phenylene chain. (as in (Raza, 2021; Shooshtari et al., 2022).

To derive key results, it is important to introduce certain terminologies related to square-hexagonal chains. In graph theory, a kink refers to a point in the graph where there is a sudden change in direction or slope. More precisely, a kink is a vertex whose degree is greater than the degrees of its neighboring vertices, resulting in a bend or angular deviation in the graph’s structure.

Kinks play a significant role in graph analysis, as they often highlight structural changes or key points within the graph. They can influence various graph properties and algorithms, including traversal methods, connectivity analysis, and the identification of critical nodes or hubs in networks. In network analysis, for instance, identifying kinks or high-degree vertices can reveal essential nodes that contribute significantly to the network’s connectivity or exert considerable influence. Moreover, the presence of kinks can affect processes like random walks, as high-degree vertices are more likely to attract repeated visits, thereby altering the overall dynamics of the system.

A polygon at one end of a chain, typically lacking a neighboring polygon on one of its sides, is referred to as a terminal polygon. In contrast, a polygon located within the chain, with neighboring polygons on both sides and not positioned at the chain’s ends, is termed a non-terminal polygon.

If the centers of two adjacent non-terminal polygons are not collinear, the polygon is described as kinked in the chain. There are two types of square-hexagonal kinks, denoted as ⊤ 1 and ⊤ 2 (Alraqad et al., 2022). In type ⊤ 1 , the kink is formed by a hexagon, while in type ⊤ 2 , square occurs as a kink. A non-terminal hexagon is considered kinked if and only if it contains two consecutive vertices of degree two. Similarly, a non-terminal square is considered kinked if and only if it has a single vertex with a degree of two. Following (Alraqad et al., 2022), we focus on square-hexagonal chains related to the kinks described below:

(1) K i n k s 1 ⊤ ; A non-terminal hexagon that has exactly two vertices with a degree of two.

In graph theory, the expected values of topological indices serve as statistical measures of a graph’s structural properties, capturing key characteristics such as connectivity, distances, and vertex degrees. These values are particularly valuable for analyzing and comparing random graphs, optimizing network designs, and predicting behaviors in fields like chemistry, biology, and social networks. Additionally, they enable the development of efficient algorithms for large-scale graph analysis by reducing the computational complexity of calculating indices across various graph models. Our work is motivated from our previous work on the kink chains introduced in (Chunsong et al., 2024). We considered three types of kink chains, ⊤ 2 2 , which are categorized as ⊤ 2 1 2 , ⊤ 2 2 2 , and ⊤ 2 3 2 , based on the specific way squares and hexagons are concatenated. For generality, we divided our analysis into two cases: odd and even numbered kink chains, addressing their orders, sizes, and corresponding vertex and edge partitions. Additionally, we calculated the topological indices defined earlier and demonstrated that the second Gourava topological index is a maximizing index, while the redefined first Zagreb index is a minimizing index in both cases. Now, we will determine the expected values of topological descriptors for the newly identified kink chains ⊤ 2 1 2 , ⊤ 2 2 2 , and ⊤ 2 3 2 . These kink chains are defined as;

• Kink chains of. T y p e ⊤ 2 2

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 kink chains ⊤ 2 1 2 , ⊤ 2 2 2 and ⊤ 2 3 2 respectively, except at terminal polygons. Let κ ́ represents the number of kinks (squares forming a kink). It is observed that even-numbered kink chains result when two squares are joined at the terminal, while odd-numbered kink chains occur when a hexagon is attached at the terminal. Consequently, an odd-numbered chain corresponds to a square terminal, whereas an even-numbered chain corresponds to a hexagonal terminal. These cases are mathematically represented as κ ́ = 2 n − 1 for odd-numbered chains and κ ́ = 2 n for even-numbered chains, where n ∈ N . The order p of each kink chain is the same across these types and follows the formula p = 6 + 4 κ ́ . However, the size q of the chain differs: for κ ́ = 2 n − 1 (terminal square), q = 11 κ ́ + 13 2 , and for κ ́ = 2 n (terminal hexagon), q = 11 κ ́ + 12 2 .

