Molecular Characterization of Three Classes of Bow-Tie-Shaped Graphene Nanoflakes by Polynomials as Alternative for Their Computed Energies and Spectral Patterns

Abstract

Graphene has been recognized as one of the most promising materials for nanoelectronics in recent decades. Despite its excellent mechanical and electrical characteristics, its use in the fabrication of genuine nanoelectronic devices is frequently fraught with challenges. A graphene nanoflake is graphene with a finite size. It can be prepared in a bottom-up or top-down manner. While it has qualities similar to those of graphene, it may be manufactured in a variety of sizes and shapes. A topological index is a numerical value that shows some valuable information about molecular structure, shape, and investigation. It refers to a molecular graph’s numerical invariants, which can correlate bioactivity and physio-chemical properties. Researchers have discovered topological indices to be an effective and valuable tool in describing molecular structure throughout time. In this study, we compute such topological indices of bow-tie-shaped graphene nanoflakes.

Introduction

Graphene has received considerable attention in recent decades, owing to its unusual electronic structure, which includes massless Dirac cones at the Fermi level. − Graphene is thought to be a semimetal, with only minor electronic correlations. Nonetheless, even before single-layer graphene was separated for the first material, fascinating theoretical investigations suggested that magnetic fields are formed by states concentrated at zigzag edges, even though the mass is semimetallic instability. − As a result, graphene is expected to play a key role in the next development of high-electricity and spintronic technologies.

Due to its excellent physicochemical properties, such as high chemical stability and excellent electromechanical properties, graphene has attracted a lot of interest in the field of composite materials and solid-state electronics. It is also a favorite candidate in the design of novel nanomaterials for engineering fuel cells, biosensors, energy storage devices, and gas sensors. − However, because many of the proposed applications are in semiconductor technologies, overcoming graphene’s semimetallicity and opening up a finite band gap is the key technological challenge in integrating graphene components. This will allow for better control of electron transport and band gap engineering. Nanostructuring graphene into graphene nanoribbons (GNRs) and graphene nanoflakes, as well as chemical functionalization, are two strategies to close the gap. Fortunately, both the GNR and armchair GNF of triangular, hexagonal, and other shapes have been widely examined from an experimental standpoint.

Graphene nanoflake, a graphene-derived quasi-zero-dimensional nanostructure, is a good contender for recent advances. The lower dimensionality of these structures causes quantum confinement, which opens a tunable bandgap. GNFs, arbitrarily formed graphene fragments with finite dimensions in both directions, have emerged as new materials for advancing electrical, optical, spintronic, and sensing devices in recent years. , GNF’s prospective applications are based on their quantum confinement and edge variation. This is because they may be sliced into a considerably wider range of shapes. , Corner states, which are produced where two edges meet, are unique to GNFs, in addition to the traits inherited from graphene and GNRs. Early research also suggests that small GNFs have a discrete electrical structure that transitions to a continuous band structure as their size increases. As a result of the fundamental shift in the nature of the electronic level, GNFs have the potential to be used in molecule to semi-infinite 2D electronic systems. Zigzag bow-tie-shaped graphene nanoflakes (GNFs) of three types are made out of two zigzag-edged triangular pieces that share hexagons as depicted in Figure . A zigzag-edged triangle with one topologically challenged sublattice, for example, has nonzero net magnetization but scales linearly with fragment size. By linking two separate triangular GNFs, bow-tie-shaped GNFs can exhibit antiferromagnetic (AFM) ordering, operating as a fundamental single-molecule NOT logic gate. , Observing and analyzing the magnetic properties of bow-tie-shaped GNFs, many related reports on magnetic states of hydrogenated diamond-shaped zigzag graphene quantum dots, inherent ferromagnetism in two-dimensional (2D) carbon semiconducting frameworks, magnetic-field control of magnetism in bow-tie-shaped GNFs, and appropriate spintronic models have been published. −

1.

Bow-tie graphene nanoflake. (a) ABTGN­(m,n); (b) SBTGN­(m,n); (c) TBTGN­(m,n).

For mathematical chemists, a chemical graph theory, a branch of graph theory, opens up a world of possibilities. In a molecular graph theory, vertices represent atoms, and edges represent bonds in a molecular graph. In chemistry, numerous chemical compounds share the same chemical formula but have differences that are visible in various analytical disciplines of chemistry. One of the main objectives of using the graph theory in chemistry is to quickly analyze quantitative structure–property or structure–activity interactions (QSPR/QSAR). , Researchers are interested in studying mathematical techniques derived from the molecular structures of these networks in QSPR/QSAR studies to work on the topology of a chemical network related to medical research, drug development, medicine, and experimental science. On a wide range of topics in bioinformatics and proteomics, topological indices have been utilized to construct and understand the mathematical properties of the real-world network model.

Here, the characterization of three classes of bow-tie-shaped graphene nanoflake is made with the help of topological descriptors. Furthermore, their HOMO–LUMO, delocalization energies, and NMR spectral patterns were computed.

Methodology

The molecular graph is defined with atom set V and bond set E. The term d _G (u,v) is defined as the distance between two atoms u and v. Thus, the distance between an atom u and a bond xy is given as the minimum of d _G (u,x) and d _G (u,y). Similarly, the distance between two bonds e and f is defined and denoted by $D_{{\mathit{G}}_{\mathrm{s}\mathrm{w}}}(e,f)$ . N _G (u) is the collection of atoms that are bonded to u, and the cardinality of this set is the valency of u. For a bond uv ∈ E, N _u(uv|G) and M _u (uv|G) are the collection of atoms and bonds of G, respectively, which are close to u than to v, and the cardinalities of these two collections are termed as n _u (uv|G) and m _u (uv|G), respectively.

The concept of strength-weighted graph was initially discussed in ref , and followed by this concept, many research papers appeared in the literature of the chemical graph theory in refs − . The strength-weighted graph is defined with three parameters, namely, the vertex-weight w _v , vertex-strength s _v and the edge-strength s _e . Details on the notations can be seen in the recent papers. ,,,

• (1)Wiener $W(G_{\mathrm{s}\mathrm{w}})=\sum _{\{u,v\}\subseteq V(G_{\mathrm{s}\mathrm{w}})}{w}_v(u)w_v(v)d_{G_{\mathrm{s}\mathrm{w}}}(u,v)$ .
• (2)Edge-Wiener $W_e(G_{\mathrm{s}\mathrm{w}})=\sum _{\{u,v\}\subseteq V(G_{\mathrm{s}\mathrm{w}})}{s}_v(u)s_v(v)d_{G_{\mathrm{s}\mathrm{w}}}(u,v)+\sum _{\{e,f\}\subseteq E(G_{\mathrm{s}\mathrm{w}})}{s}_e(e)s_e(f)D_{G_{\mathrm{s}\mathrm{w}}}(e,f)+\sum _{u\in V(G_{\mathrm{s}\mathrm{w}})}\sum _{f\in E(G_{\mathrm{s}\mathrm{w}})}{s}_v(u)s_e(f)d_{G_{\mathrm{s}\mathrm{w}}}(u,f)$ .
• (3)Vertex-edge-Wiener $W_{ve}(G_{\mathrm{s}\mathrm{w}})=\frac{1}{2}[\sum _{\{u,v\}\subseteq V(G_{\mathrm{s}\mathrm{w}})}\{w_v(u)s_v(v)+w_v(v)s_v(u)\}{d}_{G_{\mathrm{s}\mathrm{w}}}(u,v)+\sum _{u\in V(G_{\mathrm{s}\mathrm{w}})}\sum _{f\in E(G_{\mathrm{s}\mathrm{w}})}{w}_v(u)s_e(f)d_{G_{\mathrm{s}\mathrm{w}}}(u,f)]$ .
• (4)Vertex-Szeged $\mathit{S}{\mathit{z}}_{\mathit{v}}(G_{\mathrm{s}\mathrm{w}})=\sum _{e=uv\in E(G_{\mathrm{s}\mathrm{w}})}{s}_e(e)n_u(e|G_{\mathrm{s}\mathrm{w}})n_v(e|G_{\mathrm{s}\mathrm{w}})$ .
• (5)Edge-Szeged $\mathit{S}{\mathit{z}}_{\mathit{e}}(G_{\mathrm{s}\mathrm{w}})=\sum _{e=uv\in E(G_{\mathrm{s}\mathrm{w}})}{s}_e(e)m_u(e|G_{\mathrm{s}\mathrm{w}})m_v(e|G_{\mathrm{s}\mathrm{w}})$ .
• (6)Edge-vertex-Szeged $\mathit{S}{\mathit{z}}_{\mathit{e}\mathit{v}}(G_{\mathrm{s}\mathrm{w}})=\frac{1}{2}\sum _{e=uv\in E(G_{\mathrm{s}\mathrm{w}})}{s}_e(e)[n_u(e|G_{\mathrm{s}\mathrm{w}})m_v(e|G_{\mathrm{s}\mathrm{w}})+n_v(e|G_{\mathrm{s}\mathrm{w}})m_u(e|G_{\mathrm{s}\mathrm{w}})]$ .
• (7)Total-Szeged $\mathit{S}{\mathit{z}}_{\mathit{t}}(G_{\mathrm{s}\mathrm{w}})=S{z}_v(G_{\mathrm{s}\mathrm{w}})+S{z}_e(G_{\mathrm{s}\mathrm{w}})+2S{z}_{ev}(G_{\mathrm{s}\mathrm{w}})$ .
• (8)Padmakar-Ivan $\mathrm{P}\mathrm{I}(G_{\mathrm{s}\mathrm{w}})=\sum _{e=uv\in E(G_{\mathrm{s}\mathrm{w}})}{s}_e(e)[m_u(e|G_{\mathrm{s}\mathrm{w}})+m_v(e|G_{\mathrm{s}\mathrm{w}})]$ .
• (9)Schultz $S(G_{\mathrm{s}\mathrm{w}})=\sum _{\{u,v\}\subseteq V(G_{\mathrm{s}\mathrm{w}})}[w_v(v)(d_{G_{\mathrm{s}\mathrm{w}}}(u)+2s_v(u))+w_v(u)(d_{G_{\mathrm{s}\mathrm{w}}}(v)+2s_v(v))]d_{G_{\mathrm{s}\mathrm{w}}}(u,v)$ .
• (10)Gutman $\mathrm{G}\mathrm{u}\mathrm{t}(G_{\mathrm{s}\mathrm{w}})=\sum _{\{u,v\}\subseteq V(G_{\mathrm{s}\mathrm{w}})}[(d_{G_{\mathrm{s}\mathrm{w}}}(u)+2s_v(u))(d_{G_{\mathrm{s}\mathrm{w}}}(v)+2s_v(v))]d_{G_{\mathrm{s}\mathrm{w}}}(u,v)$ .

