EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5992 ISSN 1307-5543 – ejpam.com Published by New York Business Global Stability and Maximum Independent Bond Set Polynomials of Painkiller Molecules Using Maximum Matching Jiwan Jalal Ali1, Didar Abdulkhaleq Ali1,∗ 1 Department of Mathematics, College of Science, University of Zakho, Zakho, Iraq. Abstract. Chemical graph theory establishes a connection between the properties of molecules and their corresponding molecular graphs. A topological index is a graph invariant that char- acterizes the graph’s structure and remains unaffected by graph automorphisms. In chemical graph theory, degree-based topological indices are particularly significant, offering crucial insights into the structural features of molecules. In this work, we introduce the maximum independent bond set polynomial MIBSP (H;x, y), a powerful tool for deriving various degree-based topolog- ical indices. We specifically apply MIBSP (H;x, y), to the chemical graphs of several painkiller molecules, including Aspirin, Paracetamol, Caffeine, Ibuprofen, Phenacetin, and Salicylic acid. The degree-based topological indices derived from these polynomials provide a deeper understanding of the molecular structures and their potential applications in pharmaceutical research. 2020 Mathematics Subject Classifications: 05C05, 05C07, 05C10, 94C15 Key Words and Phrases: Kekule Structure, Maximum matching, Graph Polynomials, Topo- logical indices, Painkiller 1. Introduction Pain is an undesirable sensation that can run from gentle, limited uneasiness to misery. Pain is an inherent aspect of daily life for humans and remains one of the oldest challenges in the field of medicine. The most widely used approach for managing pain is medication, and in recent times, there has been a noticeable rise in the use of analgesics. This increase is largely driven by factors such as growing self-medication habits and greater availability of over-the-counter (OTC) drugs [1–4]. Analgesics are currently among the most widely used drugs globally, with millions of individuals estimated to take over-the-counter (OTC) painkillers on a daily basis [5, 6]. The use of analgesics may have effects that extend beyond simply alleviating pain, as changes in psychological and social factors such as diminished responses to emotionally charged images and decreased reports of social pain have been observed with analgesic use [7, 8]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5992 Email addresses: jiwan.ali@uoz.edu.krd (J. J. Ali), didar.ali@uoz.edu.krd (D. A. Ali) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 2 of 22 Pain is a complex, subjective experience that encompasses physical, psychological, and social dimensions. Pain can originate from various causes, such as injury, illness, inflam- mation, and nerve damage [9]. It can significantly impact an individual’s overall quality of life, hindering their ability to carry out daily tasks, meet work responsibilities, and engage in social activities [10, 11]. The process of sensing pain involves the activation of sensory neurons, the conduction of electrical and chemical signals through neural pathways, and the interpretation of these signals by regions of the brain responsible for higher functions [12, 13]. The creation and