In this section, we will calculate some topological descriptors of ⊤ 2 1 2 , ⊤ 2 2 2 and ⊤ 2 3 2 using Tables 1, 2. Let p denotes the kink chain for kinks of type ⊤ 2 2 , where p varies from 1 to 3.

Lemma 1

For n ∈ N the first Revan index of Kink chain ⊤ 2 p 2 is given as;

R 1 ⊤ 2 p 2 = 33 κ ́ + 49 if  p = 1,2       ;   f o r    κ ́ = 2 n − 1 R 1 ⊤ 2 p 2 = 33 κ ́ + 48 if  p = 1,2       ;   f o r    κ ́ = 2 n R 1 ⊤ 2 p 2 = 32 κ ́ + 50            if  p = 3 Proof. For κ ́ = 2 n − 1

• p = 1

For κ ́ = 2 n

• p = 1

Lemma 2

For n ∈ N the 2 n d Revan, 1 s t and 2 n d Redefined Zagreb, Hyper-Zagreb, 1 s t and 2 n d Gourava descriptors of kink chains ⊤ 2 1 2 , ⊤ 2 2 2 and ⊤ 2 3 2 are are presented in Table 3;

In this section, we compared the above calculated topological descriptors using graphical representation of ⊤ 2 1 2 , ⊤ 2 2 2 and ⊤ 2 3 2 at different values of κ ́ for odd and even numbered kink chains. From the Figures 5–11 we conclude that G O 2 descriptor of ⊤ 2 1 2 , ⊤ 2 2 2 and ⊤ 2 3 2 hits a highest value for both κ ́ = 2 n − 1 and κ ́ = 2 n . It follows that G O 2 is a maximizing descriptor. On the other hand R e Z 1 descriptor of ⊤ 2 1 2 , ⊤ 2 2 2 and ⊤ 2 3 2 reaches a lowest value for both κ ́ = 2 n − 1 and κ ́ = 2 n , thus R e Z 1 is a minimizing descriptor.

As we know that there are only three possible kink chains ( ⊤ 2 1 2 , ⊤ 2 2 2 and ⊤ 2 3 2 ) or arrangements of type ⊤ 2 2 , holding the conditions to make kink at each step. The kink chains for κ ́ = 1 and κ ́ = 2 are shown in Figure 12. For κ ́ ≥ 2 , terminal polygons are attachable in three different ways, resulting in three types ⊤ 2 1 2 , ⊤ 2 2 2 and ⊤ 2 3 2 . Considering that γ represents the probability of attaching terminal polygons in the first or second kind of arrangement, 1 − 2 γ represents the probability of attaching the terminal polygon in the third type of arrangement. Possible arrangements of kink chains of type ⊤ 2 2 are shown in Figure 12.