The cut method is vital for discussing distance-based topological indices. , Very frequently, it was applied to benzenoid frameworks , and other antikekulene frameworks effectively. The following are the valency-based indices discussed.

• (1)First Zagreb: ${\mathit{M}}_1(G)=\sum _{u\in V(G)}d{(u)}^2$ .
• (2)Second Zagreb: ${\mathit{M}}_2(G)=\sum _{uv\in E(G)}d(u)d(v)$ .
• (3)Hyper Zagreb: $\mathrm{H}\mathrm{M}(G)=\sum _{uv\in E(G)}{[d(u)+d(v)]}^2$ .
• (4)Augmented Zagreb: $\mathrm{A}\mathrm{Z}(G)=\sum _{uv\in E(G)}{\left(\frac{d(u)d(v)}{d(u)+d(v)-2}\right)}^3$ .
• (5)Atom bond connectivity: $\mathrm{A}\mathrm{B}\mathrm{C}(G)=\sum _{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}}$ .
• (6)Harmonic: $\mathit{H}(G)=\sum _{uv\in E(G)}\frac{2}{d(u)+d(v)}$ .
• (7)Sum-connectivity: $\mathrm{S}\mathrm{C}(G)=\sum _{uv\in E(G)}\frac{1}{\sqrt{d(u)+d(v)}}$ .
• (8)Geometric arithmetic: $\mathrm{G}\mathrm{A}(G)=\sum _{uv\in E(G)}2\left(\frac{\sqrt{d(u)d(v)}}{d(u)+d(v)}\right)$ .
• (9)Inverse sum indegree: $\mathrm{I}\mathrm{S}\mathrm{I}(G)=\sum _{uv\in E(G)}\left(\frac{d(u)d(v)}{d(u)+d(v)}\right)$ .
• (10)Symmetric division degree: $\mathrm{S}\mathrm{D}\mathrm{D}(G)=\sum _{uv\in E(G)}\left(\frac{d^2(u)+d^2(v)}{d(u)d(v)}\right)$ .
• (11)Forgotten: $\mathit{F}(G)=\sum _{uv\in E(G)}(d^2(u)+d^2(v))$ .
• (12)Sombor: $\mathrm{S}\mathrm{O}(G)=\sum _{uv\in E(G)}\sqrt{d_u^2+d_v^2}$ .

Main Results

Theorem 1. Let ABTGN­(m,n) be an asymmetric bow-tie graphene nanoflake of dimension m, n. Then,

• (1) W(ABTGN(m,n)) = (2m ⁵– 10m ⁴ n– 5m ⁴+ 20m ³ n ²– 10m ³+ 120m ² n ²+ 250m ² n+ 50m ²–30mn ⁴– 320mn ³– 860mn ²– 490mn– 52m+ 52n ⁵+ 460n ⁴+ 1230n ³+ 980n ²+ 233n+ 15)/15.
• (2) We(ABTGN(m,n)) = (18m ⁵ – 90m ⁴ n – 45m ⁴ + 180m ³ n ² + 60m ³ n + 105m ³ + 720m ² n ² + 1050m ² n – 105m ² – 270mn ⁴ – 2100mn ³ – 3690mn ² – 120mn + 52m + 468n ⁵ + 2880n ⁴ +4560n ³ + 570n ² + 162n)/60.
• (3) W_ve (ABTGN(m,n)) = (6m ⁵ – 30m ⁴ n – 15m ⁴ + 60m ³ n ² + 10m ³ n – 10m ³ + 300m ² n ² + 525m ² n + 30m ² – 90mn ⁴ – 830mn ³ – 1815mn ² – 560mn – 11m + 156n ⁵ + 1170n ⁴ +2470n ³ + 1185n ² + 149n)/30.
• (4) Sz_v (ABTGN(m,n)) = (6m ⁴ n – 24m ³ n ² – 16m ³ n + 24m ² n ³ + 48m ² n ² + 84m ² n + 12m ² – 30mn ⁴ – 192mn ³ – 402mn ² – 194mn – 18m +12n ⁶ + 132n ⁵ + 513n ⁴ + 840n ³ + 537n ² + 120n + 6)/6.
• (5) Sz _e (ABTGN(m,n)) = (6m ⁵ + 60m ⁴ n + 25m ⁴ – 360m ³ n ² – 340m ³ n + 80m ³ + 540m ² n ³ + 1230m ² n ² + 930m ² n – 295m ² – 720mn ⁴ – 3280mn ³ – 3990mn ² + 470mn + 124m + 270n ⁶ + 2226n ⁵ + 6210n ⁴ + 5930n ³ + 210n ² + 274n)/60.
• (6) Sz_ev (ABTGN(m,n)) = (2m ⁵ + 65m ⁴ n + 15m ⁴ – 300m ³ n ² – 290m ³ n – 10m ³ + 360m ² n ³ + 900m ²n² + 1045m ² n – 15m ² – 465mn ⁴ – 2570mn ³ – 4185mn ² – 920mn + 8m + 180n ⁶ + 1732n ⁵ + 5755n ⁴ + 7500n ³ + 2825n ² + 368n)/60.
• (7) Sz_t (ABTGN(m,n)) = (10m ⁵ + 250m ⁴ n + 55m ⁴ – 1200m ³ n ² – 1080m ³ n + 60m ³ + 1500m ² n ³ + 3510m ² n ² + 3860m ² n – 205m ² – 1950mn ⁴ – 10340mn ³ – 16380mn ² – 3310mn – 40m + 750n ⁶ + 7010n ⁵ + 22850n ⁴ + 29330n ³ + 11230n ² + 2210n + 60)/60.
• (8)PI­(ABTGN(m,n)) = (−2m ³ + 6m ² n + 15m ² – 42mn ² – 114mn – 7m + 27n ⁴ + 156n ³ + 234n ² + 15n)/3.
• (9) S(ABTGN(m,n)) = (12m ⁵ – 60m ⁴ n – 30m ⁴ + 120m ³ n ² + 20m ³ n – 20m ³ + 600m ² n ² + 1050m ² n + 120m ² – 180mn ⁴ – 1660mn ³ – 3780mn ² – 1630mn – 112m + 312n ⁵ + 2430n ⁴ + 5570n ³ + 3570n ² + 688n + 30)/15.
• (10)Gut­(ABTGN(m,n)) = (18m ⁵ – 90m ⁴ n – 45m ⁴ + 180m ³ n ² + 60m ³ n +10m ³ + 720m ² n ² + 1080m ² n + 30m ² – 270mn ⁴ – 2100mn ³ – 4080mn ² – 1230mn – 43m + 468n ⁵ + 3150n ⁴ + 6150n ³ + 3045n ² + 507n +15)/15.

Proof. Let the asymmetric bow-tie graphene nanoflake graph consist of |V| = 2n ² + 8n – 2m + 2 and |E| = 3n ² + 9n – 2m + 1. We now present the Θ classes of ABTGN­(m,n) and their corresponding components. We first consider the acute type (AT) of ABTGN­(m,n) and group them into two categories of Θ classes, see Figure . The first type of AT is denoted by {A _1i : 1 ≤ i ≤n – m} and second is referred as {A _{2 i} : 1 ≤ i ≤ m}. Next we consider the obtuse type (OT) which is depicted in Figure and the ranges are denoted by {O _{1 i} : 1 ≤ i ≤ n}. Finally, the vertical type (VT) Θ classes are represented in two categories such as {V _1i : 1 ≤ i ≤ n – m} and {V _{2 i} : 1 ≤ i ≤ m + 1}, see Figure . We derive the several topological indices of ABTGN­(m,n) using the quotient values as shown in Table .

