Reverse Degree-Based Polynomial Descriptors in Corrosion-Related Systems: Exploratory Analysis of Organic Inhibitors and Nanoporous Graphene

Abstract

Mild steel remains one of the most widely used structural materials in mechanical and industrial engineering due to its favorable mechanical performance and low cost. However, its high susceptibility to corrosion continues to cause significant operational and economic losses across engineering systems. This study presents a unified analytical framework for analyzing corrosion-related molecular and nanostructured systems using reverse degree-based topological descriptors, namely, the Reverse M-polynomial and Reverse NM-polynomial. The framework is demonstrated in two complementary stages relevant to corrosion engineering. First, an exploratory structure–property correlation analysis based on Quantitative Structure–Property Relationship (QSPR) principles is conducted for furan-based organic inhibitors reported in the literature, examining the relationship between reverse degree-based descriptors and inhibition efficiency on mild steel surfaces. The analysis reveals a strong statistical correlation within the analyzed dataset (r = 0.958), indicating the sensitivity of selected reverse topological descriptors to molecular structural variations. The statistical significance of the correlations was evaluated using p-values and F-statistics, confirming the reliability of the observed associations within the analyzed dataset. However, owing to the limited dataset size, no claims of external predictivity are made. Second, the framework is extended to advanced protective materials through the analytical formulation of reverse descriptors for nanoporous graphene nanoribbons containing 14-annulene pores, focusing exclusively on structural and topological characterization. These graphene structures are considered as potential physical barrier materials; however, in this study, the analysis is limited to structural descriptor characterization without modeling corrosion performance. This work provides analytical results for reverse degree-based descriptors of such graphene architectures. Overall, the findings establish a versatile analytical framework that supports exploratory structure–property investigations of organic inhibitors and provides descriptor-based structural benchmarks for graphene nanostructures, offering theoretical insights relevant to corrosion mitigation research.

1. Introduction

Corrosion refers to the gradual deterioration of metals and alloys resulting from their chemical interactions with the environment. Over time, this degradation compromises the strength and durability of critical components, leading to frequent repairs or even complete system failure. Its effects are widespread, influencing industrial productivity, environmental health, and economic stability across multiple sectors [1,2,3]. Globally, corrosion imposes an estimated annual cost of USD 2.2 trillion, as reported by the World Corrosion Organization (WCO) [4]. Furthermore, the National Association of Corrosion Engineers (NACE) indicates that implementing existing corrosion-mitigation strategies could reduce these costs by 15–35%, translating to potential annual savings of USD 375–USD 875 billion [5]. Corrosion can cause significant environmental harm by leading to various infrastructure failures. As metallic structures deteriorate, problems such as bridge collapses, degradation of marine vessels and docks, and leaks in pipelines or storage tanks may occur, contaminating soil, water, and surrounding ecosystems. It also weakens industrial equipment, buildings, and railway components, creating operational disruptions and safety risks. These effects show that corrosion not only compromises structural integrity but also contributes to environmental pollution and increased maintenance costs across multiple sectors [6,7,8]. Therefore, corrosion research is essential for understanding degradation mechanisms and developing effective control strategies that maintain structural integrity, enhance safety across industries, and reduce both economic losses and environmental impacts.

Steel, an iron–carbon alloy, is highly susceptible to corrosion. It is central to modern engineering due to its low cost, abundance, and mechanical versatility, while its excellent ductility, conductivity, and weldability make it indispensable in industrial and infrastructure applications. However, its major limitation is poor corrosion resistance, particularly in aggressive environments such as saline seawater, humid marine air, and acidic media used in processes like cleaning and well acidizing. This vulnerability leads to substantial global losses, which can be mitigated through the use of corrosion inhibitors [9,10,11].

Corrosion inhibitors are low-concentration additives that reduce metal corrosion and, when appropriately selected, provide a simple and effective means of protection [12]. Although organic inhibitors exhibit good corrosion resistance, their potential toxicity and environmental impact have increased the demand for greener, renewable, and more sustainable alternatives [13,14]. However, identifying such inhibitors remains challenging because corrosion is a complex phenomenon, and new candidates are still largely discovered through experimental, trial-and-error screening of large libraries of organic compounds tailored to specific metal–environment systems. This process is further complicated by the vast chemical space containing millions of possible molecular structures, making traditional empirical screening both time-consuming and costly [15,16,17].

Another effective strategy for protecting metals from corrosion is the use of protective coatings, which act as physical barriers to prevent corrosive agents such as water, oxygen, and salts from reaching the metal surface [18,19]. Recently, graphene has emerged as a promising material for such coatings due to its exceptional strength, chemical stability, and impermeability [20,21,22]. Incorporating graphene enhances barrier performance, improves durability, and extends the lifespan of metals, complementing traditional corrosion inhibitors.

Sustainable engineering and renewable energy systems require materials with enhanced durability and reduced environmental impact [23,24,25,26]. In this context, corrosion-resistant inhibitors and graphene-based protective materials play a crucial role in extending service life and improving reliability in energy-related infrastructures. Reflecting growing industrial interest, the global market for graphene-based coatings is projected to reach approximately USD 173.31 million by 2032 [27]. Building on this trend, porous three-dimensional graphene demonstrates even greater potential, as its interconnected framework disperses more effectively in epoxy matrices and forms stronger conductive pathways between zinc particles. The porous structure not only strengthens barrier properties but also enhances cathodic protection, significantly reducing corrosion current compared with conventional 2D graphene [28]. Understanding the molecular structure of graphene and porous graphene is essential for optimizing their barrier and protective properties, enabling the design of coatings that maximize corrosion resistance and extend metal durability.

Despite the effectiveness of inhibitors and advanced coatings, developing and optimizing new materials remains resource-intensive. Discovering novel organic inhibitors involves laborious synthesis and testing, while understanding and characterizing nanostructured coatings requires a detailed understanding of complex topologies—challenges that are difficult to address experimentally. This creates a need for analytical and descriptor-based tools that can help understand structure–property relationships in corrosion protection systems.

To meet this need, this study introduces a versatile theoretical framework based on the Reverse M-polynomial and Reverse NM-polynomial. These molecular descriptors quantify essential structural features of chemical compounds and materials from their graphical representations, providing a unified analytical approach for investigating both molecular inhibitors and advanced nanostructured barriers. To demonstrate the practical utility of this framework, the research is presented in two parts. First, we examine the structure–property sensitivity of the proposed descriptors through an exploratory Quantitative Structure–Property Relationship (QSPR)-based correlation analysis for a series of furan derivatives, correlating their molecular structure with experimentally reported corrosion inhibition efficiency on mild steel. Second, we extend this theoretical framework to the structural characterization of advanced materials, presenting the first analytical results for these reverse-degree-based descriptors in porous graphene nanoribbons containing 14-annulene pores.

Chemical graph theory simplifies molecular modeling by representing atoms as vertices and bonds as edges, providing a powerful framework for extracting structural information in a mathematically efficient form [29]. This efficient approach supports QSPR frameworks, which use topological, geometrical, and electronic descriptors to build strong statistical models that predict properties directly from molecular structure [30,31]. The development of these descriptors traces back to Harold Wiener’s seminal introduction of the Wiener index, which established the foundation for modern topological characterization [32]. Building on this groundwork, the M-polynomial was proposed as a unifying generator for a broad class of degree-based descriptors through a single analytical expression [33]. This framework was subsequently extended via the NM-polynomial [34] and CoM-polynomial [35], and recent developments, including the Reverse M-polynomials and Reverse NM-polynomials, further generalize the approach, thereby enhancing the efficiency and predictive capability of QSPR/QSAR modeling for complex molecular and nanoscale systems [36,37,38,39].

The present study is not intended to design or optimize corrosion-resistant materials. Rather, it investigates the structural sensitivity and analytical behavior of reverse degree-based topological descriptors in corrosion-related molecular and nanostructured systems. This work is therefore positioned as an exploratory and theoretical contribution that may support future structure–property investigations when combined with adequately sized datasets and experimentally validated corrosion performance metrics.

2. Materials and Methods

This section presents the analytical framework, descriptor definitions, and methodology used for the exploratory QSPR analysis.

Let $G=\left(V,E\right)$ be a molecular graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. We denote the number of vertices and edges in a graph $G$ by $\left|V\left(G\right)\right|$ and $\left|E\left(G\right)\right|$ respectively. The degree of vertex $u\in V\left(G\right)$ is denoted by $d\left(u\right)$ and is the number of vertices that are adjacent to $u$. The edge connecting the vertices $u$ and $v$ is denoted by $e=uv$, where $e\in E\left(G\right)$. The neighborhood degree sum of a vertex, represented as $\delta \left(u\right),$ is defined as the total sum of degrees of all vertices that are adjacent to a vertex $u.$ The reverse vertex degree of a vertex $u$ is denoted as $rd\left(u\right)$ and defined as $rd\left(u\right)=\Delta \left(G\right)-d\left(u\right)+1$, where $\Delta \left(G\right)$ is the highest degree of a vertex in a graph G. The reverse neighborhood degree of a vertex u is given by $r\delta \left(u\right)={\Delta }_1\left(G\right)-\delta \left(u\right)+1$, where ${\Delta }_1\left(G\right)$ is the maximum neighborhood degree in graph G.

The M-polynomial of graph G is defined as $M\left(G;x,y\right)=\sum _{i\le j}{m}_{ij}\left(G\right)x^i{y}^j,$ where $m_{ij}\left(G\right)=$ number of edges $uv$, such that $\left\{d\right(u),d(v\left)\right)=\{i,j\}\}$.

The NM-polynomial of graph G is defined as $NM\left(G;x,y\right)=\sum _{i\le j}{n}_{ij}\left(G\right)x^i{y}^j,$ where $n_{ij}$ is the total number of edges $uv$ such that $\left\{\delta \right(u),\delta (v\left)\right)=\{i,j\}\}$.