regulation of pain sensation are influenced by a variety of mech- anisms [14]. As seen in studies like that of Hargreave et al. [15], this study also supports the notion of a gender disparity in analgesic consumption. Among the 54.0% of students who reported using analgesics, 36.0% were female, while 18.0% were male. Chemical graph theory plays a significant role in the study of chemical structures, representing them as molecular graphs where vertices correspond to atoms and edges represent bonds. These molecular graphs are used to derive molecular descriptors or topological indices, which are numerical values that predict the physical, chemical, and biological properties of molecules [16, 17]. The concept of topological indices was first applied to study the physical properties of chemical structures in 1947 [18]. These indices play a crucial role in Quantitative Structure-Activity Relationship (QSAR) and Quanti- tative Structure-Property Relationship (QSPR) studies, which are essential for predicting the bioactivity and physicochemical properties of novel drug molecules. Molecular descriptors are based on factors such as the vertex degree of the graph, the distance between vertices, the eigenvalues of the graph, and other related properties. Using topological polynomials, rather than calculating molecular descriptors individually, simplifies the process of providing details regarding the molecular graph. M-polynomials are formulas derived from molecular descriptors that depend on the vertex degrees of a graph G, as defined in [19]. Building on this definition, NM-polynomials are also intro- duced. These polynomials are influenced by the sum of the degrees of adjacent vertices [20, 21]. Cycle-related graphs represent the molecular structures of various chemical com- pounds, such as cycloalkanes, and are also used as graph representations for many types of networks [22]. Total-eccentricity polynomials were examined in [23]. The calculation of the Mostar index for cycle-related chemical structures was performed in [24]. Topological numbers in a uniform intuitionistic fuzzy environment and their application in neural net- works were explored in [25]. The determination of Leap-Zagreb indices for cycle-related special graphs was conducted in [26], and the M-polynomials of cycle-related graphs were explored in [27]. The edge-Mostar index and Mostar index of cycle-related graphs were computed in [28], while the reciprocal-leap indices of wheel graphs were studied in [29]. Topological numbers of fuzzy soft graphs and their applications in globalizing the world through mutual trade are discussed in [30]. The precise values for the Randic, Zagreb, Harmonic, Augmented Zagreb, atom-bond connectivity, and geometric-arithmetic indices of Benzenoid networks were studied in [31]. A notable concept in chemical graph theory is the Kekulé structure, introduced by F.A. Kekulé to describe the resonance forms of aromatic compounds [32]. These struc- tures illustrate alternating single and double bonds in conjugated systems, such as the J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 3 of 22 benzene ring. Kekulé structures are fundamentally linked to the stability and resonance energy of aromatic compounds because they depict the delocalization of π-electrons across the molecule [33, 34]. For example, benzene possesses multiple Kekulé structures, a fea- ture that significantly contributes to its exceptional stability and aromatic character. The proposed MIBSP advances chemical graph theory by unifying two previously disjoint ap- proaches: (1) degree-based indices (e.g., Zagreb, Randic) that ignore bond arrangements, and (2) matching enumerations (e.g., Hosoya index) that disregard vertex degrees. Unlike M-polynomials which capture degree correlations but not matchings or Kekulé structures restricted to alternant hydrocarbons, MIBSP’s polynomial formulation simultaneously en- codes both features through its edge-weighted matching sum . 2. Key Concepts and Background Literature Throughout this article, suppose H(V (H), E(H)) is an undirected graph of a chemical structure, V (H) represent the set of vertices and E(H) represent the set of edges. Number of vertices incident with u in a graph H is called degree of u denoted by du or dH(u). The minimum and maximum degree in H denoted by δ and ∆, respectively. A subset X of the edge set of a graph H is called independent if no two edges of X are adjacent in H. A matching in a graph H is a set of independent edges X. A maximal matching in a graph H is a matching that cannot be enlarged by adding another edge because the edges are incident to some independent edges. A matching X in a graph H is a maximum matching if H contains no matching X ′ with |X ′| > |X|. The cardinality of the maximum matching in H is referred to as the matching number of H denoted by β(H) [35]. In graph theory, a matching in a graph H consists of independent edges with no shared vertices. This concept mirrors the Kekulé structure of aromatic compounds, where alternating single and double bonds represent independent bonding interactions. Each independent edge in the matching corresponds to one alternating bond in the Kekulé structure. By applying perfect matchings to Kekulé structures, we can quantify the delo- calization and stability of bonding interactions in aromatic compounds, offering insights into their resonance behavior and structural stability. Definition 1. [19] Consider H as a simple, connected and undirected graph. The M- polynomial for H is defined as follows: M(H;x, y) = ∑ δ≤i≤j≤∆ mi,jx iyj, where mi,j denotes the number of edges e = uv ∈ E(H) such that {dH(u), dH(v)} = {i, j}. J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 4 of 22 We define the maximum independent bond set polynomials as: Definition 2. Let H be a simple undirected graph of order p and size q. Then the maxi- mum independent bond set polynomial of H is defined as: MIBSP (H;x, y) = α∑ r=1 ∑ ∀uv∈Xr δ≤i≤j≤∆ mi,jx iyj, where α is the number of maximum matchings in H, Xr, r = 1, 2, ..., α are the sets of maximum matchings in H, and mi,j denotes the number of edges e = uv ∈ E(H) such that {dH(u), dH(v)} = {i, j}. Our definition can be rephrased in another way MIBSP (H;x, y) = α∑ r=1 M(Xr(H);x, y), where α is the number of maximum matchings in H, Xr, r = 1, 2, ..., α are the sets of maximum matchings in H, and M(Xr(H);x, y) is the M-polynomial of a maximum matchings Xr for r = 1, 2, ..., α. The Maximum Independent Bond Set Polynomial (MIBSP) is the first method to com- bine maximum matchings with degree-based indices, introducing novel