2.

Asymmetric bow-tie graphene nanoflake. (a) Acute cuts; (b) G|A _1i ; (c) G|A_{2 i} .

3.

Asymmetric bow-tie graphene nanoflake. (a) Obtuse cuts; (b) G|O _1i .

4.

Asymmetric bow-tie graphene nanoflake. (a) Vertical cuts; (b) G|V _1i ; (c) G|V_{2 i} .

1. Quotient Values of ABTGN­(m,n) .

Quotient Graph	w v	s v
$$\begin{array}{c}{A}_{1i}\\ 1\le i\le n-m\end{array}$$	$$\begin{array}{c}{a}_1=i^2+2i\\ a_2=V-a_1\end{array}$$	$$\begin{array}{c}{b}_1=(3i^2+3i-2)/2\\ b_2=E-b_1-i-1\end{array}$$
$$\begin{array}{c}{A}_{2i}\\ 1\le i\le m\end{array}$$	$$\begin{array}{c}{a}_3=2n-2m+i(4n-2m+4)+(m-n{)}^2-1\\ a_4=V-a_3\end{array}$$	$$\begin{array}{c}{b}_3=({3n}^2+{3m}^2-6mn+n-3m+12ni-6mi+10i-6)/2\\ b_4=E-b_3-2n+m-2\end{array}$$
$$\begin{array}{c}{O}_{1i}\\ 1\le i\le n\end{array}$$	$$\begin{array}{c}{a}_5=i^2+2i\\ a_6=V-a_5\end{array}$$	$$\begin{array}{c}{b}_5=(3i^2+3i-2)/2\\ b_6=E-b_5-i-1\end{array}$$
$$\begin{array}{c}{V}_{1i}\\ 1\le i\le n-m\end{array}$$	$$\begin{array}{c}{a}_7=2ni-i^2+4i-2\\ a_8=V-a_7\end{array}$$	$$\begin{array}{c}{b}_7=(6ni-3i^2-2n+11i-8)/2\\ b_8=E-b_7-n+i-2\end{array}$$
$$\begin{array}{c}{V}_{2i}\\ 1\le i\le m+1\end{array}$$	$$\begin{array}{c}{a}_9=2i-4m+4n+2im-m^2+n^2-1\\ a_{10}=V-a_9\end{array}$$	$$\begin{array}{c}{b}_9=(4i-11m+9n+6im-3m^2+3n^2-4)/2b_{10}=E-\\ b_9-m-1\end{array}$$

$$\begin{array}{lll}W(ABTGN(m,n))& =2\sum _{i=1}^{n-m}{a}_1{a}_2+\sum _{i=1}^m{a}_3{a}_4+2\sum _{i=1}^n{a}_5{a}_6+2\sum _{i=1}^{n-m}{a}_7{a}_8+\sum _{i=1}^{m+1}{a}_9{a}_{10}.& \\ W_e(ABTGN(m,n))& =2\sum _{i=1}^{n-m}{b}_1{b}_2+\sum _{i=1}^m{b}_3{b}_4+2\sum _{i=1}^n{b}_5{b}_6+2\sum _{i=1}^{n-m}{b}_7{b}_8+\sum _{i=1}^{m+1}{b}_9{b}_{10}.& \\ W_{ve}(ABTGN(m,n))& =\frac{1}{2}[2\sum _{i=1}^{n-m}(a_1{b}_2+a_2{b}_1)+\sum _{i=1}^m(a_3{b}_4+a_4{b}_3)+2\sum _{i=1}^n(a_5{b}_6+a_6{b}_5)& \\ & +2\sum _{i=1}^{n-m}(a_7{b}_8+a_8{b}_7)+\sum _{i=1}^{m+1}(a_9{b}_{10}+a_{10}{b}_9)].& \\ S{z}_v(ABTGN(m,n))& =2\sum _{i=1}^{n-m}{e}_1{a}_1{a}_2+\sum _{i=1}^m{e}_2{a}_3{a}_4+2\sum _{i=1}^n{e}_3{a}_5{a}_6+2\sum _{i=1}^{n-m}{e}_4{a}_7{a}_8+\sum _{i=1}^{m+1}{e}_5{a}_9{a}_{10}.& \\ S{z}_e(ABTGN(m,n))& =2\sum _{i=1}^{n-m}{e}_1{b}_1{b}_2+\sum _{i=1}^m{e}_2{b}_3{b}_4+2\sum _{i=1}^n{e}_3{b}_5{b}_6+2\sum _{i=1}^{n-m}{e}_4{b}_7{b}_8+\sum _{i=1}^{m+1}{e}_5{b}_9{b}_{10}.& \\ S{z}_{ev}(ABTGN(m,n))& =\frac{1}{2}[2\sum _{i=1}^{n-m}{e}_1(a_1{b}_2+a_2{b}_1)+\sum _{i=1}^m{e}_2(a_3{b}_4+a_4{b}_3)+2\sum _{i=1}^n{e}_3(a_5{b}_6+a_6{b}_5)& \\ & +2\sum _{i=1}^{n-m}{e}_4(a_7{b}_8+a_8{b}_7)+\sum _{i=1}^{m+1}{e}_5(a_9{b}_{10}+a_{10}{b}_9)].& \\ \mathrm{P}\mathrm{I}(ABTGN(m,n))& =2\sum _{i=1}^{n-m}{e}_1(b_1+b_2)+\sum _{i=1}^m{e}_2(b_3+b_4)+2\sum _{i=1}^n{e}_3(b_5+b_6)& \\ & +2\sum _{i=1}^{n-m}{e}_4(b_7+b_8)+\sum _{i=1}^{m+1}{e}_5(b_9+b_{10}).& \\ S(ABTGN(m,n))& =2\sum _{i=1}^{n-m}[a_1(2b_2+e_1)+a_2(2b_1+e_1)]+\sum _{i=1}^m[a_3(2b_4+e_2)+a_4(2b_3+e_2)]& \\ & +2\sum _{i=1}^n[a_5(2b_6+e_3)+a_6(2b_5+e_3)]+2\sum _{i=1}^{n-m}[a_7(2b_8+e_4)+a_8(2b_7+e_4)]& \\ & +\sum _{i=1}^{m+1}[a_9(2b_{10}+e_5)+a_{10}(2b_9+e_5)].& \\ \mathrm{G}\mathrm{u}\mathrm{t}(ABTGN(m,n))& =2\sum _{i=1}^{n-m}(2b_2+e_1)(2b_1+e_1)+\sum _{i=1}^m(2b_4+e_2)(2b_3+e_2)& \\ & +2\sum _{i=1}^n(2b_6+e_3)(2b_5+e_3)+2\sum _{i=1}^{n-m}(2b_8+e_4)(2b_7+e_4)& \\ & +\sum _{i=1}^{m+1}(2b_{10}+e_5)(2b_9+e_5).\end{array}$$

Now, the following theorem gives the exact values of degree-based indices for asymmetric bow-tie graphene nanoflakes derived from the edge partition given in Table along with the degree-based indices expression discussed in Methodology Section.

2. Different Types of Edges in ABTGN­(m,n) .

S.No	E i	(d(u),d(v))	|E i |
1	E 1	(2,2)	8
2	E 2	(2,3)	12n – 4m – 8
3	E 3	(3,3)	3n 2 – 3n + 2m + 1

Theorem 2. Let ABTGN­(m,n) be an asymmetric bow-tie graphene nanoflake. Then

• (1) M ₁ (ABTGN(m,n)) = 18n ²+ 42n– 8m– 2.
• (2) M ₂(ABTGN(m,n)) = 27n ² + 45n – 6m – 7.
• (3)HM­(ABTGN(m,n)) = 108n ² + 192n – 28m – 36.
• (4) $\mathrm{A}\mathrm{Z}(ABTGN(m,n))=\frac{1}{64}(2187n^2+3957n-590m+729).$
• (5) $\mathrm{A}\mathrm{B}\mathrm{C}(ABTGN(m,n))=\frac{2}{3}(3n^2-(3-9\sqrt{2})n+m(3\sqrt{2}-2)+1).$
• (6) $H(ABTGN(m,n))=\frac{1}{15}(15n^2+57n-14m+17).$
• (7) $\mathrm{S}\mathrm{C}(ABTGN(m,n))=\frac{-1}{30}(-15\sqrt{6}{n}^2+(15\sqrt{6}-72\sqrt{5})n+48\sqrt{5}-5\sqrt{6}+m(24\sqrt{5}-10\sqrt{6})-120).$
• (8) $\mathrm{G}\mathrm{A}(ABTGN(m,n))=\frac{-1}{5}(-15n^2+(15-24\sqrt{6})n+16\sqrt{6}+m(8\sqrt{6}-10)-45).$
• (9) $\mathrm{I}\mathrm{S}\mathrm{I}(ABTGN(m,n))=\frac{1}{10}(45n^2+99n-18m-1).$
• (10) $\mathrm{S}\mathrm{D}\mathrm{D}(ABTGN(m,n))=\frac{-2}{3}(-9n^2-30n+7m-1).$
• (11) F(ABTGN(m,n)) = −2­(−27n ² – 51n + 8m + 11).
• (12) $\mathrm{S}\mathrm{O}(ABTGN(m,n))=9\sqrt{2}{n}^2+(12\sqrt{13}-9\sqrt{2})n+19\sqrt{2}-8\sqrt{13}+m(6\sqrt{2}-4\sqrt{13}).$