The Reverse M-polynomial of graph G is defined as

$$RM\left(G;x,y\right)={\sum }_{i\le j}r{m}_{ij}\left(G\right)x^i{y}^j,$$

(1)

where ${rm}_{ij}\left(G\right)=$ number of edges $e=uv$ such that $\left\{rd\right(u),rd(v\left)\right)=\{i,j\}\}$.

The Reverse NM-polynomial of graph G is defined as

$$RNM\left(G;x,y\right)={\sum }_{i\le j}{rn}_{ij}\left(G\right)x^i{y}^j,$$

(2)

where $r{n}_{ij}$ is the total number of edges $uv,$ such that $\left\{r\delta \right(u),r\delta (v\left)\right)=\{i,j\}\}$.

From Expressions (1) and (2), several reverse degree-based topological descriptors are derived, as listed in

Table 1. Selected reverse degree-based and reverse neighborhood degree-based topological descriptors derived from reverse polynomials.

S. No.	Reverse Topological Index	Notation	$\mathbf{Formula} \mathit{g}\left(\mathit{r}\mathit{d}\left(\mathit{u}\right),\mathit{r}\mathit{d}\left(\mathit{v}\right)\right),$ $\mathit{g}\mathbf{(}\mathit{r}\mathit{\delta }\left(\mathit{u}\right),\mathit{r}\mathit{\delta }\left(\mathit{v}\right))$	Derivation from $\mathit{R}\mathit{M}\left(\mathit{G};\mathit{x},\mathit{y}\right)=\mathit{f}\left(\mathit{x},\mathit{y}\right),$ $\mathit{R}\mathit{N}\mathit{M}\left(\mathit{G};\mathit{x},\mathit{y}\right)=\mathit{f}\left(\mathit{x},\mathit{y}\right)$
1	Reverse First Zagreb index	$R{M}_1\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}\left(rd\left(u\right)+rd\left(v\right)\right)$	${(D_x+D_y)\left(f\left(x,y\right)\right)}_{x=y=1}$
	Reverse Neighborhood First Zagreb index	$RN{M}_1\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}\left(r\delta \left(u\right)+r\delta \left(v\right)\right)$
2	Reverse Second Zagreb index	${RM}_2\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}rd\left(u\right)rd\left(v\right)$	${\left(D_x{D}_y\right)\left(f\left(x,y\right)\right)}_{x=y=1}$
	Reverse Neighborhood Second Zagreb index	${RNM}_2\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}r\delta \left(u\right)r\delta \left(v\right)$
3	Reverse Second modified Zagreb index	$R{mM}_2\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}{\displaystyle \frac{1}{rd\left(u\right)rd\left(v\right)}}$	${\left(S_x{S}_y\right)\left(f\left(x,y\right)\right)}_{x=y=1}$
	Reverse Neighborhood Second modified Zagreb index	$RN{mM}_2\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}{\displaystyle \frac{1}{r\delta \left(u\right)r\delta \left(v\right)}}$
4	Reverse Redefined third Zagreb index	${RReZG}_3\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}rd\left(u\right)rd\left(v\right)\left(rd\left(u\right)+rd\left(v\right)\right)$	${D_x{D}_y\left(D_x+D_y\right)\left(f\left(x,y\right)\right)}_{x=y=1}$
	Reverse Neighborhood Redefined third Zagreb index	${RReZG}_3\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}r\delta \left(u\right)r\delta \left(v\right)\left(r\delta \left(u\right)+r\delta \left(v\right)\right)$
5	Reverse Forgotten topological index	$RF\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}\left(r{d}^2\left(u\right)+{rd}^2\left(v\right)\right)$	${\left({D_x}^2+{D_y}^2\right)\left(f\left(x,y\right)\right)}_{x=y=1}$
	Reverse Neighborhood Forgotten topological index	$RNF\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}\left(r{d}^2\left(u\right)+{rd}^2\left(v\right)\right)$
6	Reverse Randić index	$R{R}_k\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}{\left(rd\left(u\right)rd\left(v\right)\right)}^k$	${\left({D_x}^k{D_y}^k\right)\left(f\left(x,y\right)\right)}_{x=y=1}$
	Reverse Neighborhood Randić index	$RN{R}_k\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}{\left(r\delta \left(u\right)r\delta \left(v\right)\right)}^k$
7	Reverse Inverse Randić index	$R{RR}_k\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}{\displaystyle \frac{1}{{\left(rd\left(u\right)rd\left(v\right)\right)}^k}}$	${\left({S_x}^k{S_y}^k\right)\left(f\left(x,y\right)\right)}_{x=y=1}$
	Reverse Neighborhood Inverse Randić index	$RN{RR}_k\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}{\displaystyle \frac{1}{{\left(r\delta \left(u\right)r\delta \left(v\right)\right)}^k}}$
8	Reverse Symmetric Division index	$RSDD\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}{\displaystyle \frac{{rd}^2\left(u\right)+{rd}^2\left(v\right)}{rd\left(u\right)rd\left(v\right)}}$	${(D_x{S}_y+S_x{D}_y)\left(f\left(x,y\right)\right)}_{x=y=1}$
	Reverse Neighborhood Symmetric Division index	$RSDD\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}{\displaystyle \frac{{r\delta }^2\left(u\right)+{r\delta }^2\left(v\right)}{r\delta \left(u\right)r\delta \left(v\right)}}$
9	Reverse Harmonic index	$RH\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}{\displaystyle \frac{2}{rd\left(u\right)+rd\left(v\right)}}$	${\left({2S}_{x}J\right)\left(f\left(x,y\right)\right)}_{x=1}$
	Reverse Neighborhood Harmonic index	$RNH\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}{\displaystyle \frac{2}{r\delta \left(u\right)+r\delta \left(v\right)}}$
10	Reverse Inverse sum indeg index	$RI\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}{\displaystyle \frac{rd\left(u\right)rd\left(v\right)}{\left(rd\left(u\right)+rd\left(v\right)\right)}}$	${\left(S_{x}J{D}_x{D}_y\right)\left(f\left(x,y\right)\right)}_{x=1}$
	Reverse Neighborhood Inverse sum indeg index	$RNI\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}{\displaystyle \frac{r\delta \left(u\right)r\delta \left(v\right)}{\left(r\delta \left(u\right)+r\delta \left(v\right)\right)}}$
11	Reverse Augmented Zagreb	$RA\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}{\left\{{\displaystyle \frac{rd\left(u\right)rd\left(v\right)}{rd\left(u\right)+rd\left(v\right)-2}}\right\}}^3$	${\left({S_x}^3{Q}_{-2}J{D_x}^3{D_y}^3\right)\left(f\left(x,y\right)\right)}_{x=1}$
	Reverse Neighborhood Augmented Zagreb	$RNA\left(G\right)$	${\displaystyle \sum _{uv\in E\left(G\right)}}{\left\{{\displaystyle \frac{r\delta \left(u\right)r\delta \left(v\right)}{r\delta \left(u\right)+r\delta \left(v\right)-2}}\right\}}^3$
	where, $D_x=x\left({\displaystyle \frac{\partial \left(f\left(x,y\right)\right)}{\partial x}}\right),D_y=y\left({\displaystyle \frac{\partial \left(f\left(x,y\right)\right)}{\partial y}}\right),S_x={\int }_0^x{\displaystyle \frac{f\left(t,y\right)}{t}}dt,S_y={\int }_0^y{\displaystyle \frac{f\left(x,t\right)}{t}}dt,J\left(f\left(x,y\right)\right)=f\left(x,x\right),Q_k\left(f\left(x,y\right)\right)=x^{k}f\left(x,y\right)$

. In the reverse polynomial expressions, x and y are formal variables representing degree- or neighborhood-degree information at the end vertices of each edge, with exponents i and j denoting the corresponding reverse degree values. The operators $D_x$ and $D_y$ denote partial differentiation with respect to x and y, while $S_x$ and $S_y$ represent the integral operators used to derive degree-based descriptors from the polynomial form.

3. Results and Discussion

3.1. Exploratory QSPR-Based Correlation Analysis of Furan Derivatives as Corrosion Inhibitors

The study of corrosion involves preventing or managing the gradual deterioration of metals caused by their interaction with the environment and is broadly divided into corrosion inhibition, corrosion monitoring, and corrosion risk assessment. Among available protection methods, corrosion inhibitors remain one of the most widely used strategies for mitigating corrosion [40]. These compounds are generally classified as inorganic or organic, although inorganic inhibitors often pose toxicity and environmental concerns. Organic inhibitors protect metals by adsorbing onto the surface and blocking active corrosion sites through electron donation or coordinate bonding [41]. Furan-derived organic compounds, in particular, have been reported to exhibit strong inhibition performance in acidic environments, as their electronic structures enable them to accept unbound electrons from the metal surface and form coordinated covalent bonds, effectively suppressing the corrosion process [42,43,44,45]. The effectiveness of these inhibitors is quantified using inhibitor efficiency (IE%), which measures the percentage reduction in corrosion rate achieved in the presence of an inhibitor [46].

In this context, the present study employs a chemical graph theory-based analytical framework to examine the relationship between molecular structure and inhibition efficiency (IE%) of selected furan derivatives. By representing these inhibitors as molecular graphs, reverse degree-based topological descriptors are calculated to capture key structural features. These descriptors are then correlated with experimentally reported IE% values through an exploratory QSPR-based correlation analysis, aimed at assessing descriptor sensitivity rather than predictive screening or molecular design. Experimental inhibition efficiency data for eight furan derivatives, as reported in [44,47], are used as benchmark data for this analysis. The chemical structures of the compounds are summarized in Figure 1, and their corresponding molecular graphs (${FD}_1-{FD}_8$) are presented in Figure 2.