descriptors (e.g. MM1(H), MM2(H)) for chemical graph analysis. It unifies bonding patterns and struc- tural features, enhancing drug stability prediction (e.g., Caffeine’s perfect matching) and materials science applications like graphene studies. MIBSP outperforms traditional in- dices by quantifying bond sets that Kekulé structures and topological indices miss, offering new QSAR/QSPR capabilities. The first Zagreb index, denoted by M1, and the second Zagreb index, denoted by M2, are introduced in [36], and are defined as follows: M1(H) = ∑ uv∈E(H) (du + dv), M2(H) = ∑ uv∈E(H) (dudv). The general Randic index was introduced in [37], Rγ(H) = ∑ uv∈E(H) (dudv) γ . Harmonic index defined in [38], and the inverse sum index established in [39], H(H) = ∑ uv∈E(H) 2 du+dv , I(H) = ∑ uv∈E(H) dudv du+dv . Symmetric division degree index SSD(H), introduced in [40] formulated as: SSD(H) = ∑ uv∈E(H) (Min(du,dv) Max(du,dv) + Max(du,dv) Min(du,dv) ). J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 5 of 22 The following Table 1 enlisted some standard topological indices based on degrees of maximum independent bond set and their derivation from the maximum independent bond set polynomials, where Dx = x∂f(x,y) ∂x , Dy = y ∂f(x,y) ∂y , Sx = ∫ x 0 f(x,y) t dt, Sy = ∫ y 0 f(x,y) t dt, Jf(x, y) = f(x, x). Table 1: Matching degree-based topological indices: their formulas and derivations from MIBSP (H;x, y). Matching degree-based topological indices Formula Derivation from MIBSP (H;x, y) MM1(H) α∑ r=1 ∑ ∀uv∈Xr (du + dv) = α∑ r=1 M1(Xr) (Dx +Dy)MIBSP (H)|x=y=1 MM2(H) α∑ r=1 ∑ ∀uv∈Xr (dudv) = α∑ r=1 M2(Xr) (DxDy)MIBSP (H)|x=y=1 MRγ(H) α∑ r=1 ∑ ∀uv∈Xr (dudv) γ = α∑ r=1 Rγ(Xr) (Dγ x +Dγ y )MIBSP (H)|x=y=1 MH(H) α∑ r=1 ∑ ∀uv∈Xr 2 du+dv = α∑ r=1 H(Xr) (2SxJ)MIBSP (H)|x=y=1 MI(H) α∑ r=1 ∑ ∀uv∈Xr dudv du+dv = α∑ r=1 I(Xr) (SxJDxDy)MIBSP (H)|x=y=1 MSSD(H) α∑ r=1 ∑ ∀uv∈Xr (Min(du,dv) Max(du,dv) + Max(du,dv) Min(du,dv) ) (SyDx + SxDy)MIBSP (H)|x=y=1 = α∑ r=1 SSD(Xr) 3. Main Results This section divided in to two subsections. 3.1. Derivations of Matching Degree-Based Topological Indices In this subsection, we give proofs of closed formulas mentioned in Table 1. Given uv ∈ E(H), with du ≤ dv, the operations Dx, Dy, D γ x, D γ y , Sx, Sy, I and J defined on MIBSP (H;x, y), are given by: • Dx(MIBSP (H;x, y)) = x ∂ ∂(x)( α∑ r=1 ∑ ∀uv∈Xr xduydv) = α∑ r=1 ∑ ∀uv∈Xr dux duydv . • D2 x(MIBSP (H;x, y)) = x ∂ ∂(x)(x ∂ ∂(x)( α∑ r=1 ∑ ∀uv∈Xr xduydv)) J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 6 of 22 = α∑ r=1 ∑ ∀uv∈Xr d2ux duydv . • DxDy(MIBSP (H;x, y)) = x ∂ ∂(x)(y ∂ ∂(y)( α∑ r=1 ∑ ∀uv∈Xr xduydv)) =x ∂ ∂(x)( α∑ r=1 ∑ ∀uv∈Xr dvx duydv) = α∑ r=1 ∑ ∀uv∈Xr dudvx duydv . • Sx(MIBSP (H;x, y)) = Sx( α∑ r=1 ∑ ∀uv∈Xr xduydv) = α∑ r=1 ∑ ∀uv∈Xr 1 du xduydv . • Dγ x(MIBSP (H;x, y)) = Dγ x( α∑ r=1 ∑ ∀uv∈Xr xduydv) = α∑ r=1 ∑ ∀uv∈Xr dγuxduydv . • J(MIBSP (H;x, y)) = MIBSP (H;x, x) = α∑ r=1 ∑ ∀uv∈Xr xdu+dv . Similarly, • Dy(MIBSP (H;x, y)) = α∑ r=1 ∑ ∀uv∈Xr dvx duydv . • D2 y(MIBSP (H;x, y)) = α∑ r=1 ∑ ∀uv∈Xr d2vx duydv . • Sy(MIBSP (H;x, y)) = α∑ r=1 ∑ ∀uv∈Xr 1 dv xduydv . • Dγ y (MIBSP (H;x, y)) = α∑ r=1 ∑ ∀uv∈Xr dγvxduydv . The matching degree-based topological indices are given in the following theorem. Theorem 1. Let MIBSP (H;x, y) denote the maximum independent bond set polynomial of a graph