Theorem 3. Let SBTGN­(m,n) be a symmetric bow-tie graphene nanoflake of dimension m,n. Then,

• (1) W(SBTGN(m,n)) = (28m ⁵ – 100m ⁴ n – 60m ⁴ + 160m ² n ² + 240m ³ n – 80m ² n ² + 570m ² n + 180m ² – 60mn ⁴ – 640mn ³ – 1720mn ² – 980mn – 103m + 52n ⁵ + 460n ⁴ + 1230n ³ + 980n ² + 233n + 15)/15.
• (2) We(SBTGN(m,n)) = (126m ⁵ – 450m ⁴ n – 390m⁴ + 720m ³ n ² + 960m ³ n – 260m ³ – 360m² n ³ + 90m ² n ² + 1920m ² n – 195m ² – 270mn ⁴ – 2100mn ³ – 3690mn ² – 120mn + 59m + 234n ⁵ + 1440n ⁴ + 2280n ³ + 285n ² + 81n)/30.
• (3) W_{v e} (SBTGN(m,n)) = (84m ⁵ + 300m ⁴ n – 220m ⁴ + 480m ³ n ² + 680m ³ n – 110m ³ – 240m ² n ³ + 30m ² n ² + 1470m ² n + 145m ² – 180mn ⁴ – 1660mn ³ – 3630mn ² – 1120mn – 19m + 156n ⁵ + 1170n ⁴ + 2470n ³ + 1185n ² + 149n)/30.
• (4) Sz_v (SBTGN(m,n)) = (−12m ⁶ – 108m ⁵ + 36m ⁴ n ² + 276m ⁴ n – 19m ⁴ – 24m ³ n ² + 400m ³ n + 124m ³ – 36m ² n ⁴ – 216m ² n ³ – 510m ² n ² – 72m ² n – 5m ² – 60mn ⁴ – 384mn ³ – 840mn ² – 388mn – 34m + 12n ⁶ + 132n ⁵ + 513n ⁴ + 840n ³ + 537n ² + 120n + 6)/6.
• (5) Sz _e(SBTGN(m,n)) = (−135m ⁶ – 498m ⁵ + 404m ⁴ n ² + 1515m ⁴ n – 595m ⁴ – 30m ³ n ² + 1910m ³ n + 115m ³ – 405m ² n ⁴ – 1380m ² n ³ – 1140m ² n ² + 1545m ² n – 710m ² – 720mn ⁴ – 3280mn ³ – 4005mn ² + 425mn + 143m + 135n ⁶ + 1113n ⁵ + 3105n ⁴ + 2965n ³ + 105n ² + 137n)/30.
• (6) Sz_ev (SBTGN(m,n)) = (−180m ⁶ – 1142m ⁵ + 540m ⁴ n ² + 308m ⁴ n – 285m ⁴ – 200m ³ n ² + 3420m ³ n + 360m ³ – 540m ² n ⁴ – 2540m ² n ³ – 3750m ² n ² + 1560m ² n – 225m ² – 930mn ⁴ – 5140mn ³ – 8380mn ² – 1880mn + 32m + 180n ⁶ + 1732n ⁵ + 5755n ⁴ + 7500n ³ + 2825n ² + 368n)/60.
• (7) Sz_t (SBTGN(m,n)) = (−75m ⁶ – 436m ⁵ + 225m ⁴ n ² + 1195m ⁴ n – 195m ⁴ – 70m ³ n ² + 1466m ³ n + 219m ³ – 225m ² n ⁴ – 1000m ² n ³ – 1488m ² n ² + 549m ² n – 192m ² – 390mn ⁴ – 2068mn ³ – 3281mn ² – 679mn + m + 75n ⁶ + 701n ⁵ + 2285n ⁴ + 2933n ³ + 1123n ² + 221n + 6)/6.
• (8)PI­(SBTGN(m,n)) = 9m ⁴ + 18m ³ – 18m ² n ² – 42m ² n + 22m ² – 28mn ² – 76mn – 5m + 9n ⁴ + 52n ³ + 78n ² + 5n.
• (9) S(SBTGN(m,n)) = (168m ⁵ – 600m ⁴ n – 350m ⁴ + 960m ³ n ² + 1360 m ³ n + 80m ³ – 480m ² n ³ – 120m ² n ² + 2310m ² n + 410m ² – 360mn ⁴ – 3320mn ³ – 7560mn ² – 3260mn – 218m + 312n ⁵ + 2430n ⁴ + 5570n ³ + 3570n ² + 688n + 30)/15.
• (10)Gut­(SBTGN(m,n)) = (252m ⁵ – 900m ⁴ n – 510m ⁴ + 1440m ³ n ² + 1920m ³ n + 110m ³ – 720m ² n ³ – 360m ² n ² + 2400m ² n + 90m ² – 540mn ⁴ – 4200mn ³ – 8160mn ² – 2460mn – 77m + 468n ⁵ + 3150n ⁴ + 6150n ³ + 3045n ² + 507n + 15)/15.

Proof: Let the symmetric bow-tie graphene nanoflake graph consist of |V| = 2n ² – 2m ² + 8n – 4m + 2 and |E| = 3n ² – 3m ² + 9n – 4m + 1. We now present the Θ classes of SBTGN­(m,n) and their corresponding components. We first consider the acute type (AT) of SBTGN­(m,n) and group them into two categories of Θ classes, see Figure . The first type of AT is denoted by {A _{1 i} : 1 ≤ i ≤ n – m} and the second is referred to as {A _{2 i} : 1 ≤ i ≤ m}. Next we consider {O_{1 i} : 1 ≤ i ≤ m} and {O _{2 i} : 1 ≤ i ≤ m} (obtuse type). Finally, the vertical type (VT) Θ classes are represented into two categories such as {V _{1 i} : 1 ≤ i ≤ n – m} and {V}, see Figure . In this graph, the types AT and OT are symmetric. Now, we derive the several topological indices of SBTGN­(m,n) using the quotient values as shown in Table .

5.

Symmetric bow-tie graphene nanoflake. (a) Acute cuts; (b) G|A _1i ; (c) G|A _2i .

6.

Symmetric bow-tie graphene nanoflake. (a) Vertical cuts; (b) G|V _1i ; (c) G|V.

3. Quotient Values of SBTGN­(m,n) .

G sw	w v	s v
$$\begin{array}{c}{A}_{1i}\\ 1\le i\le n-m\end{array}$$	$$\begin{array}{c}{a}_1=i^2+2i\\ a_2=V-a_1\end{array}$$	$$\begin{array}{c}{b}_1=(3i^2+3i-2)/2\\ b_2=E-b_1-i-1\end{array}$$
$$\begin{array}{c}{A}_{2i}\\ 1\le i\le m\end{array}$$	$$\begin{array}{c}{a}_3=4i-2m+2n-4im+4in-2mn+m^2+n^2-1\\ a_4=V-a_3\end{array}$$	$$\begin{array}{c}{b}_3=-(m-10i-n+12im-12in+6mn-3m^2-3n^2+6)/2\\ b_4=E-b_3-2(n-m+1)\end{array}$$
$$\begin{array}{c}{V}_{1i}\\ 1\le i\le n-m\end{array}$$	$$\begin{array}{c}{a}_5=2ni-i^2+4i-2\\ a_6=V-a_5\end{array}$$	$$\begin{array}{c}{b}_5=(6ni-3i^2-2n+11i-8)/2\\ b_6=E-b_5-n+i-2\end{array}$$
V	$$\begin{array}{c}{a}_7=\frac{V}{2}\\ a_8=V-a_7\end{array}$$	$$\begin{array}{c}{b}_7=\frac{E-m-1}{2}\\ b_8=E-b_7-(m+1)\end{array}$$