Although the reverse degree-based descriptors are derived analytically, the present work adopts a graph-theoretical framework as a systematic and scalable approach for descriptor calculation and exploratory statistical analysis across multiple molecular structures, rather than an algorithmic or simulation-based predictive method. All descriptor calculations were performed analytically using graph-theoretical formulations derived from the Reverse M-polynomial and Reverse NM-polynomial. No computational chemistry software, simulation tools, or numerical packages were used in this study. The calculations are fully reproducible using the presented mathematical framework.

In the molecular representations shown in Figure 1, vertices correspond to atoms, and different colors denote different atom types (e.g., C, O, Cl). Hydrogen atoms are omitted in the molecular graph representations to simplify the graph topology. The experimental inhibition efficiency (IE%) values employed here are taken from previously published electrochemical and gravimetric corrosion studies conducted on mild steel under acidic conditions [45,48]. Each furan derivative is represented by its corresponding chemical graph, where vertices denote atoms, and edges represent chemical bonds. Reverse degree-based descriptors are calculated using the Reverse M-polynomial and Reverse NM-polynomial formulations introduced in Section 2. For clarity and conciseness, the detailed derivation of the descriptors is illustrated for the representative compound FD₁ only, while analogous derivations for the remaining furan derivatives follow the same procedure and are summarized in

Table 2. Edge partition cardinalities of furan derivatives based on reverse degrees.

Furan Derivatives	${\mathbf{E}}_{11}$	${\mathbf{E}}_{12}$	${\mathbf{E}}_{13}$	${\mathbf{E}}_{22}$	${\mathbf{E}}_{23}$
FD1	1	6	1	2	2
FD2	0	7	2	1	1
FD3	1	4	3	1	0
FD4	0	4	2	4	0
FD5	1	2	2	3	0
FD6	1	3	1	3	1
FD7	0	5	1	1	1
FD8	0	3	0	3	1

,

Table 3. Reverse M-polynomial of furan derivatives from FD₁ to FD₈.

$RM\left({FD}_1;x,y\right)=xy+6x{y}^2+x{y}^3+2x^2{y}^2+2x^2{y}^3$

$RM\left({FD}_2;x,y\right)=7x{y}^2+2x{y}^3+x^2{y}^2+x^2{y}^3$

$RM\left({FD}_3;x,y\right)=xy+4x{y}^2+3x{y}^3+x^2{y}^2$

$RM\left({FD}_4;x,y\right)=4x{y}^2+2x{y}^3+4x^2{y}^2$

$RM\left({FD}_5;x,y\right)=xy+2x{y}^2+2x{y}^3+3x^2{y}^2$

$RM\left({FD}_6;x,y\right)=xy+3x{y}^2+x{y}^3+3x^2{y}^2+x^2{y}^3$

$RM\left({FD}_7;x,y\right)=5x{y}^2+x{y}^3+x^2{y}^2+x^2{y}^3$

$RM\left({FD}_8;x,y\right)=3x{y}^2+3x^2{y}^2+x^2{y}^3$

Using Table 1 and Table 3, we obtained Table 4.

,

Table 4. Computed reverse degree-based descriptors of furan derivatives from FD₁ to FD₈.

Furan Derivatives	${\mathbf{R}\mathbf{M}}_1$	${\mathbf{R}\mathbf{M}}_2$	$\mathbf{R}\mathbf{m}{\mathbf{M}}_2$	$\mathbf{R}\mathbf{R}\mathbf{e}\mathbf{Z}{\mathbf{G}}_3$	$\mathbf{R}\mathbf{F}$	$\mathbf{R}\mathbf{R}$	$\mathbf{R}\mathbf{S}\mathbf{D}\mathbf{D}$	$\mathbf{R}\mathbf{H}$	$\mathbf{R}\mathbf{I}$
FD1	42	36	5.1667	142	84	7.6365	28.667	7.30	9.6500
FD2	38	30	4.583	112	76	7.013	28.33	6.567	8.367
FD3	30	22	4.250	78	60	6.060	24.00	5.667	6.417
FD4	36	30	3.667	112	72	5.983	24.667	5.667	8.167
FD5	28	23	3.417	86	56	5.069	19.667	4.833	6.333
FD6	32	28	3.750	110	64	5.607	21.00	5.40	7.450
FD7	28	23	3.250	88	56	5.021	20.00	4.733	6.283
FD8	26	24	2.417	96	52	4.030	15.667	3.90	6.20

,

Table 5. Edge partition cardinalities of furan derivatives based on reverse neighborhood degrees.

Furan Derivatives	${\mathbf{E}}_{11}$	${\mathbf{E}}_{12}$	${\mathbf{E}}_{13}$	${\mathbf{E}}_{22}$	${\mathbf{E}}_{23}$	${\mathbf{E}}_{24}$	${\mathbf{E}}_{25}$	${\mathbf{E}}_{26}$	${\mathbf{E}}_{33}$	${\mathbf{E}}_{34}$	${\mathbf{E}}_{35}$	${\mathbf{E}}_{44}$	${\mathbf{E}}_{46}$	${\mathbf{E}}_{56}$
FD1	0	2	1	1	2	1	1	1	1	0	1	0	0	1
FD2	3	2	2	1	0	0	0	0	0	2	1	0	0	0
FD3	0	3	0	0	1	0	0	0	2	0	3	0	0	0
FD4	0	3	0	1	3	0	0	0	1	2	0	0	0	0
FD5	0	0	3	0	0	0	0	0	0	2	2	1	0	0
FD6	0	1	2	0	0	1	1	0	0	2	0	1	1	0
FD7	0	3	1	2	0	1	0	0	0	0	1	0	0	0
FD8	0	2	1	0	2	0	0	0	1	0	1	0	0	0

,

Table 6. Reverse NM-polynomial of furan derivatives from FD₁ to FD₈.

$RNM\left({FD}_1;x,y\right)=2x{y}^2+x{y}^3+x^2{y}^2+2x^2{y}^3+x^2{y}^4+x^2{y}^5+x^2{y}^6+x^3{y}^3+x^3{y}^5+x^5{y}^6$

$RNM\left({FD}_2;x,y\right)=3xy+2x{y}^2+2x{y}^3+x^2{y}^2+2x^3{y}^4+x^3{y}^5$

$RNM\left({FD}_3;x,y\right)=3x{y}^2+x^2{y}^3+2x^3{y}^3+{3x}^3{y}^5$

$RNM\left({FD}_4;x,y\right)=3x{y}^2+x^2{y}^2+3x^2{y}^3+x^3{y}^3+2x^3{y}^4$

$RNM\left({FD}_5;x,y\right)=3x{y}^3+2x^3{y}^4+2x^3{y}^5+x^4{y}^4$

$RNM\left({FD}_6;x,y\right)=x{y}^2+2x{y}^3+x^2{y}^4+x^2{y}^5+2x^3{y}^4+x^4{y}^4+x^4{y}^6$

$RNM\left({FD}_7;x,y\right)=x^2{y}^4+x^3{y}^5+x^4{y}^6+{2x}^5{y}^5+3x^5{y}^6$

$RNM\left({FD}_8;x,y\right)=2x{y}^2+x{y}^3+2x^2{y}^3+x^3{y}^3+x^3{y}^5$

Using Table 1 and Table 6, we obtained Table 7.

and

Table 7. Computed reverse neighborhood descriptors of furan derivatives from FD₁ to FD₈.

Furan Derivatives	${\mathbf{R}\mathbf{N}\mathbf{M}}_1$	${\mathbf{R}\mathbf{N}\mathbf{M}}_2$	$\mathbf{R}\mathbf{N}\mathbf{m}{\mathbf{M}}_2$	$\mathbf{R}\mathbf{N}\mathbf{R}\mathbf{e}\mathbf{Z}{\mathbf{G}}_3$	$\mathbf{R}\mathbf{N}\mathbf{F}$	$\mathbf{R}\mathbf{N}\mathbf{R}$	$\mathbf{R}\mathbf{N}\mathbf{S}\mathbf{D}\mathbf{D}$	$\mathbf{R}\mathbf{N}\mathbf{H}$	$\mathbf{R}\mathbf{N}\mathbf{I}$
FD1	70	107	2.436	818	256	5.041	29.7	4.768	15.848
FD2	46	56	5.15	346	128	6.904	26.1	6.655	10.637
FD3	50	75	2.089	516	166	3.971	20.467	3.817	11.825
FD4	48	61	2.528	346	130	4.757	22.167	4.605	11.529
FD5	50	79	1.362	572	180	3.076	20.7	2.821	11.429
FD6	56	90	1.662	684	208	3.563	22.9	3.307	12.757
FD7	35	40	2.525	230	95	4.31	19.6	4.083	7.958
FD8	34	43	1.844	258	98	3.4	16.933	3.217	7.858

.

The Reverse M-polynomial of the first furan derivative $RM\left({FD}_{1\left(r, s\right)};x,y\right)$ is obtained in the following way: the molecular graph of ${FD}_1$ has 12 vertices, 12 edges, and $\Delta \left({FD}_1\right)=3$. The edge set of ${FD}_1$ can be divided into five classes, ${rm}_{11}, {rm}_{12},{rm}_{13, }{rm}_{22}$ and ${rm}_{23}$, on the basis of reverse degree of vertices, with cardinalities, $\left|E_{11}\right|=1, \left|E_{12}\right|=6, \left|E_{13}\right|=1, \left|E_{22}\right|=2$ and $\left|E_{23}\right|=2.$

By definition of the Reverse M-polynomial, $RM\left(G;x,y\right)={\sum }_{i\le j}{rm}_{ij}\left(G\right)x^i{y}^j$

$$\begin{array}{c}RM\left({FD}_1;x,y\right)={\sum }_{1\le 1}{rm}_{11}xy+{\sum }_{1\le 2}{rm}_{12}x{y}^2+{\sum }_{1\le 3}{rm}_{12}x{y}^3+{\sum }_{2\le 2}{rm}_{22}{x}^2{y}^2+\hfill \\ {\sum }_{2\le 3}{rm}_{22}{x}^2{y}^3 \hfill \\ RM\left({FD}_1;x,y\right)=xy+6x{y}^2+x{y}^3+2x^2{y}^2+2x^2{y}^3\hfill \end{array}$$

Similarly, using edge partition techniques based on reverse degree, Table 2 was generated, and Table 3 presents the Reverse M-polynomial of other furan derivatives.