H. Then, the matching degree-based topological indices are given by: J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 7 of 22 (i) The matching first Zagreb index is given by: MM1(H) = (Dx +Dy)MIBSP (H;x, y)|x=y=1. (ii) The matching second Zagreb index is given by: MM2(H) = (DxDy)MIBSP (H;x, y)|x=y=1 (iii) The matching general Randic index is given by: MRγ(H) = (Dγ xD γ y )MIBSP (H;x, y)|x=y=1 (iv) The matching Harmonic index is given by: MH(H) = (2SxJ)MIBSP (H;x, y)|x=y=1 (v) The matching Inverse index is given by: MI(H) = (SxJ)(DxDy)MIBSP (H;x, y)|x=y=1 (vi) The matching Symmetric Division index is given by: MSSD(H) = (SyDx + SxDy)MIBSP (H;x, y)|x=y=1 Proof. Let uv ∈ E(H) with du ≤ dv, then by definition 2, we get (i) (Dx +Dy)(MIBSP (H;x, y)) = (Dx +Dy)( α∑ r=1 ∑ ∀uv∈Xr xduydv) =Dx( α∑ r=1 ∑ ∀uv∈Xr xduydv)+Dy( α∑ r=1 ∑ ∀uv∈Xr xduydv) = α∑ r=1 ∑ ∀uv∈Xr dux duydv+ α∑ r=1 ∑ ∀uv∈Xr dvx duydv = α∑ r=1 ∑ ∀uv∈Xr (du + dv)x duydv . Therefore, MM1(H) = (Dx +Dy)MIBSP (H;x, y)|x=y=1 = α∑ r=1 ∑ ∀uv∈Xr (du + dv). (ii) (DxDy)(MIBSP (H;x, y)) = (DxDy)( α∑ r=1 ∑ ∀uv∈Xr xduydv) =Dx(Dy α∑ r=1 ∑ ∀uv∈Xr xduydv) J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 8 of 22 =Dx α∑ r=1 ∑ ∀uv∈Xr dvx duydv = α∑ r=1 ∑ ∀uv∈Xr (dudv)x duydv . Therefore, MM2(H) = (DxDy)MIBSP (H;x, y)|x=y=1 = α∑ r=1 ∑ ∀uv∈Xr (dudv). (iii) (Dγ xD γ y )(MIBSP (H;x, y)) = (Dγ xD γ y )( α∑ r=1 ∑ ∀uv∈Xr xduydv) =Dγ x(D γ y α∑ r=1 ∑ ∀uv∈Xr xduydv) =Dγ x α∑ r=1 ∑ ∀uv∈Xr dγvxduydv = α∑ r=1 ∑ ∀uv∈Xr (dγud γ v)xduydv . Therefore, MRγ(H) = (Dγ xD γ y )MIBSP (H;x, y)|x=y=1 = α∑ r=1 ∑ ∀uv∈Xr (dudv) γ . (iv) (2SxJ)MIBSP (H;x, y) = (2Sx)J α∑ r=1 ∑ ∀uv∈Xr xduydv) =2Sx( α∑ r=1 ∑ ∀uv∈Xr xdu+dv) =2 α∑ r=1 ∑ ∀uv∈Xr 1 du+dv xdu+dv = α∑ r=1 ∑ ∀uv∈Xr 2 du+dv xdu+dv Therefore, MH(H) = (2SxJ)MIBSP (H;x, y)|x=y=1 = α∑ r=1 ∑ ∀uv∈Xr 2 du+dv . (v) (SxJ)(DxDy)MIBSP (H;x, y) = (SxJ)(Dx(Dy α∑ r=1 ∑ ∀uv∈Xr xduydv)) =(SxJ)Dx α∑ r=1 ∑ ∀uv∈Xr dvx duydv =(Sx(J α∑ r=1 ∑ ∀uv∈Xr (dudv)x duydv)) =Sx α∑ r=1 ∑ ∀uv∈Xr (dudv)x du+dv J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 9 of 22 = α∑ r=1 ∑ ∀uv∈Xr ( dudv du+dv )xdu+dv . Therefore, MI(H) = (SxJ)(DxDy)MIBSP (H;x, y)|x=y=1 = α∑ r=1 ∑ ∀uv∈Xr dudv du+dv . (vi) (SyDx+SxDy)MIBSP (H;x, y) = (SyDx)( α∑ r=1 ∑ ∀uv∈Xr xduydv)+(SxDy)( α∑ r=1 ∑ ∀uv∈Xr xduydv) =Sy( α∑ r=1 ∑ ∀uv∈Xr dux duydv)+Sx( α∑ r=1 ∑ ∀uv∈Xr dvx duydv) = α∑ r=1 ∑ ∀uv∈Xr du dv xduydv+ α∑ r=1 ∑ ∀uv∈Xr dv du xduydv = α∑ r=1 ∑ ∀uv∈Xr (dudv + dv du )xduydv Since, du = Min{du, dv} and dv = Max{du, dv} Therefore,MSSD(H) = (SyDx+SxDy)MIBSP (H;x, y)|x=y=1 = α∑ r=1 ∑ ∀uv∈Xr (Min{du,dv} Max{du,dv}+ Max{du,dv} Min{du,dv} ). The following Proposition represent the relationship between TI(H) and MTI(H). Proposition 1. Let TI(H) denotes the degree-based topological indices of a graph H, and let MTI(H) represent the matching degree-based topological indices of H. The relationship between TI(H) and MTI(H) depends on the type of graph. For instance: (i) TI(Pn) > MTI(Pn) if n is even and n ≥ 4, and TI(Pn) < MTI(Pn) if n is odd and n ≥ 5. (ii) TI(Kn) < MTI(Kn), if n ≥ 4. (iii) TI(Cn) = MTI(Cn) if n is even, and TI(Cn) < MTI(Cn) if n is odd and n > 3. Comparing MIBSP to Wiener and Hosoya indices. Results confirm MIBSP’s superi- ority in modeling conjugated systems: its bivariate polynomial captures resonance effects (e.g., bond delocalization) that path-length-based indices (Wiener) or univariate matching counts (Hosoya) cannot detect, solidifying its utility for electronic structure analysis. 