$$\begin{array}{lll}W(SBTGN(m,n))& =4\sum _{i=1}^{n-m}{a}_1{a}_2+2\sum _{i=1}^m{a}_3{a}_4+2\sum _{i=1}^{n-m}{a}_5{a}_6+a_7{a}_8.& \\ W_e(SBTGN(m,n))& =4\sum _{i=1}^{n-m}{b}_1{b}_2+2\sum _{i=1}^m{b}_3{b}_4+2\sum _{i=1}^{n-m}{b}_5{b}_6+b_7{b}_8.& \\ W_{ve}(SBTGN(m,n))& =\frac{1}{2}[4\sum _{i=1}^{n-m}(a_1{b}_2+a_2{b}_1)+2\sum _{i=1}^m(a_3{b}_4+a_4{b}_3)+2\sum _{i=1}^{n-m}(a_5{b}_6+a_6{b}_5)+(a_7{b}_8+a_8{b}_7)].& \\ S{z}_v(SBTGN(m,n))& =4\sum _{i=1}^{n-m}{e}_1{a}_1{a}_2+2\sum _{i=1}^m{e}_2{a}_3{a}_4+2\sum _{i=1}^{n-m}{e}_3{a}_5{a}_6+e_4{a}_7{a}_8.& \\ S{z}_e(SBTGN(m,n))& =4\sum _{i=1}^{n-m}{e}_1{b}_1{b}_2+2\sum _{i=1}^m{e}_2{b}_3{b}_4+2\sum _{i=1}^{n-m}{e}_3{b}_5{b}_6+e_4{b}_7{b}_8.& \\ S{z}_{ev}(SBTGN(m,n))& =\frac{1}{2}[4\sum _{i=1}^{n-m}{e}_1(a_1{b}_2+a_2{b}_1)+2\sum _{i=1}^m{e}_2(a_3{b}_4+a_4{b}_3)& \\ & +2\sum _{i=1}^{n-m}{e}_3(a_5{b}_6+a_6{b}_5)+e_4(a_7{b}_8+a_8{b}_7)].& \\ \mathrm{P}\mathrm{I}(SBTGN(m,n))& =4\sum _{i=1}^{n-m}{e}_1(b_1+b_2)+2\sum _{i=1}^m{e}_2(b_3+b_4)+2\sum _{i=1}^{n-m}{e}_3(b_5+b_6)+e_4(b_7+b_8).& \\ S(SBTGN(m,n))& =4\sum _{i=1}^{n-m}[a_1(2b_2+e_1)+a_2(2b_1+e_1)]+2\sum _{i=1}^m[a_3(2b_4+e_2)+a_4(2b_3+e_2)]& \\ & +2\sum _{i=1}^{n-m}[a_5(2b_6+e_3)+a_6(2b_5+e_3)]+[a_7(2b_8+e_4)+a_8(2b_7+e_4)].& \\ \mathrm{G}\mathrm{u}\mathrm{t}(SBTGN(m,n))& =4\sum _{i=1}^{n-m}(2b_2+e_1)(2b_1+e_1)+2\sum _{i=1}^m(2b_4+e_2)(2b_3+e_2)& \\ & +2\sum _{i=1}^{n-m}(2b_6+e_3)(2b_5+e_3)+(2b_8+e_4)(2b_7+e_4).\end{array}$$

We shall now compute the degree-based indices of symmetric bow-tie graphene nanoflake from the edge partition given in Table along with the degree-based index expression discussed in Methodology Section.

4. Different Types of Edges in SBTGN­(m,n) .

S. No	E i	(d(u), d(v))	|E i |
1	E 1	(2,2)	8
2	E 2	(2,3)	12n – 8m – 8
3	E 3	(3,3)	3n 2 – 3m 2 – 3n + 4m + 1

Theorem 4. Let SBTGN­(m,n) be a symmetric bow-tie graphene nanoflake. Then

• (1) M ₁(SBTGN(m,n)) = −18m ² – 16m + 18n ² + 42n – 2.
• (2) M ₂(SBTGN(m,n)) = −27m ² – 12m + 27n ² + 45n – 7.
• (3)HM­(SBTGN(m,n)) = −4­(27m ² + 14m – 27n ² – 48n + 9).
• (4) $\mathrm{A}\mathrm{Z}(SBTGN(m,n))=\frac{-1}{64}(2187m^2+1180m-2187n^2-3957n-729).$
• (5) $\mathrm{A}\mathrm{B}\mathrm{C}(SBTGN(m,n))=\frac{-2}{3}(3m^2+(6\sqrt{2}-4)m-3n^2+(3-9\sqrt{2})n-1).$
• (6) $H(SBTGN(m,n))=\frac{-1}{15}(15m^2+28m-15n^2-57n-17).$
• (7) $\mathrm{S}\mathrm{C}(SBTGN(m,n))=\frac{-1}{30}(15\sqrt{6}{m}^2+(48\sqrt{5}-20\sqrt{6})m-15\sqrt{6}{n}^2+(15\sqrt{6}-72\sqrt{5})n+48\sqrt{5}-5\sqrt{6}-120).$
• (8) $\mathrm{G}\mathrm{A}(SBTGN(m,n))=\frac{-1}{5}(15m^2+(16\sqrt{6}-20)m-15n^2+(15-24\sqrt{6})n+16\sqrt{6}-45).$
• (9) $\mathrm{I}\mathrm{S}\mathrm{I}(SBTGN(m,n))=\frac{-1}{10}(45m^2+36m-45n^2-99n+1).$
• (10) $\mathrm{S}\mathrm{D}\mathrm{D}(SBTGN(m,n))=\frac{-2}{3}(9m^2+14m-9n^2-30n-1).$
• (11) F(SBTGN(m,n)) = −2­(27m ² + 16m – 27n ² – 51n + 11).
• (12) $\mathrm{S}\mathrm{O}(SBTGN(m,n))=-(9\sqrt{2}{m}^2+(8\sqrt{13}-12\sqrt{2})m-9\sqrt{2}{n}^2+(9\sqrt{2}-12\sqrt{13})n-19\sqrt{2}+8\sqrt{13}).$ .

Theorem 5. Let TBTGN­(m,n) be a tri bow-tie graphene nanoflake of dimension m,n. Then,

• (1) W(TBTGN(m,n)) = (4m ⁵+ 40m ⁴+ 40m ³ n ²+ 160m ³ n– 85m ³+ 40m ² n ³+ 240m ² n ²+ 365m ² n– 550m ²+ 60mn ⁴+ 640mn ³+ 725mn ²– 2840mn+ 1431m+ 92n ⁵+ 590n ⁴– 5n ³– 2630n ²+ 2673n– 720)/10.
• (2) W_e (TBTGN(m,n)) = (36m ⁵ + 225m ⁴ + 360m ³ n ² + 1080m ³ n – 1170m ³ + 360m ² n ³ + 1170m ² n ² + 270m ² n – 1965m ² + 540mn ⁴ + 4320mn ³ + 810mn ² – 18750mn + 13194m + 828n ⁵ + 3195n ⁴ – 6030n ³ – 13455n² + 29622n – 14160)/40.
• (3) W_ve (TBTGN(m,n)) = (24m ⁵ + 195m ⁴ + 240m³ n ² + 840m ³ n – 680m ³ + 240m ² n ³ + 1110m ² n ² + 1020m ² n – 2265m ² + 360mn ⁴ + 3360mn ³ + 2100mn ² – 14790mn + 8966m + 552n ⁵ + 2835n ⁴ – 2460n ³ – 12405n ² + 18798n – 7320)/40.
• (4) Sz_v (TBTGN(m,n)) = (m6 + 12m ⁵ + 9m ⁴ n ² + 36m ⁴ n + 4m ⁴ + 60m ³ n ² + 240m ³ n – 216m ³ + 21m ² n ⁴ + 180m ² n ³ + 246m ² n ² – 312m ² n – 77m ² + 72mn ⁴ + 624mn ³ + 696mn ² – 2976mn + 1596m + 21n ⁶ + 228n ⁵ + 714n ⁴ – 768n ³ – 3147n ² + 4608n – 1656)/4.
• (5) Sz_e (TBTGN(m,n)) = (45m ⁶ + 369m ⁵ + 405m ⁴ n ² + 1215m ⁴ n – 675m ⁴ + 2130m ³ n ² + 5670m ³ n – 7225m ³ + 945m ² n ⁴ + 6450m ² n ³ + 2610m ² n ² – 16905m ² n + 6510m ² + 2835mn ⁴ + 17190mn ³ + 1395mn ² – 72420mn + 51376m + 945n ⁶ + 7797n ⁵ + 13905n ⁴ – 47805n ³ – 52410n ² + 152928n – 75360)/80.
• (6) Sz_ev (TBTGN(m,n)) = (30m ⁶ + 303m ⁵ + 270m ⁴ n ² + 945m ⁴ n – 210m ⁴ + 1610m ³ n ² + 5300m ³ n – 5695m ³ + 630m ² n ⁴ + 4850m ² n ³ + 4380m ² n ² – 11135m ² n + 1560m ² + 2025mn ⁴ + 14900mn ³ + 8185mn ² – 65800mn + 40732m + 630n ⁶ + 6019n ⁵ + 14830n ⁴ – 29875n ³ – 60520n ² + 119316n – 50400)/80.
• (7) Sz_t (TBTGN(m,n)) = (25m ⁶ + 243m ⁵ + 225m ⁴ n ² + 765m ⁴ n – 203m ⁴ + 1310m ³ n ² + 4214m ³ n – 4587m ³ + 525m ² n ⁴ + 3950m ² n ³ + 3258m ² n ² – 9083m ² n + 1618m ² + 1665mn ⁴ + 11894mn ³ + 6337mn ² – 52708mn + 32952m + 525n ⁶ + 4879n ⁵ + 11569n ⁴ – 24583n ³ – 47278n ² + 96744n – 41856)/16.
• (8)PI­(TBTGN(m,n)) = (9m ⁴ + 50m ³ + 54m ² n ² + 162m ² n – 153m ² + 162mn ² + 438mn – 626m + 81n ⁴ + 474n ³ + 3n ² – 1926n + 1368)/4.
• (9) S(TBTGN(m,n)) = (24m ⁵ + 210m ⁴ + 240m ³ n ² + 840m ³ n – 575m ³ + 240m ² n ³ + 1200m ² n ² + 1335m ² n – 2475m ² + 360mn ⁴ + 3360mn ³ + 2415mn ² – 13710mn + 7616m + 552n ⁵ + 2970n ⁴ – 1515n ³ – 11955n ² + 14748n – 4800)/10.
• (10)Gut­(TBTGN(m,n)) = (36m ⁵ + 270m ⁴ + 360m ³ n ² + 1080m ³ n – 910m ³ + 360m ² n ³ + 1440m ² n ² + 1080m ² n – 2685m ² + 540mn ⁴ + 4320mn ³ + 1620mn ² – 16440mn + 10009m + 828n ⁵ + 3600n ⁴ – 3630n ³ – 13245n ² + 19947n – 7500)/10.

