In this section, the proper analytical expressions of neighbourhood degree sum-based indices and entropy measures are computed using the M-polynomial for β-Graphene. Graphene is a carbon allotrope, a two-dimensional hexagonal network in which the carbon atoms form vertices with sp
2 hybridisation. Graphene has many exceptional properties, including mechanical strength, optical transparency, and electric and thermal conductivity. Furthermore, the one-atomic layer structure of graphene makes it ultralight and super thin. Graphene has a thickness of about 0.35 nm, which is approximately 1/200,000th of the thickness of human hair. However, the closely arranged carbon atoms and the sp
2 orbital hybridisation provide exceptional stability to the graphene structure. Thus, graphene shows extraordinary transparency of 97.7 percent, which means that it only absorbs 2.3 percent of visible light [
22].
In terms of its structure, graphene can be considered the basic unit of graphite, fullerene [
25], carbon nanotube [
26], graphyne [
27], and other related materials such as amorphous carbon, carbon fibre, charcoal [
28], as well as aromatic molecules such as polycyclic aromatic hydrocarbons. As they all have the same structure, they all have some properties in common, even though their different sizes and shapes make them very different. Thus, the structural study of graphene helps to understand the above-listed materials. It can be observed from
Figure 1 that the
-Graphene contains seven graphene layers stacked next to each other. In three dimensions, the
-Graphene contains seven layers stacked on top of each other. Thus, it can be inferred that the
-Graphene consists of vertex set
and edge set
.
Proof of Theorem 2. Let
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.