Let the K i n k c h a i n ⊤ 2 2 with κ ́ number of kinks and probability γ is represented by ( ⊤ 2 γ 2 ) κ ́ . We compute Revan, Hyper-Zagreb, redefined Zagreb and Guorava descriptors of possible square-hexagonal kink chains ( ⊤ 2 γ 2 ) κ ́ . Let | E i j | = β i j denotes the number of edges for ( ⊤ 2 γ 2 ) κ ́ with end vertices of degree i and j respectively. There are only β 22 , β 23 , β 24 , β 34 and β 44 -type of edges in ( ⊤ 2 γ 2 ) κ ́ . Here Δ ⊤ 2 2 and δ ⊤ 2 2 are 4 and 2 implies that (Revan degrees) r s = 4,3,2 for d s = 2,3 and 4 respecively. From the definition, the topological descriptors can be expressed as R 1 ⊤ 2 γ 2 κ ́ = 8 β 22 ⊤ 2 γ 2 κ ́ + 7 β 23 ⊤ 2 γ 2 κ ́ + 6 β 24 ⊤ 2 γ 2 κ ́ + 5 β 34 ⊤ 2 γ 2 κ ́ + 4 β 44 ⊤ 2 γ 2 κ ́ R 2 ⊤ 2 γ 2 κ ́ = 16 β 22 ⊤ 2 γ 2 κ ́ + 12 β 23 ⊤ 2 γ 2 κ ́ + 8 β 24 ⊤ 2 γ 2 κ ́ + 6 β 34 ⊤ 2 γ 2 κ ́ + 4 β 44 ⊤ 2 γ 2 κ ́ R e Z 1 ⊤ 2 γ 2 κ ́ = β 22 ⊤ 2 γ 2 κ ́ + 5 6 β 23 ⊤ 2 γ 2 κ ́ + 3 4 β 24 ⊤ 2 γ 2 κ ́ + 7 12 β 34 ⊤ 2 γ 2 κ ́ + 1 2 β 44 ⊤ 2 γ 2 κ ́ R e Z 2 ⊤ 2 γ 2 κ ́ = β 22 ⊤ 2 γ 2 κ ́ + 6 5 β 23 ⊤ 2 γ 2 κ ́ + 4 3 β 24 ⊤ 2 γ 2 κ ́ + 12 7 β 34 ⊤ 2 γ 2 κ ́ + 2 β 44 ⊤ 2 γ 2 κ ́ H Z ⊤ 2 γ 2 κ ́ = 16 β 22 ⊤ 2 γ 2 κ ́ + 25 β 23 ⊤ 2 γ 2 κ ́ + 36 β 24 ⊤ 2 γ 2 κ ́ + 49 β 34 ⊤ 2 γ 2 κ ́ + 64 β 44 ⊤ 2 γ 2 κ ́ G O 1 ⊤ 2 γ 2 κ ́ = 8 β 22 ⊤ 2 γ 2 κ ́ + 11 β 23 ⊤ 2 γ 2 κ ́ + 14 β 24 ⊤ 2 γ 2 κ ́ + 19 β 34 ⊤ 2 γ 2 κ ́ + 24 β 44 ⊤ 2 γ 2 κ ́ G O 2 ⊤ 2 γ 2 κ ́ = 16 β 22 ⊤ 2 γ 2 κ ́ + 30 β 23 ⊤ 2 γ 2 κ ́ + 48 β 24 ⊤ 2 γ 2 κ ́ + 84 β 34 ⊤ 2 γ 2 κ ́ + 128 β 44 ⊤ 2 γ 2 κ ́

As ( ⊤ 2 γ 2 ) κ ́ is a random kink chain, it follows R 1 ( ⊤ 2 γ 2 ) κ ́ , R 2 ( ⊤ 2 γ 2 ) κ ́ , R e Z 1 ( ⊤ 2 γ 2 ) κ ́ , R e Z 2 ( ⊤ 2 γ 2 ) κ ́ , H Z ( ⊤ 2 γ 2 ) κ ́ , G O 1 ( ⊤ 2 γ 2 ) κ ́ and G O 2 ( ⊤ 2 γ 2 ) κ ́ are random variables. Let us denote by E n R 1 = E [ R 1 ( ⊤ 2 γ 2 ) κ ́ ] , E n R 2 = E [ R 2 ( ⊤ 2 γ 2 ) κ ́ ] , E n R e Z 1 = E [ R e Z 1 ( ⊤ 2 γ 2 ) κ ́ ] , E n R e Z 2 = E [ R e Z 2 ( ⊤ 2 γ 2 ) κ ́ ] , E n H Z = E [ H Z ( ⊤ 2 γ 2 ) κ ́ ] , E n G O 1 = E [ G O 1 ( ⊤ 2 γ 2 ) κ ́ ] and E n G O 2 = E [ G O 2 ( ⊤ 2 γ 2 ) κ ́ ] the expected values of these descriptors respectively.

Note that if κ ́ is odd in kink chains ⊤ 2 1 2 , ⊤ 2 2 2 and ⊤ 2 3 2 , then at ( κ ́ − 1 ) t h step, even numbered kink chains are formed, and at ( κ ́ − 2 ) t h step, odd numbered kink chains are formed. Similarly, if κ ́ is even in kink chains ⊤ 2 1 2 , ⊤ 2 2 2 and ⊤ 2 3 2 at ( κ ́ − 1 ) t h step, odd numbered kink chains are obtained and at ( κ ́ − 2 ) t h step, even numbered kink chains are obtained again. So, we furthur divide our results in two possible stages;

(1) At ( κ ́ − 1 ) t h stage and (2) At ( κ ́ − 2 ) t h stage.