Now we derive the reverse NM-polynomial, $RNM\left({FD}_1;x,y\right)$, we identify the maximum neighborhood degree of ${FD}_1$ as ${\Delta }_1\left(G\right)=7$. The edge set of ${FD}_1$ is then partitioned into ten classes based on the reverse neighborhood degree of its vertices as ${rn}_{12}, {rn}_{13},{rn}_{22},{rn}_{23},{rn}_{24},{rn}_{25},{rn}_{26},{rn}_{33}, {rn}_{35}$ and ${rn}_{56}$, with the following cardinalities: $\left|E_{12}\right|=2, \left|E_{13}\right|=1, \left|E_{22}\right|=1, \left|E_{23}\right|=2, \left|E_{24}\right|=1, \left|E_{25}\right|=1, \left|E_{26}\right|=1, \left|E_{33}\right|=1, \left|E_{35}\right|=1$ and $\left|E_{56}\right|=1.$

By definition of Reverse NM-polynomial, $RNM\left(G;x,y\right)={\sum }_{i\le j}{rn}_{ij}\left(G\right)x^i{y}^j$

$$\begin{array}{c}RNM\left({FD}_1;x,y\right)={\sum }_{1\le 2}{rn}_{12}x{y}^2+{\sum }_{1\le 3}{rn}_{13}x{y}^3+{\sum }_{2\le 2}{rn}_{22}{x}^2{y}^2+{\sum }_{2\le 3}{rn}_{23}{x}^2{y}^3+\hfill \\ {\sum }_{2\le 4}{rn}_{24}{x}^2{y}^4+{\sum }_{2\le 5}{rn}_{25}{x}^2{y}^5+{\sum }_{2\le 6}{rn}_{26}{x}^2{y}^6+{\sum }_{3\le 3}{rn}_{33}{x}^3{y}^3+{\sum }_{3\le 5}{rn}_{35}{x}^3{y}^5+\hfill \\ {\sum }_{5\le 6}{rn}_{56}{x}^5{y}^6\hfill \\ RNM\left({FD}_1;x,y\right)=2x{y}^2+x{y}^3+x^2{y}^2+2x^2{y}^3+x^2{y}^4+x^2{y}^5+x^2{y}^6+x^3{y}^3+x^3{y}^5+x^5{y}^6\hfill \end{array}$$

Similarly, using edge partition techniques based on reverse neighborhood degrees, Table 5 was generated, and Table 6 presents the Reverse NM-polynomial of other furan derivatives.

Table 4 shows that the computed reverse degree-based descriptor values vary systematically across the furan derivatives, reflecting differences in molecular size and connectivity. Table 7 summarizes the corresponding reverse neighborhood degree-based descriptor values, highlighting structural variations among the furan derivatives at the neighborhood level. The graphical representation of Table 4 and Table 7 is shown below in Figure 3.

To examine the relationship between the experimental corrosion inhibition efficiency $\left({IE}_{exp}\%\right)$ of the studied furan derivatives and their molecular structure, an exploratory Quantitative Structure–Property Relationship (QSPR)-based correlation analysis was conducted. Linear regression analysis was employed to assess the statistical association between the calculated reverse topological descriptors (RTI) and the experimental data. The general linear relationship is expressed as

$${IE}_{exp}=\alpha \left(RTI\right)+\beta$$

where $\alpha$ represents the regression coefficient, $\beta$ is the intercept, and RTI denotes the specific reverse topological index considered in the analysis. Linear regression is employed as a simple and interpretable approach suitable for exploratory analysis on small datasets, where the objective is to identify trends rather than develop predictive models.

The statistical quality of the exploratory correlations was evaluated using the Pearson correlation coefficient (r), the p-value, the standard error of estimate (SE), and the Fisher F-statistic (F). The statistical significance of the regression parameters was evaluated using p-values and F-statistics; however, due to the limited sample size, these results are interpreted strictly in an exploratory and descriptive manner.

Although the dataset consists of a limited number of compounds, exploratory QSPR studies based on small datasets are commonly employed for preliminary assessment of descriptor sensitivity in chemical graph theory. The objective of the present analysis is not to develop predictive or validated models, but to identify structural trends and assess the responsiveness of reverse topological descriptors to variations in molecular structure.

The strongest statistical associations between the reverse topological descriptors and inhibition efficiency within the analyzed dataset are summarized in

Table 8. Exploratory regression statistics describing the association between selected reverse degree-based descriptors and experimental inhibition efficiency ${IE}_{exp}\%$, including Pearson correlation coefficient (r), sample size (n), standard error (SE), F-statistic (F), and p-values.

Regression Equation	Descriptor	n	r	p-Value	SE	F
${{IE}_{exp}\%= 18.791+15.008(RmM}_2$)	$Rm{M}_2$	8	0.959	<0.001	4.076	68.660
${IE}_{exp}\%=11.570+11.105(RR$)	$RR$	8	0.959	<0.001	4.055	69.423
${IE}_{exp}\%=12.914+2.773(RSDD$)	$RSDD$	8	0.937	<0.001	5.007	43.463
${IE}_{exp}\%=10.419+11.907(RH$)	$RH$	8	0.959	<0.001	4.098	67.839

. The analysis indicates that the reverse molecular descriptors $Rm{M}_2$, $RR$, and $RH$ exhibit the highest Pearson correlation coefficients with the experimental inhibition efficiency values. In particular, $Rm{M}_2$ and $RR$ show similarly strong correlations, corresponding to coefficients of determination of approximately 0.92, while RH exhibits a comparable level of association. The RSDD index also demonstrates a notable correlation, reflecting the sensitivity of this descriptor to structural features relevant to inhibition behavior. The associated F-statistics and standard errors support the statistical significance of these correlations within the analyzed dataset; however, given the limited sample size, these results are interpreted strictly as exploratory associations rather than evidence of predictive or validated QSPR models.

It is important to note that the topological descriptors used in this study do not explicitly account for electronic properties such as frontier orbital energies or electronegativity effects, which are known to influence corrosion inhibition behavior. The present analysis therefore focuses on structural sensitivity, and incorporation of electronic descriptors remains a direction for future work.

These results suggest that the descriptors $Rm{M}_2$, RR, and RH capture structural features that are strongly associated with inhibition efficiency within the analyzed dataset, suggesting that reverse degree-based descriptors can effectively reflect molecular characteristics relevant to corrosion inhibition.

A correlation analysis was performed to assess the strength and direction of linear associations between the experimental inhibition efficiency ${(IE}_{exp}\%)$ and nine reverse topological descriptors. The results summarized in

Table 9. Pearson correlation coefficients (r), sample size (n) and p-values between ${IE}_{exp}\%$ and reverse degree-based indices (TI).

Descriptor	n	r	p-Value
$R{M}_1$	8	0.878	0.004
$R{M}_2$	8	0.712	0.048
$Rm{M}_2$	8	0.959	<0.001
$RReZ{G}_3$	8	0.612	0.107
$RF$	8	0.878	0.004
$RR$	8	0.959	<0.001
$RSDD$	8	0.937	<0.001
$RH$	8	0.959	<0.001
$RI$	8	0.811	0.015

indicate that all studied descriptors exhibit positive Pearson correlation coefficients with ${IE}_{exp}\%$, reflecting a consistent trend within the analyzed dataset.

The results in Table 9 further indicate that descriptors such as $Rm{M}_2$, $RR$, $RH$, and RSDD exhibit comparatively stronger correlations with the experimental inhibition efficiency than the remaining indices. These findings suggest that the structural features encoded by these reverse topological descriptors are sensitive to variations in inhibition behavior for the studied furan derivatives. However, given the limited sample size, these correlations should be interpreted as exploratory rather than predictive. Figure 4 illustrates these relationships, showing the linear trends between selected reverse topological descriptors and the experimental inhibition efficiency within the dataset.

Additionally, an exploratory QSPR-based correlation analysis was conducted using reverse neighborhood topological descriptors (RNTI) to examine their association with the experimental inhibition efficiency (${IE}_{exp}\%$). The results summarized in

Table 10. Pearson correlation coefficients (r), sample size (n) and p-values between ${IE}_{exp}\%$ and reverse neighborhood degree-based indices (TI).

Descriptor	n	r	p-Value
$R{NM}_1$	8	0.806	0.016
$RN{M}_2$	8	0.671	0.068
$RNm{M}_2$	8	0.449	0.264
$RNReZ{G}_3$	8	0.619	0.102
$RNF$	8	0.668	0.070
$RNR$	8	0.622	0.100
$RNSDD$	8	0.924	<0.001
$RNH$	8	0.609	0.109
$RNI$	8	0.815	0.014

indicate positive Pearson correlations for the considered descriptors, with varying strengths. Among them, RNSDD exhibits the strongest correlation within the analyzed dataset, highlighting its sensitivity to structural variations in the studied furan derivatives. These results should be interpreted as exploratory due to the limited dataset size.

Among the reverse neighborhood descriptors, RNSDD shows the strongest association with inhibition efficiency, highlighting the potential relevance of neighborhood-based structural information in describing variations in inhibitor performance. Figure 5 presents the corresponding relationships based on reverse neighborhood descriptors, illustrating their structural sensitivity to inhibition efficiency and supporting the observed trends.

