Expected values of topological descriptors for possible kink chains of type 2⊤2

In this paper, we investigate square-hexagonal chains, a class of systems where the inner dual of a structure with a square-hexagon shape forms a path graph. The specific configuration of square and hexagonal polygons, and how they are concatenated, leads to different types of square-hexagonal chains. A square containing a vertex of degree 2 is classified as having a kink, and the resulting kink is referred to as a type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mmultiscripts><mml:mrow><mml:mo>⊤</mml:mo></mml:mrow> <mml:none></mml:none> <mml:none></mml:none> <mml:mprescripts></mml:mprescripts> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> <mml:none></mml:none></mml:mmultiscripts> </mml:mrow> </mml:math> kink. This kink is further subdivided into three types: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mmultiscripts><mml:mrow><mml:mo>⊤</mml:mo></mml:mrow> <mml:mrow><mml:mn>1</mml:mn></mml:mrow> <mml:none></mml:none> <mml:mprescripts></mml:mprescripts> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> <mml:none></mml:none></mml:mmultiscripts> </mml:mrow> </mml:math> , ₂ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>⊤</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> , and ₂ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>⊤</mml:mo> <mml:mn>3</mml:mn></mml:mrow> </mml:math> . We focus on the kink chain of type ₂ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>⊤</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> and compute various topological descriptors for this configuration. By deriving analytical expressions, we determine the maximizing and minimizing values of these descriptors. Additionally, we provide a comprehensive analysis of the expected values for these descriptors and offer a comparison of their behaviors through analytical, numerical, and graphical methods. These results offer insights into the structural properties and behavior of square-hexagonal chains, particularly in relation to the optimization of topological descriptors.