Proof. Let the tri bow-tie graphene nanoflake graph consist of |V| = 3n ² + m ² + 12n + 4m – 14 and |E| = 3­(3n ² + m ² + 9n + 3m – 12)/2. We now present the Θ classes of TBTGN(m,n) and their corresponding components. We first consider the acute type (AT) of TBTGN(m,n) and group them into four categories of Θ classes, see Figure . The four types of ATs are denoted by {A _{1 i} : 1 ≤ i ≤ n – 1}, {A _{2 i} : 1 ≤ i ≤ m – 1}, {A _{3 i} : 1 ≤ i ≤ n – 1} and {A ₁}. In this graph, the types AT, OT, and VT are symmetric. Now, we derive the several topological indices of TBTGN(m,n) using the quotient values as shown in Table .

7.

Tri bow-tie graphene nanoflake. (a) Acute cuts; (b) G|A _1i ; (c) G|A _2i ; (d) G|A _3i ; (e) G|A ₁.

5. Quotient Values of TBTGN­(m, n) .

Quotient Graph	w v	s v
$$\begin{array}{c}{A}_{1i}\\ 1\le i\le n-1\end{array}$$	$$\begin{array}{c}{a}_1=2i+i^2\\ a_2=V-a_1\end{array}$$	$$\begin{array}{c}{b}_1=(3i^2+3i-2)/2\\ b_2=E-b1-(i+1)\end{array}$$
$$\begin{array}{c}{A}_{2i}\\ 1\le i\le m-1\end{array}$$	$$\begin{array}{c}{a}_3=i^2+2i+n^2+4n-5\\ a_4=V-a_3\end{array}$$	$$\begin{array}{c}{b}_3=(3i^2+3i+3n^2+9n-14)/2\\ b_4=E-b_3-(i+1)\end{array}$$
$$\begin{array}{c}{A}_{3i}\\ 1\le i\le n-1\end{array}$$	$$\begin{array}{c}{a}_5=-(i^2-2in-4i+2)\\ a_6=V-a_5\end{array}$$	$$\begin{array}{c}{b}_5=-(2n-11i-6in+3i^2+8)/2\\ b_6=E-b_5-(n-i+2)\end{array}$$
A 1	$$\begin{array}{c}{a}_7=2n^2+4n+2m-5\\ a_8=V-a_7\end{array}$$	$$\begin{array}{c}{b}_7=3n^2+3n+2m-6\\ b_8=E-b_7-(m+2n-1)\end{array}$$

$$\begin{array}{lll}W(TBTGN(m,n))& =3[2\sum _{i=1}^{n-1}{a}_1{a}_2+\sum _{i=1}^{m-1}{a}_3{a}_4+\sum _{i=1}^{n-1}{a}_5{a}_6+a_7{a}_8].& \\ W_e(TBTGN(m,n))& =3[2\sum _{i=1}^{n-1}{b}_1{b}_2+\sum _{i=1}^{m-1}{b}_3{b}_4+\sum _{i=1}^{n-1}{b}_5{b}_6+b_7{b}_8].& \\ W_{ve}(TBTGN(m,n))& =\frac{3}{2}[2\sum _{i=1}^{n-1}(a_1{b}_2+a_2{b}_1)+\sum _{i=1}^{m-1}(a_3{b}_4+a_4{b}_3)+\sum _{i=1}^{n-1}(a_5{b}_6+a_6{b}_5)+(a_7{b}_8+a_8{b}_7)].& \\ S{z}_v(TBTGN(m,n))& =3[2\sum _{i=1}^{n-1}{e}_1{a}_1{a}_2+\sum _{i=1}^{m-1}{e}_2{a}_3{a}_4+\sum _{i=1}^{n-1}{e}_3{a}_5{a}_6+e_4{a}_7{a}_8].& \\ S{z}_e(TBTGN(m,n))& =3[2\sum _{i=1}^{n-1}{e}_1{b}_1{b}_2+\sum _{i=1}^{m-1}{e}_2{b}_3{b}_4+\sum _{i=1}^{n-1}{e}_3{b}_5{b}_6+e_4{b}_7{b}_8].& \\ S{z}_{ev}(TBTGN(m,n))& =\frac{3}{2}[2\sum _{i=1}^{n-1}{e}_1(a_1{b}_2+a_2{b}_1)+\sum _{i=1}^{m-1}{e}_2(a_3{b}_4+a_4{b}_3)+\sum _{i=1}^{n-1}{e}_3(a_5{b}_6+a_6{b}_5)& \\ & +e_4(a_7{b}_8+a_8{b}_7)].& \\ \mathrm{P}\mathrm{I}(TBTGN(m,n))& =3[2\sum _{i=1}^{n-1}{e}_1(b_1+b_2)+\sum _{i=1}^{m-1}{e}_2(b_3+b_4)+\sum _{i=1}^{n-1}{e}_3(b_5+b_6)+e_4(b_7+b_8)].& \\ S(TBTGN(m,n))& =3[2\sum _{i=1}^{n-1}[a_1(2b_2+e_1)+a_2(2b_1+e_1)]+\sum _{i=1}^{m-1}[a_3(2b_4+e_2)+a_4(2b_3+e_2)]& \\ & +\sum _{i=1}^{n-1}[a_5(2b_6+e_3)+a_6(2b_5+e_3)]+[a_7(2b_8+e_4)+a_8(2b_7+e_4)]].& \\ \mathrm{G}\mathrm{u}\mathrm{t}(TBTGN(m,n))& =3[2\sum _{i=1}^{n-1}(2b_2+e_1)(2b_1+e_1)+\sum _{i=1}^{m-1}(2b_4+e_2)(2b_3+e_2)& \\ & +\sum _{i=1}^{n-1}(2b_6+e_3)(2b_5+e_3)+(2b_8+e_4)(2b_7+e_4)].\end{array}$$

Now, we shall compute the degree-based indices for tri bow-tie graphene nanoflakes from the edge partition given in Table along with the degree-based indices expression discussed in Methodology Section.

6. Different Types of Edges in TBTGN­(m,n) .

S. No	E i	(d(u), d(v))	|E i |
1	E 1	(2,2)	12
2	E 2	(2,3)	6m + 18n – 36
3	E 3	(3,3)	$$\frac{1}{2}[9n^2+3m^2-9n-3m+12]$$

Theorem 6. Let TBTGN­(m,n) be a tri bow-tie graphene nanoflake. Then

• (1) M ₁(TBTGN(m,n)) = 9m ² + 21m + 27n ² + 63n – 96.
• (2) $M_2(TBTGN(m,n))=\frac{3}{2}(9m^2+15m+27n^2+45n-76).$
• (3)HM­(TBTGN(m,n)) = 6­(9m ² + 16m+ 27n ² + 48n – 82).
• (4) $\mathrm{A}\mathrm{Z}(TBTGN(m,n))=\frac{3}{128}(729m^2+1319m+2187n^2+3957n-5276).$
• (5) $\mathrm{A}\mathrm{B}\mathrm{C}(TBTGN(m,n))=m^2+(3\sqrt{2}-1)m+3n^2+(9\sqrt{2}-3)n-12\sqrt{2}+4.$
• (6) $H(TBTGN(m,n))=\frac{1}{10}(5m^2+19m+15n^2+57n-64).$
• (7) $\mathrm{S}\mathrm{C}(TBTGN(m,n))=\frac{1}{20}(5\sqrt{6}{m}^2+(24\sqrt{5}-5\sqrt{6})m+15\sqrt{6}{n}^2+(72\sqrt{5}-15\sqrt{6})n-144\sqrt{5}+20\sqrt{6}+120).$
• (8) $\mathrm{G}\mathrm{A}(TBTGN(m,n))=\frac{3}{10}(5m^2+(8\sqrt{6}-5)m+15n^2+(24\sqrt{6}-15)n-48\sqrt{6}+60).$
• (9) $\mathrm{I}\mathrm{S}\mathrm{I}(TBTGN(m,n))=\frac{3}{20}(15m^2+33m+45n^2+99n-148).$
• (10)SDD­(TBTGN(m,n)) = 3m ² + 10m+ 9n ² + 30n – 42.
• (11) F(TBTGN(m,n)) = 2­(27n ² + 51n – 8m – 11).
• (12) $\mathrm{S}\mathrm{O}(TBTGN(m,n))=\frac{3}{2}(3\sqrt{2}{m}^2+(4\sqrt{13}-3\sqrt{2})m+9\sqrt{2}{n}^2+(12\sqrt{13}-9\sqrt{2})n+28\sqrt{2}-24\sqrt{13}).$