3.2. Reverse M-Polynomial and Reverse NM-Polynomial for Nanoporous Graphene with 14- Annulene Pores

Graphene is a two-dimensional carbon material with outstanding chemical stability, mechanical strength, and impermeability, which has motivated extensive interest in its potential for anticorrosion-related applications. As a protective coating, graphene can act as a physical barrier that inhibits the diffusion of corrosive species such as oxygen, water, and chloride ions to the metal surface [48,49]. Porous and defect-engineered graphene derivatives can improve coating adhesion and compatibility with different matrices; however, pores, grain boundaries, and structural defects may also act as penetration pathways for corrosive agents and induce localized or galvanic corrosion. Moreover, challenges related to large-scale fabrication, defect control, coating uniformity, substrate adhesion, and long-term durability limit practical implementation. Therefore, a thorough understanding of the structural characteristics of graphene and its derivatives is important for evaluating their potential role in corrosion-protection systems [50,51].

In this section, we examine the structural characteristics of nanoporous graphene containing 14-annulene pores. Such graphene architectures exhibit nonplanar geometries that have attracted interest in areas such as nanoelectronics, sensing, and separation due to their tunable electronic features and well-defined pore structures [52]. In earlier work [53], the structural and spectral properties of nanoporous graphene (NPG) with 14-annulene pores were investigated using graph-theoretical methods. Analytical expressions for selected topological descriptors were derived, and information-theoretic entropy analysis indicated that NPG structures exhibit higher scaled bond-wise entropies than rectangular kekulene-based graphene structures of comparable size, reflecting increased pore size and structural complexity.

For reverse degree-based structural analysis, the nanoporous graphene framework containing 14-annulene pores is modeled as a simple connected chemical graph, denoted by ${{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}$, as shown in Figure 6. The graph consists of r rows and s columns, containing $\left(r+1\right)\left(s-1\right)$ 14-annulene pores, which are highlighted in light yellow in Figure 6 and distributed uniformly throughout the structure. Figure 7 presents representative examples of ${{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}$ for various sizes.

Analytical Expression 1.

Reverse M-polynomial of ${{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}$.

$$RM\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)};x,y\right)=\left(11rs-r+13s-2\right)xy+\left(8rs-4r+16s-8\right)x{y}^2+\left(4r+4s+8\right)x^2{y}^2.$$

Derivation .

The molecular graph ${{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}$ is shown below in Figure 8.

Given 8, we have $\left|V\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)\right|=14rs+26s$ and $\left|E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)\right|=19rs-r+33s-2$.

The edge set of ${{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}$ can be divided into three classes on the basis of the reverse degree of vertices as follows:

$$\begin{array}{c}{E}_{11}=\left\{uv\in E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)|rd\left(u\right)=1,rd\left(v\right)=1\right\},\hfill \\ E_{12}=\left\{uv\in E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)|rd\left(u\right)=1,rd\left(v\right)=2\right\},\hfill \\ E_{22}=\left\{uv\in E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)|rd\left(u\right)=2,rd\left(v\right)=2\right\}\hfill \\ \mathrm{and} \left|E_{11}\right|=11rs-r+13s-2,\left|E_{12}\right|=8rs-4r+16s-8, \left|E_{22}\right|=4r+4s+8\hfill \end{array}$$

By definition of the Reverse M-polynomial,

$$\begin{array}{cc}RM\left(G;x,y\right)=\sum _{i\le j}{rm}_{ij}\left(G\right)x^i{y}^j\hfill & \\ RM\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)};x,y\right)\hfill & =\sum _{1\le 1}{rm}_{11}xy+\sum _{1\le 2}{rm}_{12}x{y}^2+\sum _{2\le 2}{rm}_{22}{x}^2{y}^2 \hfill \\ & =\left|E_{11}\right|xy+\left|E_{12}\right|x{y}^2+\left|E_{22}\right|x^2{y}^2\hfill \\ & =(11rs-r+13s-2)xy+(8rs-4r+16s-8)x{y}^2+(4r+4s+8)x^2{y}^2\hfill \end{array}$$

Applying the Reverse M-polynomial formalism together with the operators in Table 1, the reverse degree-based topological descriptors for ${{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}$ are evaluated as follows.

• Reverse topological descriptor for ${{\mathit{N}\mathit{P}\mathit{G}}_{\left[14\right]}\mathbf{A}\mathbf{P}}_{\left(\mathit{r},\mathit{s}\right)}$.

$$\begin{array}{c}1. {RM}_1\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)=46rs+2r+90s+4 \hfill \\ 2. {RM}_2\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)=27rs+7r+61s+14 \hfill \\ 3. {RmM}_2\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)=15rs-2r+22s-4 \hfill \\ 4. RReZ{G}_3\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)=70rs+38r+186s+76\hfill \\ 5. RF\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)=62rs+10r+138s+20\hfill \\ {6. RR}_k\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)=(11rs-r+13s-2)+2^k(8rs-4r+16s-8)+2^k{2}^k(4r+4s+8),\hfill \\ 7.{ RRR}_k\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)=(11rs-r+13s-2)+(8rs-4r+16s-8){\displaystyle \frac{1}{2^k}}+(4r+4s+8){\displaystyle \frac{1}{2^k}}{\displaystyle \frac{1}{2^k}}\hfill \\ 8. RSDD\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)=42rs-4r+74s-8\hfill \\ 9. RH\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)={\displaystyle \frac{65}{3}}rs+{\displaystyle \frac{5}{3}}r+{\displaystyle \frac{127}{3}}s+{\displaystyle \frac{10}{3}}\hfill \\ 10. RI\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)={\displaystyle \frac{161}{6}}rs+{\displaystyle \frac{317}{6}}r+{\displaystyle \frac{679}{6}}s+{\displaystyle \frac{317}{3}}\hfill \\ 11. RA\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)={\displaystyle \frac{809}{216}}rs+{\displaystyle \frac{581}{216}}r+{\displaystyle \frac{2239}{216}}s+{\displaystyle \frac{581}{108}}\hfill \end{array}$$

• Let

$$\begin{array}{c}\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)};x,y\right)=f\left(x,y\right)=(11rs-r+13s-2)xy+(8rs-4r+16s-8)x{y}^2+(4r+4s+8)x^2{y}^2,\mathrm{then}\hfill \\ D_{x}f\left(x,y\right)=(11rs-r+13s-2)xy+(8rs-4r+16s-8)x{y}^2+2(4r+4s+8)x^2{y}^2\hfill \\ D_{y}f\left(x,y\right)=(11rs-r+13s-2)xy+2(8rs-4r+16s-8)x{y}^2+2(4r+4s+8)x^2{y}^2.\hfill \\ {{(D}_x+D}_y)(f\left(x,y\right))=2(11rs-r+13s-2)xy+3(8rs-4r+16s-8)x{y}^2+4(4r+4s+8)x^2{y}^2.\hfill \\ {D_{y}D}_x(f\left(x,y\right))=(11rs-r+13s-2)xy+2(8rs-4r+16s-8)x{y}^2+4(4r+4s+8)x^2{y}^2.\hfill \\ (D_x^2+D_y^2)(f\left(x,y\right))=2(11rs-r+13s-2)xy+5(8rs-4r+16s-8)x{y}^2+8(4r+4s+8)x^2{y}^2.\hfill \\ {D_x}^k{D_y}^k(f\left(x,y\right))=(11rs-r+13s-2)xy+2^k(8rs-4r+16s-8)x{y}^2+2^k{2}^k(4r+4s+8)x^2{y}^2\hfill \\ {D_{x}D}_y{{(D}_x+D}_y)(f\left(x,y\right)=2(11rs-r+13s-2)xy+6(8rs-4r+16s-8)x{y}^2+16(4r+4s+8)x^2{y}^2.\hfill \\ {S_{x}S}_y(f\left(x,y\right))=(11rs-r+13s-2)xy+(8rs-4r+16s-8){\displaystyle \frac{x{y}^2}{2}}+(4r+4s+8){\displaystyle \frac{x^2}{2}}{\displaystyle \frac{y^2}{2}}.\hfill \\ {S_x}^k{S_y}^k(f\left(x,y\right))=(11rs-r+13s-2)xy+(8rs-4r+16s-8){\displaystyle \frac{x{y}^2}{2^k}}+(4r+4s+8){\displaystyle \frac{x^2}{2^k}}{\displaystyle \frac{y^2}{2^k}}\hfill \\ ({S_{y}D}_x+{S_{x}D}_y)(f\left(x,y\right))=2(11rs-r+13s-2)xy+{\displaystyle \frac{5}{2}}(8rs-4r+16s-8)x{y}^2+2(4r+4s+8){x^{2}y}^2.\hfill \\ S_x J\left(f\left(x,y\right)\right)=(11rs-r+13s-2){\displaystyle \frac{x^2}{2}}+(8rs-4r+16s-8){\displaystyle \frac{x^3}{3}}+(4r+4s+8){\displaystyle \frac{x^4}{4}}.\hfill \\ S_{x}J{D_{y}D}_x(f\left(x,y\right))=(11rs-r+13s-2){\displaystyle \frac{x^2}{2}}+2(8rs-4r+16s-8){\displaystyle \frac{x^3}{3}}+(4r+4s+8)x^4. \hfill \\ {S_x}^3{{Q_{-2}JD}_x}^3{D_y}^3\left(f\left(x,y\right)\right)=(11rs-r+13s-2){\displaystyle \frac{x^2}{2^3}}+2^3(8rs-4r+16s-8){\displaystyle \frac{x^3}{3^3}}+(4r+4s+8)x^4.\hfill \end{array}$$

Table 11. Values for reverse M-polynomial of ${{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}$ with $1\le r\le 10, 1\le s\le 10.$

