Several distance and degree-based molecular structural attributes of cove-edged graphene nanoribbons

A carbon-based material with a broad scope of favourable developments is called graphene. Recently, a graphene nanoribbon with cove-edged was integrated by utilizing a bottom-up liquid-phase procedure, and it can be geometrically viewed as a hybrid of the armchair and the zigzag edges. It is indeed a type of nanoribbon containing asymmetric edges made up of sequential hexagons with impressive mechanical and electrical characteristics. Topological indices are numerical values associated with the structure of a chemical graph and are used to predict various physical, chemical, and biological properties of molecules. They are derived from the graph representation of molecules, where atoms are represented as vertices and bonds as edges. In this article, we derived the exact topological expressions of cove-edged graphene nanoribbons based on the graph-theoretical structural measures that help reduce the number of repetitive laboratory tasks necessary for studying the physicochemical characteristics of graphene nanoribbons with curved edges.


Introduction
Graphene has attracted tremendous consideration because of its unprecedented blend of electronic, mechanical, thermal and optical characteristics [1].Lately, graphene research has been extraordinarily dynamic, and the evolution in the manufacture and portrayal of graphene has been exceptionally critical [2].Among the numerous qualities of the 2D-material graphene, its high chargecarrier portability is one of the most significant as it permits graphene a guarantee of fantastic execution in field-effect transistors (FETs) [3][4][5].It has quickly grown to prominence in the field of science and technology.Because of its recognized electrical, compound and optical exhibitions, graphene and its subordinates have additionally created enormous interest in analytical chemistry.The vacancy of an energy bandgap, which is a sign of semiconductor materials, prevents these materials from being used in optoelectronic S. Prabhu, G. Murugan, M. Imran et al.   (5,7).
devices [6,7].Reducing one of the graphene sheet dimensions till it influences atomic sizes is one way to design a gap opening.Graphene nanoribbons (GNRs) are to bring in the next level of semiconductor applications [8].
Graphene nanoribbons appear as a cutting edge transporter for improving nano dimensional symptomatic gadgets and medication delivery frameworks because of the exciting and forefront electronic, warm, mechanical and optical properties related with graphene [9,10].They are exceptionally evolved graphenes with a broad significance because of their peculiar properties, such as enormous surface territory, upgraded mechanical quality, and improved electro-conductivity.These nanoribbons are the best transporter for anticancer medications and other exceptionally aromatic medications [11].The semi one-dimensional extended monolayer segments of graphenes have a high length to width proportion.The length and breadth estimations can be used to communicate the dimensions of the GNRs [12].They are synthetical  2 hybridised carbon structures with a honeycomb grid geometry.The edge structure is the important component in GNR characteristics, with armchair-edge GNRs (AGNRs) semiconducting along width-subordinate bandgaps and zigzag-edge GNRs (ZGNRs) probably appealing owing to their edge-restricted states, which can be turned polarised.Furthermore, armchair and zigzag edge configurations, alternative geometries such as cove edges [13,14] that result in non-planarity owing to the aversion amid nearby hydrogen molecules [15] can also be considered.See Fig. 1.
Recently [16][17][18][19], bottom-up synthesis of graphene permitted a cove edged graphene nanoribbons (CGNRs) as shown in Fig. 2 which is the combination of an armchair and zigzag peripheries.This construction was described as structurally explicit and remarkably long (> 200).Exchanging electron rich and electron-poor subunits tune the frontier orbitals of CGNRs, which are molecularly defined and soluble.In solution-made cove-edged nanoribbons have the added benefit of a strangely shaped, distorted  surface which is deeply dissolvable [20].When properly functionalized, solvent nanoribbons are admirable electron-transporting materials.These structures have been used in various fascinating classes containing electronic [21], chemical [22] and mechanical applications [23].
Cheminformatics assumes a vital part to keep up and accesses the gigantic measure of chemical data created by chemists utilizing a legitimate database.Furthermore, research requires a creative approach for extracting information from data to demonstrate complicated interconnections amid the structure of biological activity and a chemical molecule, as well as the impact of reaction conditions on chemical reactivity [24].Structure portrayal manages reaction characterization, structure descriptors and searching, molecular modelling, and computer-assisted structure elucidation.The interactions between countless chemical and, in particular, biological facts of substances and their structure are far too complicated to be reliably anticipated using fundamental principles.Structure descriptors (Topological indices) must be determined for the structures of a dataset [25].At that stage, data analysis or a model-building method must be used to create a scientific model for the connection between the topological indices and the explored property.For the two stages, a lot of strategies have been created.A large group of techniques, including in the large numbers, for computing structure descriptors, is accessible [26].Increasingly more consideration presently moves to the utilization of molecular indices which can be deciphered and consequently give a model that builds experiences into the connection between a compound's structure and properties.The atomic descriptors are helpful in portraying the 1D, 2D, 3D structure, or the sub-atomic surface properties [27].Furthermore, the representation of chemical molecules is given more attention than just a subatomic characterisation [28].
Inspired by the work of Wiener and Randić, the distance [29][30][31][32] based and degree [33][34][35][36][37] based indices are investigated by many researchers in the domain of molecular science.Also, several other topological indices for molecular graphs have been detailed.Wiener and Szeged type indices are instances of distance-based topological indices.Schultz and Gutman indices are degree-distance indices, were given this way.For recent research on this topic the reader can refer [38][39][40][41][42][43][44][45][46][47].There are also connectivity-based indices, such as the generalised connectivity index, Zagreb index, Estrada's ABC index, harmonic, sum-connectivity, and geometric-arithmetic indices, that follow Randić connectivity index.The dermal penetrations and octanol segment coefficients of aromatic compounds have been shown in [48] to correlate strongly with topological connectivity indices.Melting temperatures, boiling points, vapour pressures, chromatographic retention indices, toxicities, reported bioactivities, osmotic coefficients, and diffusion constants, on the other hand, have all been linked to both vertex-degree and edge-based topological indices.For more on the application part of topological indices, the reader can refer [49][50][51].This article presents the mathematical expressions of several distances and degree-based indices of a graphene nanoribbon with cove-type edges.