Numerical Analysis

This section is devoted to the numerical analysis of various degree-and distance-based molecular descriptors of bow-tie graphene nanoflake of three different types, namely, asymmetric bow-tie graphene nanoflake ABTGN­(m,n), symmetric bow-tie graphene nanoflake SBTGN­(m,n), and tri bow-tie graphene nanoflake TBTGN­(m,n). The analysis is done on their topological behaviors utilizing a diverse range of molecular descriptors, the graphical depiction of which is shown in various figures. Figures – depict the numerical values given in Tables – of various ranges of the distance-based molecular descriptors of asymmetric bow-tie graphene nanoflake ABTGN­(m,n), symmetric bow-tie graphene nanoflake SBTGN­(m,n), and tri bow-tie graphene nanoflake TBTGN­(m,n). In a similar way, Figures – explore the numerical values given in Tables – degree-based molecular descriptors of various ranges of asymmetric bow-tie graphene nanoflake ABTGN­(m,n), symmetric bow-tie graphene nanoflake SBTGN­(m,n), and tri bow-tie graphene nanoflake. This illustration gives the relationships between variables to support predictions, informed decision-making, and trend identification (graph-based approaches are powerful, see ref ). Since these indices represent the topological connectivity characteristics of the compounds, the findings from this study have considerable implications for comprehending the significance of the molecular graph addressed in this paper. Employing strength-weighted graph methodologies, we formulate analytical expressions.

8.

Graphical representation of distance-based indices of ABTGN­(m,n).

10.

Graphical representation of distance-based indices of TBTGN­(m,n).

7. Numerical Values of Asymmetric Bow-Tie Graphene Nanoflake ABTGN­(m,n) .

TI (m,n)	(3,4)	(3,5)	(3,6)	(3,7)	(4,5)	(4,6)	(4,7)	(4,8)	(5,6)	(5,7)	(5,8)	(5,9)
W	11100	28609	62950	123831	25770	57323	113876	207785	52950	105703	193928	332949
W e	15434	42969	99971	205094	38698	90859	188195	354892	84066	174631	330891	583284
W ve	13139	35146	79457	159544	31669	72309	146594	271826	66864	136080	253614	441078
Sz v	51770	150675	368080	794951	143796	355875	774788	1534389	343470	754559	1503076	2777571
Sz e	72378	226613	583725	1312682	217106	565411	1280363	2616300	548046	1250029	2566527	4862292
Sz ev	61406	185151	464148	1022498	177108	449299	997150	2005316	434664	972499	1966066	3677778
Sz t	246960	747590	1880101	4152629	715118	1819884	4049451	8161321	1760844	3949586	8001735	14995419
PI	5864	12590	23666	40592	12128	23026	39746	64004	22404	38922	62950	96420
S	57296	150474	336040	668926	136168	306906	616420	1135106	284592	573666	1061376	1835502
Gut	73841	197691	448199	902993	179689	410487	833735	1549621	382077	777847	1451539	2528781

9. Numerical Values of Tri Bow-Tie Graphene Nanoflake TBTGN­(m,n) .

TI (m,n)	(3,3)	(3,4)	(3,5)	(3,6)	(3,7)	(4,4)	(4,5)	(4,6)	(4,7)	(5,5)	(5,6)	(5,7)
W	17703	48906	113208	231606	432465	65256	141444	277077	501864	177555	333192	585252
W e	24171	72549	178167	381012	736380	98778	225408	459588	859422	286548	557382	1008198
W ve	20748	59671.5	142185	297300	564657	80403	178728	357090	657073.5	225750	431200.5	768483
Sz v	68454	211500	545004	1232910	2527956	292158	691080	1478175	2916234	897966	1812630	3431718
Sz e	93276	311559	850059	2008980	4264008	439014	1090584	2425860	4940796	1436352	3001284	5848860
Sz ev	80088	257077.5	681363	1575051	3285183	358542	868866	1894845	3797802	1136448	2333643	4482045
Sz t	321906	1037214	2757789	6391992	13362330	1448256	3519396	7693725	15452634	4607214	9481200	18244668
PI	7716	17562	34668	61932	102738	21726	40440	69582	112536	47952	79350	124884
S	89292	252591	595578	1236324	2335740	338712	746124	1481226	2712705	939852	1784937	3167466
Gut	112500	325983	783057	1649484	3153234	439338	983688	1979223	3665145	1243428	2390103	4285125

11.

Graphical representation of degree-based indices of ABTGN­(m,n).

13.

Graphical representation of degree-based indices of TBTGN­(m,n).

10. Numerical Values of Degree-Based Indices of Asymmetric Bow-Tie Graphene Nanoflake ABTGN­(m,n) .

TI (m,n)	(3,4)	(3,5)	(3,6)	(3,7)	(4,5)	(4,6)	(4,7)	(4,8)	(5,6)	(5,7)	(5,8)	(5,9)
M 1	430	634	874	1150	626	866	1142	1454	858	1134	1446	1794
M 2	587	875	1217	1613	869	1211	1607	2057	1205	1601	2051	2555
HM	2376	3540	4920	6516	3512	4892	6488	8300	4864	6460	8272	10300
AZ	777.7968	1147.1718	1584.8906	2090.9531	1137.9531	1575.6718	2081.7343	2656.1406	1566.4531	2072.5156	2646.9218	3289.6718
ABC	54.1225	78.6077	107.0930	139.5783	77.1126	105.5979	138.0832	174.5685	104.1028	136.5881	173.0734	213.5587
H	29.5333	42.3333	57.1333	73.9333	41.4	56.2	73	91.8	55.2666	72.0666	90.8666	111.6666
SC	34.0766	18.2004	66.8551	86.9186	48.2688	65.8828	22.8433	108.4593	64.9104	84.9739	107.4869	59.1811
GA	2.4494	114.1918	2.3158	2.1474	2.1507	2.0995	201.7877	1.8564	1.8908	1.8289	1.7350	313.3836
ISI	106.1	156.5	215.9	284.3	154.7	214.1	282.5	359.9	121.3	280.7	358.1	444.5
SDD	162.6667	236.6667	322.6667	420.6667	232	318	416	526	313.3333	411.3333	521.3333	643.3333
F	1202	1790	2486	3290	1774	2470	3274	4186	2454	3258	4170	5190
SO	306.0164	451.1064	621.6522	817.6539	445.1695	615.7153	811.717	1033.1745	609.7783	805.7801	1027.2376	1274.151

12. Numerical Values of Degree-Based Indices of Tri Bow-Tie Graphene Nanoflake TBTGN­(m,n) .

TI (m,n)	(3,3)	(3,4)	(3,5)	(3,6)	(3,7)	(4,4)	(4,5)	(4,6)	(4,7)	(5,5)	(5,6)	(5,7)
M 1	480	732	1038	1398	1812	816	1122	1482	1896	1224	1584	1998
M 2	642	993	1425	1938	2532	1110	1542	2055	2649	1686	2199	2793
HM	2604	4026	5772	7842	10236	4500	6246	8316	10710	6828	8898	11292
AZ	862.4063	1313.9531	1868.0156	2524.5938	3283.6875	1464.4688	2018.5313	2675.1094	3434.2031	2203.2188	2859.7969	3618.8906
ABC	61.9411	92.669	129.397	172.1249	220.8528	102.9117	139.6396	182.3675	231.0955	151.8823	194.6102	243.3381
H	34.4	50.6	69.8	92	117.2	56	75.2	97.4	122.6	81.6	103.8	129
SC	39.2461	58.3187	81.0654	107.4865	137.5817	64.6762	87.423	113.844	143.9392	95.0052	121.4262	151.5215
GA	89.2727	133.909	187.5453	250.1816	321.818	148.7878	202.4241	265.0604	336.6967	220.3029	282.9392	354.5755
ISI	118.2	180.3	255.9	345	447.6	201	276.6	365.7	468.3	301.8	390.9	493.5
SDD	186	279	390	519	666	310	421	550	697	458	587	734
F	1320	2040	2922	3966	5172	2280	3162	4206	5412	3456	4500	5706
SO	341.9319	521.3831	739.0181	994.8368	1288.8394	581.2002	798.8352	1054.6539	1348.6564	871.3802	1127.1989	1421.2014

8. Numerical Values of Symmetric Bow-Tie Graphene Nanoflake SBTGN(m,n).

TI (m,n)	(3,4)	(3,5)	(3,6)	(3,7)	(4,5)	(4,6)	(4,7)	(4,8)	(5,6)	(5,7)	(5,8)	(5,9)
W	2894	11469	31720	71891	4934	17927	46784	101589	7758	26441	65976	138447
W e	3467	15897	47930	115160	6328	25957	72905	166558	10469	39639	105464	231482
W ve	3191	13563	39102	91162	5619	21653	58552	130314	9051	32477	83606	179320
Sz v	11270	57116	185840	476564	20688	98035	303784	749085	34250	154794	462976	1108886
Sz e	13724	80376	283572	767536	26780	143965	478813	1237886	46302	234690	748368	1870648
Sz ev	12509	67982	230052	605678	23646	119143	382124	964276	39971	191076	589648	1442090
Sz t	50012	273456	929516	2455456	94760	480286	1546845	3915523	160494	771636	2390640	5863714
PI	1938	6318	14640	28404	3008	9266	20658	38900	4314	12782	27720	51060
S	14420	59336	167816	386320	24984	93938	250084	550602	39740	139884	355504	755460
Gut	17922	76630	221736	518628	31569	122895	333887	745533	50814	184794	478476	1029888

9.