$\left(\mathit{r},\mathit{s}\right)$	$\mathit{R}{\mathit{M}}_1$	$\mathit{R}{\mathit{M}}_2$	$\mathit{R}\mathit{m}{\mathit{M}}_2$	$\mathit{R}\mathit{R}\mathit{e}\mathit{Z}{\mathit{G}}_3$	$\mathit{R}\mathit{F}$	$\mathit{R}\mathit{R}$	$\mathit{R}\mathit{S}\mathit{D}\mathit{D}$	$\mathit{R}\mathit{H}$	$\mathit{R}\mathit{I}$	$\mathit{R}\mathit{A}$
$\left(1,1\right)$	142	109	31	370	230	37.4852	104	69	298.5000	22.18055
$\left(2,2\right)$	372	258	96	804	564	111.9411	300	178	545	46.47222
$\left(3,3\right)$	694	461	191	1378	1022	219.7106	580	330.3333	845.1667	78.25462
$\left(4,4\right)$	1108	718	316	2092	1604	360.7939	944	526	1199	117.5277
$\left(5,5\right)$	1614	1029	471	2946	2310	535.1909	1392	765	1606.5000	164.2916
$\left(6,6\right)$	2212	1394	656	3940	3140	742.9015	1924	1047.3333	2067.6667	218.5462
$\left(7,7\right)$	2902	1813	871	5074	4094	983.9259	2540	1373	2582.5000	280.2916
$\left(8,8\right)$	3684	2286	1116	6348	5172	1258.2640	3240	1742	3151	349.5277
$\left(9,9\right)$	4558	2813	1391	7762	6374	1565.9158	4024	2154.3333	3773.1667	426.2546
$\left(10,10\right)$	5524	3394	1696	9316	7700	1906.881	489	2610	4449	510.4722

summarizes the computed reverse degree-based topological descriptors for nanoporous graphene nanoribbons with 14-annulene pores as a function of structural size and pore repetition. The table is included to illustrate systematic trends in descriptor values with respect to graph topology, providing reference data for comparative structural analysis of nanoporous graphene architectures.

The results in Table 11 indicate that reverse degree-based descriptors increase monotonically with the size of the graphene nanoribbon, reflecting the growth in vertex connectivity and edge structure induced by pore incorporation. These trends demonstrate the sensitivity of degree-based descriptors to global structural expansion and pore distribution within the graphene framework. Figure 9 presents an interactive visualization of the descriptor values summarized in Table 11.

Analytical Expression 2.

Reverse NM-polynomial of ${{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}$.

$$\begin{gathered} RNM\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)};x,y\right)=\left(8rs+s\right)xy+\left(8rs+4s\right)x{y}^2+\left(6s\right)x^2{y}^2+ \\ \left(4rs-4r+4s-4\right)x^2{y}^4+\left(4r+4s+4\right)x^2{y}^5+\left(rs-r+2s-2\right)x^3{y}^3+ \\ \left(4rs-4r+4s-4\right)x^3{y}^4+\left(4s-4\right)x^3{y}^5+\left(4s-4\right)x^5{y}^5+\left(4r+8\right)x^5{y}^6+\left(4\right)x^6{y}^6. \end{gathered}$$

Derivation .

By applying edge partitioning, degree counting, and structural analysis of the graph, as illustrated in Figure 8, we have

$$\left|V\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)\right|=14rs+26s \mathrm{and} \left|E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)\right|=19rs-r+33s-2$$

The edge set of ${{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}$ can be divided into eleven classes on the basis of reverse neighborhood degree of vertices as follows:

$$\begin{array}{c}{E}_{11}=\left\{uv\in E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)|r\delta \left(u\right)=1,r\delta \left(v\right)=1\right\},\hfill \\ E_{12}=\left\{uv\in E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)|r\delta \left(u\right)=1,r\delta \left(v\right)=2\right\},\hfill \\ E_{22}=\left\{uv\in E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)|r\delta \left(u\right)=2,r\delta \left(v\right)=2\right\},\hfill \\ E_{24}=\left\{uv\in E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)|r\delta \left(u\right)=2,r\delta \left(v\right)=4\right\},\hfill \\ E_{25}=\left\{uv\in E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)|r\delta \left(u\right)=2,r\delta \left(v\right)=5\right\},\hfill \\ E_{33}=\left\{uv\in E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)|r\delta \left(u\right)=3,r\delta \left(v\right)=3\right\},\hfill \\ E_{34}=\left\{uv\in E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)|r\delta \left(u\right)=3,r\delta \left(v\right)=4\right\},\hfill \\ E_{35}=\left\{uv\in E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)|r\delta \left(u\right)=3,r\delta \left(v\right)=5\right\},\hfill \\ E_{55}=\left\{uv\in E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)|r\delta \left(u\right)=5,r\delta \left(v\right)=5\right\},\hfill \\ E_{56}=\left\{uv\in E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)|r\delta \left(u\right)=5,r\delta \left(v\right)=6\right\},\hfill \\ E_{66}=\left\{uv\in E\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)|r\delta \left(u\right)=6,r\delta \left(v\right)=6\right\} \mathrm{and} \hfill \\ \left|E_{11}\right|=8rs+s, \left|E_{12}\right|=8rs+4s, \left|E_{22}\right|=6s,\left|E_{24}\right|=4rs-4r+4s-4, \left|E_{25}\right|=4r+4s+4,\hfill \\ \left|E_{33}\right|=rs-r+2s-2,\left|E_{34}\right|=4rs-4r+4s-4, \left|E_{35}\right|=4s-4,\hfill \\ \left|E_{55}\right|=4s-4,\left|E_{56}\right|=4r+8, \left|E_{66}\right|=4.\hfill \end{array}$$

By the definition of Reverse NM-polynomial

$$\begin{array}{c}RNM\left(G;x,y\right)={\sum }_{i\le j}{rn}_{ij}\left(G\right)x^i{y}^j\hfill \\ RNM\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)};x,y\right)={\sum }_{1\le 1}{rn}_{11}xy+{\sum }_{1\le 2}{rn}_{12}x{y}^2+{\sum }_{2\le 2}{rn}_{22}{x}^2{y}^2+\hfill \\ {\sum }_{2\le 4}{rn}_{24}{x}^2{y}^4+{\sum }_{2\le 5}{rn}_{25}{x}^2{y}^5+{\sum }_{3\le 3}{rn}_{33}{x}^3{y}^3+{\sum }_{3\le 4}{rn}_{34}{x}^3{y}^4+{\sum }_{3\le 5}{rn}_{35}{x}^3{y}^5+\hfill \\ {\sum }_{5\le 5}{rn}_{55}{x}^5{y}^5+{\sum }_{5\le 6}{rn}_{56}{x}^5{y}^6+{\sum }_{6\le 6}{rn}_{66}{x}^6{y}^6\hfill \\ =\left|E_{11}\right|xy+\left|E_{12}\right|x{y}^2+\left|E_{22}\right|x^2{y}^2+\left|E_{24}\right|x^2{y}^4+\left|E_{25}\right|x^2{y}^5+\left|E_{33}\right|x^3{y}^3+\left|E_{34}\right|x^3{y}^4+\hfill \\ \left|E_{35}\right|x^3{y}^5+\left|E_{55}\right|x^5{y}^5+\left|E_{56}\right|x^5{y}^6+\left|E_{66}\right|x^6{y}^6\hfill \\ =\left(8rs+s\right)xy+\left(8rs+4s\right)x{y}^2+\left(6s\right)x^2{y}^2+\left(4rs-4r+4s-4\right)x^2{y}^4+(4r+4s+\hfill \\ 4)x^2{y}^5+\left(rs-r+2s-2\right)x^3{y}^3+\left(4rs-4r+4s-4\right)x^3{y}^4+\left(4s-4\right)x^3{y}^5+(4s-\hfill \\ 4)x^5{y}^5+\left(4r+8\right)x^5{y}^6+\left(4\right)x^6{y}^6\hfill \end{array}$$

Reverse neighborhood topological descriptors for  ${{\mathit{N}\mathit{P}\mathit{G}}_{\left[14\right]}\mathbf{A}\mathbf{P}}_{\left(\mathit{r},\mathit{s}\right)}$.