Graph-theoretical terminologies
The degree of a vertex   () is the number of edges that are connected to it, and the neighbourhood of a vertex  is the set of vertices that are adjacent to it,   ().Denote   (, ) as the minimum number of edges that connects ,  ∈  (), which is the formally called distance among them.In the same context,   (,  ) = min{  (, ),   (, )} for  ∈  ()  =  ∈ () and   (,  ) = min{  (,  ),   (,  )} for ,  ∈ (), where  =  and  = .If   (, ) =   ′ (, ) for a subgraph  ′ of , then  ′ is an isometric, and if all the graph geodesic between ,  ∈  ( ′ ) fall completely within  ′ , then  ′ is a convex.
(|) and   (|) are the subset of  () and () that are closer to , when compared to , and the cardinal number of these two sets are respectively represented by   (|) and   (|), for an edge  =  ∈ ℰ().The variables   (|) and   (|) are designated in the same way.
1. Wiener Type Indices: 2. Szeged Type Indices: () = 1 2 3. Degree and Distance-Based Type Indices: When dealing with distance-based TIs, the cut technique is quite useful [62,63].This technique is based on the Djoković-Winkler The relation  is symmetric and reflexive, although it is not transitive in general.For convenience, we denote {1, 2 … } as ℕ  .
where we use Also the numerical values presented in Tables 5 & 6 are depicted in Fig. 10.

Degree-based indices
This part of the paper gives the mathematical derivations of the degree-based TIs of CGNR based on the edge-partition of degree of end vertices that reported in Table 7.

Conclusion and future direction
Graphene nanoribbons hold great promise for advancing nanoscale technologies due to their unique combination of electronic, mechanical, and thermal properties.Continued research and development are expected to unlock new applications and improve the scalability and control of GNR production.The topological indices provide information about the structural characteristics of    the compound [64].Therefore, the results obtained in this study are crucial for comprehending the importance of these largesized aromatic compounds in various fields such as drug discovery, materials science, predictive toxicology, and astrochemistry.The expressions outlined in this article would aid in reducing the extent of repetitive laboratory tasks necessary for studying the physicochemical characteristics of graphene nanoribbons with curved edges.The future researcher could take up the other types of graphenes which has different type of fencing in their molecular structure.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Table 5
Computed numerical value for distance-based indices.

Table 9
Computed numerical value for degree-based indices.