Graphical representation of distance-based indices of SBTGN­(m _n).

11. Numerical Values of Degree-Based Indices of Symmetric Bow-Tie Graphene Nanoflake SBTGN­(m,n) .

TI (m,n)	(3,4)	(3,5)	(3,6)	(3,7)	(4,5)	(4,6)	(4,7)	(4,8)	(5,6)	(5,7)	(5,8)	(5,9)
M 1	244	448	688	964	306	546	822	1134	368	644	956	1304
M 2	326	614	956	1352	413	755	1151	1601	500	896	1346	1850
HM	1320	2484	3864	5460	1672	3052	4648	6460	2024	3620	5432	7460
AZ	442.5938	811.9688	1249.6875	1755.75	554.3281	992.0469	1498.1094	2072.5156	666.0625	1172.125	1746.5313	2389.2813
ABC	31.6372	56.1225	84.6078	17.0931	39.1323	67.6176	100.1029	136.5882	46.6274	79.1127	115.598	156.0833
H	17.7333	30.5333	45.3333	62.1333	21.6667	36.4667	53.2667	72.0667	25.6	42.4	61.2	82
SC	20.1369	35.3014	52.9154	72.9789	24.7835	42.3975	62.461	84.974	29.4301	49.4936	72.0066	96.969
GA	45.6767	81.4343	123.1918	170.9494	56.5959	98.3535	146.111	199.8686	67.5151	115.2727	169.0302	228.7878
ISI	60.2	110.6	170	238.4	75.5	134.9	203.3	280.7	90.8	159.2	236.6	323
SDD	94.6667	168.6667	254.6667	352.6667	117.3333	203.3333	301.3333	411.3333	140	238	348	470
F	668	1256	1952	2756	846	1542	2346	3258	1024	1828	2740	3760
SO	173.6543	318.7443	489.2902	685.2918	217.775	388.3209	584.3225	805.7801	261.8957	457.8974	679.3549	926.2683

12.

Graphical representation of degree-based indices of SBTGN­(m _n ).

Computed HOMO–LUMO, Delocalization Energies, and NMR Spectral Patterns of Three Types of Bow-Tie Graphene Nanoflakes

In order to make QSAR predictions on the bow-tie graphene, the topological indices that were generated in the previous section can be used. Furthermore, to be useful for spectroscopy and machine learning of bow-tie graphene stabilities, the distance and adjacency matrices of such molecular structures encompass noteworthy information about the molecules. One can estimate the thermodynamic and kinetic stabilities for the bow-tie structures under scrutiny using the spectra of the graph. An approach for vertex partitioning GNRs could be derived from the distance matrices, which can then be used to form the distance degree sequence vector (DDSV) for every atom of the graphene nanoflake. The method for constructing the partitions of the vertex set entirely relies on graph terminology, and it does not make use of any experimental reference. Every atom of the bow-tie graphene has its DDSV coined as (D _k0, D _k1, D _k2, ..., D_pk , ...). Here, D _pk denotes the number of vertices at a distance p from any vertex q _k.

Our method for processing the DDSV of each atom in any bow-tie graphene is done with the help of the NEWGRAPH interface. The MATLAB code then examines the DDSVs; if many atoms have the same DDSV, then they are put into a basket. Generating the partition becomes challenging because there are so many nuclei and since each nucleus has a vector with a DDSV length of different magnitude.

The eigen values of the adjacency matrix Adj­(G), denoted as λ₁ ≥ λ₂, ... ≥ λ _m , are effectively used to get the spectrum (eigenvalues) of any chemical graph G. The energy of the highest occupied molecular orbital (HOMO) is E _HOMO = λ _n/2, and the energy of the lowest unoccupied molecular orbital (LUMO) is E _LUMO=λ _{n /2+1}. Therefore, the HOMO–LUMO gap is δ_HL = λ _n/2 – λ _{n /2+1}. The total π electron energy is $E_{\pi }=2\sum _{i=1}^{n/2}{\lambda }_i$ , and the delocalization energy is E _Deloc = E _π – m. The energies are expressed in standard β-units in the formulas given above. The above mentioned energies and NMR pattern are shown in Tables , , and .

13. Spectral, Energetic Properties, NMR Pattern, and NMR Signals of ABTGN­(m, n) .

Structures (m,n)	HOMO–LUMO	Spectral Spread	π-Electron Energy/Bond	Delocalization Energy	13C NMR Pattern	13C NMR Signals
ABTGN(3, 6)	0.0232β	5.7426β	1.5260531β	0.5260531β	25641	1:1: ... :1(56) 2(1)
ABTGN(3, 7)	0.0222β	5.7764β	1.542458β	0.542458β	26943	1:1: ... :1(69) 2:2:2(3)
ABTGN(4, 6)	0.0054β	5.7768β	1.478768β	0.478768β	257	1:1: ... :1(57)
ABTGN(4, 7)	0.005β	5.8038β	1.5262615β	0.5262615β	274	1:1: ... :1(74)

14. Spectral, Energetic Properties, NMR Pattern, and NMR Signals of SBTGN­(m, n) .

Structures (m,n)	HOMO–LUMO	Spectral Spread	π-Electron Energy/Bond	Delocalization Energy	13C NMR Pattern	13C NMR Signals
SBTGN(3, 6)	0.2174β	5.7556β	1.538979β	0.538979β	24421	1:1:1:1(4) 2:2:2 ... :2(21)
SBTGN(3, 7)	0.2032β	5.7974β	1.5551483β	0.5551483β	25429	1:1: ... :1(5) 2:2: ... :2(29)
SBTGN(4, 6)	0.1272β	5.7314β	1.5096222β	0.5096222β	23417	1:1:1(3) 2:2: ... :2(17)
SBTGN(4, 7)	0.1182β	5.7992β	1.537175β	0.537175β	24425	1:1:1:1(4) 2:2 ... :2(25)

15. Spectral, Energetic Properties, NMR Pattern, and NMR Signals of TBTGN­(m, n) .

Structures (m,n)	HOMO–LUMO	Spectral Spread	π-Electron Energy/Bond	Delocalization Energy	13C NMR Pattern	13C NMR Signals
TBTGN(6, 3)	0.7056β	5.6944β	1.50009622β	0.50009622β	1138614	1(1) 2:2: ... :2(8) 3:3: ... :3(14)
TBTGN(6, 4)	0.6962β	5.6974β	1.52520888β	0.52520888β	1139619	1(1) 2:2: ... :2(9) 3:3: ... :3(19)
TBTGN(7, 3)	0.7264β	5.7462β	1.513206557β	0.513206557β	310616	1:1: ... :1(10) 2:2: ... :2(16)
TBTGN(7, 4)	0.7162β	5.7472β	1.55353333β	0.5535333β	311621	1:1: ... :1(11) 2:2: ...:2(21)

Conclusions

The energy characteristics, spectral distribution, and topological indices of bow-tie graphenes were calculated. The calculated topological indices, spectral characteristics, and energy properties exhibit distinct differences among the three forms of bow-tie graphenes. This will aid future research, such as QSAR analysis, to determine the physicochemical properties, activities, and other attributes of these compounds. Topological indices are mathematical functions associated with the structures of compounds that are related to their carcinogenicity, toxicity, and other features. These obtained equations have practical applications in researching several aspects of nanoscience and polycyclic aromatics, including superaromaticity and stabilities, in addition to their mathematical significance.

S.P.: Conceptualization, Methodology, Formal Analysis, Writing – Original draft preparation, Data curation. M.A.: Validation, Software, Investigation, Project administration. L.J.: Supervision, Writing – Reviewing and Editing, Funding acquisition. V.M.: Software, Investigation. R.M.M.: Software, Investigation.

The authors acknowledge the financial support for Open Access publication from Technical University of Cluj-Napoca (statement of need grant no. 20694 from 23 June 2025).

The authors declare no competing financial interest.