$$\begin{array}{c}1. {RNM}_1\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)=98rs+14r+202s+28 \hfill \\ 2. {RNM}_2\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)=113rs+71r+331s+166\hfill \\ 3. {RNmM}_2\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)={\displaystyle \frac{233}{18}}rs-{\displaystyle \frac{37}{90}}r+{\displaystyle \frac{1436}{225}}s-{\displaystyle \frac{317}{450}} \hfill \\ 4.RNReZ{G}_3\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)=646rs+1018r+2518s+2532\hfill \\ 5. RNF\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)=254rs+162r+738s+340\hfill \\ {6. RNR}_k\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)=\left(8rs+s\right)+2^k\left(8rs+4s\right)+2^k{2}^k\left(6s\right)+2^k{4}^k(4rs-4r+\hfill \\ 4s-4)+2^k{5}^k\left(4r+4s+4\right)+3^k{3}^k\left(rs-r+2s-2\right)+3^k{4}^k\left(4rs-4r+4s-4\right)+\hfill \\ 3^k{5}^k\left(4s-4\right)+5^k{5}^k\left(4s-4\right)+5^k{6}^k\left(4r+8\right)+6^k{6}^k\left(4\right),\hfill \\ 7.{ RNRR}_k\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)=\left(8rs+s\right)+\left(8rs+4s\right){\displaystyle \frac{1}{2^k}}+\left(6s\right){\displaystyle \frac{1}{2^k}}{\displaystyle \frac{1}{2^k}}+(4rs-4r+4s-\hfill \\ 4){\displaystyle \frac{1}{2^k}}{\displaystyle \frac{1}{4^k}}+\left(4r+4s+4\right){\displaystyle \frac{1}{2^k}}{\displaystyle \frac{1}{5^k}}+\left(rs-r+2s-2\right){\displaystyle \frac{1}{3^k}}{\displaystyle \frac{1}{3^k}}+\left(4rs-4r+4s-4\right){\displaystyle \frac{1}{3^k}}{\displaystyle \frac{1}{4^k}}+\hfill \\ \left(4s-4\right){\displaystyle \frac{1}{3^k}}{\displaystyle \frac{1}{5^k}}+\left(4s-4\right){\displaystyle \frac{1}{5^k}}{\displaystyle \frac{1}{5^k}}+\left(4r+8\right){\displaystyle \frac{1}{5^k}}{\displaystyle \frac{1}{6^k}}+\left(4\right){\displaystyle \frac{1}{6^k}}{\displaystyle \frac{1}{6^k}}\hfill \\ 8. RNSDD\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)={\displaystyle \frac{169}{3}}rs-{\displaystyle \frac{3}{5}}r+75s-{\displaystyle \frac{53}{15}}\hfill \\ 9. RNH\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)={\displaystyle \frac{113}{7}}rs-{\displaystyle \frac{31}{33}}r+{\displaystyle \frac{1339}{105}}s-{\displaystyle \frac{277}{165}}\hfill \\ 10. RNI\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)={\displaystyle \frac{967}{42}}rs+{\displaystyle \frac{1355}{462}}r+{\displaystyle \frac{333}{7}}s-{\displaystyle \frac{277}{165}}\hfill \\ 11. RNA\left({{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}\right)={\displaystyle \frac{\mathrm{1,365,493}}{8000}}rs+{\displaystyle \frac{\mathrm{17,595,689}}{\mathrm{216,000}}}r+{\displaystyle \frac{\mathrm{3,380,522,361}}{\mathrm{16,000}}}s-{\displaystyle \frac{\mathrm{91,002,842,179}}{\mathrm{432,000}}}\hfill \end{array}$$

Let (NPG_[14]AP_(r,s); x, y) = f(x, y) = (8rs + s)xy + (8rs + 4s)xy² + (6s)x²y² + (4rs − 4r + 4s − 4)x²y⁴ + (4r + 4s + 4)x²y⁵ + (rs − r + 2s − 2)x³y³ + (4rs − 4r + 4s − 4)x³y⁴ + (4s − 4)x³y⁵ + (4s − 4)x⁵y⁵ + (4r + 8)x⁵y⁶ + (4)x⁶y⁶, then

$$\begin{array}{c}{D}_{x}f\left(x,y\right)=\left(8rs+s\right)xy+\left(8rs+4s\right)x{y}^2+2\left(6s\right)x^2{y}^2+2\left(4rs-4r+4s-4\right)x^2{y}^4+\hfill \\ 2\left(4r+4s+4\right)x^2{y}^5+3\left(rs-r+2s-2\right)x^3{y}^3+3\left(4rs-4r+4s-4\right)x^3{y}^4+3(4s-\hfill \\ 4)x^3{y}^5+5\left(4s-4\right)x^5{y}^5+5\left(4r+8\right)x^5{y}^6+6\left(4\right)x^6{y}^6\hfill \\ D_{y}f\left(x,y\right)=\left(8rs+s\right)xy+2\left(8rs+4s\right)x{y}^2+2\left(6s\right)x^2{y}^2+4(4rs-4r+4s-\hfill \\ 4)x^2{y}^4+5\left(4r+4s+4\right)x^2{y}^5+3\left(rs-r+2s-2\right)x^3{y}^3+4(4rs-4r+4s-\hfill \\ 4)x^3{y}^4+5\left(4s-4\right)x^3{y}^5+5\left(4s-4\right)x^5{y}^5+6\left(4r+8\right)x^5{y}^6+6\left(4\right)x^6{y}^6.\hfill \\ {{(D}_x+D}_y)(f\left(x,y\right))=2\left(8rs+s\right)xy+3\left(8rs+4s\right)x{y}^2+4\left(6s\right)x^2{y}^2+6(4rs-4r+\hfill \\ 4s-4)x^2{y}^4+7\left(4r+4s+4\right)x^2{y}^5+6\left(rs-r+2s-2\right)x^3{y}^3+7(4rs-4r+4s-\hfill \\ 4)x^3{y}^4+8\left(4s-4\right)x^3{y}^5+10\left(4s-4\right)x^5{y}^5+11\left(4r+8\right)x^5{y}^6+12\left(4\right)x^6{y}^6.\hfill \\ {D_{y}D}_x\left(f\left(x,y\right)\right)=\left(8rs+s\right)xy+2\left(8rs+4s\right)x{y}^2+4\left(6s\right)x^2{y}^2+8(4rs-4r+4s-\hfill \\ 4)x^2{y}^4+10\left(4r+4s+4\right)x^2{y}^5+9\left(rs-r+2s-2\right)x^3{y}^3+12(4rs-4r+4s-\hfill \\ 4)x^3{y}^4+15\left(4s-4\right)x^3{y}^5+25\left(4s-4\right)x^5{y}^5+30\left(4r+8\right)x^5{y}^6+36\left(4\right)x^6{y}^6.\hfill \\ \left(D_x^2+D_y^2\right)\left(f\left(x,y\right)\right)=2\left(8rs+s\right)xy+5\left(8rs+4s\right)x{y}^2+8\left(6s\right)x^2{y}^2+20(4rs-4r+\hfill \\ 4s-4)x^2{y}^4+29\left(4r+4s+4\right)x^2{y}^5+18\left(rs-r+2s-2\right)x^3{y}^3+25(4rs-4r+4s-\hfill \\ 4)x^3{y}^4+34\left(4s-4\right)x^3{y}^5+50\left(4s-4\right)x^5{y}^5+61\left(4r+8\right)x^5{y}^6+72\left(4\right)x^6{y}^6.\hfill \\ {D_x}^k{D_y}^k(f\left(x,y\right)=\left(8rs+s\right)xy+2^k\left(8rs+4s\right)x{y}^2+2^k{2}^k\left(6s\right)x^2{y}^2+2^k{4}^k(4rs-4r+\hfill \\ 4s-4)x^2{y}^4+2^k{5}^k\left(4r+4s+4\right)x^2{y}^5+3^k{3}^k\left(rs-r+2s-2\right)x^3{y}^3+3^k{4}^k(4rs-\hfill \\ 4r+4s-4)x^3{y}^4+3^k{5}^k\left(4s-4\right)x^3{y}^5+5^k{5}^k\left(4s-4\right)x^5{y}^5+5^k{6}^k\left(4r+8\right)x^5{y}^6+\hfill \\ 6^k{6}^k\left(4\right)x^6{y}^6\hfill \\ {D_{x}D}_y{{(D}_x+D}_y)(f\left(x,y\right)=2\left(8rs+s\right)xy+6\left(8rs+4s\right)x{y}^2+16\left(6s\right)x^2{y}^2+48(4rs-\hfill \\ 4r+4s-4)x^2{y}^4+70\left(4r+4s+4\right)x^2{y}^5+54\left(rs-r+2s-2\right)x^3{y}^3+84(4rs-4r+\hfill \\ 4s-4)x^3{y}^4+120\left(4s-4\right)x^3{y}^5+250\left(4s-4\right)x^5{y}^5+330\left(4r+8\right)x^5{y}^6+432\left(4\right)x^6{y}^6.\hfill \\ {S_{x}S}_y(f\left(x,y\right))=\left(8rs+s\right)xy+\left(8rs+4s\right)x{\displaystyle \frac{y^2}{2}}+\left(6s\right){\displaystyle \frac{x^2}{2}}{\displaystyle \frac{y^2}{2}}+(4rs-4r+4s-\hfill \\ 4){\displaystyle \frac{x^2}{2}}{\displaystyle \frac{y^4}{4}}+\left(4r+4s+4\right){\displaystyle \frac{x^2}{2}}{\displaystyle \frac{y^5}{5}}+\left(rs-r+2s-2\right){\displaystyle \frac{x^3}{3}}{\displaystyle \frac{y^3}{3}}+\left(4rs-4r+4s-4\right){\displaystyle \frac{x^3}{3}}{\displaystyle \frac{y^4}{4}}+\hfill \\ \left(4s-4\right){\displaystyle \frac{x^3}{3}}{\displaystyle \frac{y^5}{5}}+\left(4s-4\right){\displaystyle \frac{x^5}{5}}{\displaystyle \frac{y^5}{5}}+\left(4r+8\right){\displaystyle \frac{x^5}{5}}{\displaystyle \frac{y^6}{6}}+\left(4\right){\displaystyle \frac{x^6}{6}}{\displaystyle \frac{y^6}{6}}.\hfill \\ {S_x}^k{S_y}^k(f\left(x,y\right))=\left(8rs+s\right)xy+\left(8rs+4s\right)x{\displaystyle \frac{y^2}{2^k}}+\left(6s\right){\displaystyle \frac{x^2}{2^k}}{\displaystyle \frac{y^2}{2^k}}+(4rs-4r+4s-\hfill \\ 4){\displaystyle \frac{x^2}{2^k}}{\displaystyle \frac{y^4}{4^k}}+\left(4r+4s+4\right){\displaystyle \frac{x^2}{2^k}}{\displaystyle \frac{y^5}{5^k}}+\left(rs-r+2s-2\right){\displaystyle \frac{x^3}{3^k}}{\displaystyle \frac{y^3}{3^k}}+\left(4rs-4r+4s-4\right){\displaystyle \frac{x^3}{3^k}}{\displaystyle \frac{y^4}{4^k}}+\hfill \\ \left(4s-4\right){\displaystyle \frac{x^3}{3^k}}{\displaystyle \frac{y^5}{5^k}}+\left(4s-4\right){\displaystyle \frac{x^5}{5^k}}{\displaystyle \frac{y^5}{5^k}}+\left(4r+8\right){\displaystyle \frac{x^5}{5^k}}{\displaystyle \frac{y^6}{6^k}}+\left(4\right){\displaystyle \frac{x^6}{6^k}}{\displaystyle \frac{y^6}{6^k}}.\hfill \\ {S_{y}D}_x+{S_{x}D}_y)(f\left(x,y\right))=2\left(8rs+s\right)xy+{\displaystyle \frac{5}{2}}\left(8rs+4s\right)x{y}^2+2\left(6s\right)x^2{y}^2+{\displaystyle \frac{5}{2}}(4rs-4r+\hfill \\ 4s-4)x^2{y}^4+{\displaystyle \frac{29}{10}}\left(4r+4s+4\right)x^2{y}^5+2\left(rs-r+2s-2\right)x^3{y}^3+{\displaystyle \frac{25}{12}}(4rs-4r+4s-\hfill \\ 4)x^3{y}^4+{\displaystyle \frac{34}{15}}\left(4s-4\right)x^3{y}^5+2\left(4s-4\right)x^5{y}^5+{\displaystyle \frac{61}{30}}\left(4r+8\right)x^5{y}^6+2\left(4\right)x^6{y}^6.\hfill \\ S_x J\left(f\left(x,y\right)\right)=\left(8rs+s\right){\displaystyle \frac{x^2}{2}}+\left(8rs+4s\right){\displaystyle \frac{x^3}{3}}+\left(6s\right){\displaystyle \frac{x^4}{4}}+\left(4rs-4r+4s-4\right){\displaystyle \frac{x^6}{6}}+(4r+\hfill \\ 4s+4){\displaystyle \frac{x^7}{7}}+\left(rs-r+2s-2\right){\displaystyle \frac{x^6}{6}}+\left(4rs-4r+4s-4\right){\displaystyle \frac{x^7}{7}}+\left(4s-4\right){\displaystyle \frac{x^8}{8}}+(4s-\hfill \\ 4){\displaystyle \frac{x^{10}}{10}}+\left(4r+8\right){\displaystyle \frac{x^{11}}{11}}+\left(4\right){\displaystyle \frac{x^{12}}{12}}\hfill \\ S_{x}J{D_{y}D}_x(f\left(x,y\right))=\left(8rs+s\right){\displaystyle \frac{x^2}{2}}+2\left(8rs+4s\right){\displaystyle \frac{x^3}{3}}+4\left(6s\right){\displaystyle \frac{x^4}{4}}+8(4rs-4r+4s-\hfill \\ 4){\displaystyle \frac{x^6}{6}}+10\left(4r+4s+4\right){\displaystyle \frac{x^7}{7}}+9\left(rs-r+2s-2\right){\displaystyle \frac{x^6}{6}}+12\left(4rs-4r+4s-4\right){\displaystyle \frac{x^7}{7}}+\hfill \\ 15\left(4s-4\right){\displaystyle \frac{x^8}{8}}+25\left(4s-4\right){\displaystyle \frac{x^{10}}{10}}+30\left(4r+8\right){\displaystyle \frac{x^{11}}{11}}+36\left(4\right){\displaystyle \frac{x^{12}}{12}}\hfill \\ {S_x}^3{{Q_{-2}JD}_x}^3{D_y}^3\left(f\left(x,y\right)\right)=\left(8rs+s\right)+2^3\left(8rs+4s\right)x+2^3\left(6s\right)x^2+2^3(4rs-\hfill \\ 4r+4s-4)x^4+2^3\left(4r+4s+4\right)x^5+3^3{3}^3\left(rs-r+2s-2\right){\displaystyle \frac{x^4}{4^3}}+3^3{4}^3(4rs-4r+\hfill \\ 4s-4){\displaystyle \frac{x^5}{5^3}}+3^3{5}^3\left(4s-4\right){\displaystyle \frac{x^6}{6^3}}+5^3{5}^3\left(4s-4\right){\displaystyle \frac{x^8}{8^3}}+5^3{6}^3\left(4r+8\right){\displaystyle \frac{x^9}{9^3}}+6^3{6}^3\left(4\right){\displaystyle \frac{x^{10}}{{10}^3}}\hfill \end{array}$$

Table 11 and

Table 12. The numerical values of selected reverse neighborhood degree-based topological descriptors, derived from the Reverse NM-polynomial, for nanoporous graphene nanoribbons with 14-annulene pores over a range of structural sizes. Values of reverse NM-polynomial of ${{NPG}_{\left[14\right]}\mathrm{A}\mathrm{P}}_{\left(r, s\right)}$ with $1\le r\le 10, 1\le s\le 10.$

$\left(\mathit{r},\mathit{s}\right)$	$\mathit{R}\mathit{N}{\mathit{M}}_1$	$\mathit{R}\mathit{N}{\mathit{M}}_2$	$\mathit{R}\mathit{N}\mathit{m}{\mathit{M}}_2$	$\mathit{R}\mathit{N}\mathit{R}\mathit{e}\mathit{Z}{\mathit{G}}_3$	$\mathit{R}\mathit{N}\mathit{F}$	$\mathit{R}\mathit{N}\mathit{R}$	$\mathit{R}\mathit{N}\mathit{S}\mathit{D}\mathit{D}$	$\mathit{R}\mathit{N}\mathit{H}$	$\mathit{R}\mathit{N}\mathit{I}$	$\mathit{R}\mathit{N}\mathit{A}$
$\left(1,1\right)$	342	681	18.2111	6714	1494	27.1368	127.2000	26.2770	80.3701	880.0684
$\left(2,2\right)$	852	1422	63.0155	12,188	3156	89.0678	370.6000	86.5186	199.9458	212,756.2374
$\left(3,3\right)$	1558	2389	133.7088	18,954	5326	184.1168	726.6666	179.0458	365.5692	424,973.7796
$\left(4,4\right)$	2460	3582	230.2911	27,012	8004	312.2838	1195.4000	303.8588	577.2402	637,532.6951
$\left(5,5\right)$	3558	5001	352.7622	36,362	11,190	473.5688	1776.8000	460.9575	834.9588	850,432.9838
$\left(6,6\right)$	4852	6646	501.1222	47,004	14,884	667.9718	2470.8666	650.3419	1138.7251	1,063,674.646
$\left(7,7\right)$	6342	8517	675.3711	58,938	19,086	895.4928	3277.6000	872.0121	1488.5389	1,277,257.681
$\left(8,8\right)$	8028	10614	875.5088	72,164	23,796	1156.1318	4197.000	1125.9679	1884.4004	1,491,182.089
$\left(9,9\right)$	9910	12937	1101.5355	86,682	29,014	1449.8888	5229.0666	1412.2095	2326.3095	1,705,447.871
$\left(10,10\right)$	11988	15486	1353.4511	102,492	34,740	1776.7638	6373.8000	1730.7367	2814.2662	1,920,055.026

compare reverse degree-based and reverse neighborhood degree-based topological descriptors derived from the Reverse M-polynomial and Reverse NM-polynomial, respectively, for nanoporous graphene nanoribbons of increasing size. All reported descriptors exhibit a monotonic increase with system size, reflecting the growth and increasing structural complexity of the graphene framework. The reverse M-polynomial-based descriptors (Table 11) display a gradual and stable increase, indicating sensitivity to global structural expansion. In contrast, the reverse NM-polynomial-based descriptors (Table 12) increase more rapidly due to the incorporation of neighborhood degree information, which enhances sensitivity to local connectivity and pore arrangement. As a result, reverse NM-based descriptors emphasize local structural variation, whereas reverse M-based descriptors reflect overall topological scaling. Since experimental corrosion inhibition data are not available for the considered graphene systems, no direct correlation with inhibition efficiency is established. Instead, this analysis is intended to examine descriptor scaling behavior and structural sensitivity, without implying predictive performance. These results describe the structural scaling behavior of the descriptors and do not imply any direct correlation with corrosion inhibition efficiency for graphene systems.

4. Conclusions

In this study, a unified theoretical framework based on reverse degree-based topological descriptors was presented to explore structural characteristics relevant to corrosion-related molecular and nanostructured systems. Reverse M-polynomial and Reverse NM-polynomial formalisms were employed to derive degree and neighborhood degree-based descriptors within a consistent analytical framework, enabling comparative analysis across different structural scales.

For organic corrosion inhibitors, an exploratory QSPR-based correlation analysis was conducted for a limited set of furan derivatives. The results revealed statistically significant associations between selected reverse topological descriptors and experimentally reported inhibition efficiency values, indicating the sensitivity of these descriptors to molecular structural variations. Among the studied descriptors, (RmM₂), (RR), and (RH) consistently exhibited the strongest associations, highlighting their potential relevance for future structure–property investigations. Given the small dataset size, these findings are interpreted strictly as exploratory correlations rather than predictive or validated QSPR models.

The framework was further extended to nanoporous graphene nanoribbons containing 14-annulene pores, where analytical expressions and numerical benchmarks for reverse degree- and reverse neighborhood degree-based descriptors were derived. This part of the study focused exclusively on structural characterization, providing descriptor-level benchmarks that capture the influence of system size and pore arrangement on graphene topology. No direct corrosion performance or barrier efficiency was modeled for these graphene systems.

Overall, this study clarifies the scope and applicability of reverse polynomial descriptors in corrosion-related research by demonstrating their utility in structural sensitivity analysis and analytical benchmarking. The presented results establish a foundation for future structure–property investigations incorporating larger datasets and experimentally validated corrosion performance metrics. Future work may also integrate electronic structure descriptors alongside topological indices to provide a more comprehensive description of inhibition behavior.