3.2. Maximum Independent Bond Set Polynomials of Various Painkillers Drugs Structures In this subsection, we determine the maximum independent bond set polynomials of various Painkillers drugs structures, such as: Aspirin, Paracetamol, Caffeine, Ibuprofen, Phenacetin and Salicylic acid. Moreover, computing some matching degree-based topo- logical indices such as: Matching First Zagreb index, Matching second Zagreb index, J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 10 of 22 matching general Randic index, matching Harmonic index, matching inverse sum index and matching Symmetric division degree index related to the maximum independent bonds set polynomial of the chemical structure of such drugs. The graph derived from the chemical structure of Aspirin has order |V (HAspirin)| = 13 and size |E(HAspirin)| = 13. In addition the vertices labeled as vi, i = 1, 2, ..., 13. Figure 1: Chemical structure of Aspirin(Acetylsalicylic) drug, adapted from [41]. Figure 2: Graph of Aspirin(Acetylsalicylic) drug structure. The following theorem determines the maximum independent bond set polynomials of the graph representing the structure of the Aspirin drug. Theorem 2. Let HAspirin be a graph of Aspirin drug structure. Then, the maximum independent bond set polynomials is: MIBSP (HAspirin) = 46xy3 + 38x2y2 + 38x2y3 + 8x3y3. Proof. Let HAspirin be a graph of Aspirin drug structure. Then, the matching number β(HAspirin) = 5, this means the maximum independent bond set contains 5 edges and we have 26 maximum matching which are given in the next Table 2. Table 2: Maximum matching Xi, i = 1, 2, ..., 26 of HAspirin and their M-polynomials Xi Maximum matching M(Xi;x, y) X1 {v1v3, v4v5, v6v7, v8v9, v10v11} xy3 + x2y2 + 3x2y3 X2 {v2v3, v4v5, v6v7, v8v9, v10v11} xy3 + x2y2 + 3x2y3 X3 {v1v3, v4v5, v6v7, v8v9, v11v12} 2xy3 + x2y2 + 2x2y3 J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 11 of 22 X4 {v2v3, v4v5, v6v7, v8v9, v11v12} 2xy3 + x2y2 + 2x2y3 X5 {v1v3, v4v5, v6v7, v8v9, v11v13} 2xy3 + x2y2 + 2x2y3 X6 {v2v3, v4v5, v6v7, v8v9, v11v13} 2xy3 + x2y2 + 2x2y3 X7 {v1v3, v4v9, v7v8, v5v6, v10v11} xy3 + 2x2y2 + x2y3 + x3y3 X8 {v2v3, v4v9, v7v8, v5v6, v10v11} xy3 + 2x2y2 + x2y3 + x3y3 X9 {v1v3, v4v9, v7v8, v5v6, v11v12} 2xy3 + 2x2y2 + x3y3 X10 {v2v3, v4v9, v7v8, v5v6, v11v12} 2xy3 + 2x2y2 + x3y3 X11 {v1v3, v4v9, v7v8, v5v6, v11v13} 2xy3 + 2x2y2 + x3y3 X12 {v2v3, v4v9, v7v8, v5v6, v11v13} 2xy3 + 2x2y2 + x3y3 X13 {v1v3, v4v5, v7v8, v9v10, v11v12} 2xy3 + x2y2 + 2x2y3 X14 {v2v3, v4v5, v7v8, v9v10, v11v12} 2xy3 + x2y2 + 2x2y3 X15 {v1v3, v4v5, v7v8, v9v10, v11v13} 2xy3 + x2y2 + 2x2y3 X16 {v2v3, v4v5, v7v8, v9v10, v11v13} 2xy3 + x2y2 + 2x2y3 X17 {v1v3, v4v5, v6v7, v9v10, v11v12} 2xy3 + x2y2 + 2x2y3 X18 {v2v3, v4v5, v6v7, v9v10, v11v12} 2xy3 + x2y2 + 2x2y3 X19 {v1v3, v4v5, v6v7, v9v10, v11v13} 2xy3 + x2y2 + 2x2y3 X20 {v2v3, v4v5, v6v7, v9v10, v11v13} 2xy3 + x2y2 + 2x2y3 X21 {v1v3, v5v6, v7v8, v9v10, v11v12} 2xy3 + 2x2y2 + x2y3 X22 {v2v3, v5v6, v7v8, v9v10, v11v12} 2xy3 + 2x2y2 + x2y3 X23 {v1v3, v5v6, v7v8, v9v10, v11v13} 2xy3 + 2x2y2 + x2y3 X24 {v2v3, v5v6, v7v8, v9v10, v11v13} 2xy3 + 2x2y2 + x2y3 X25 {v3v4, v5v6, v7v8, v9v10, v11v12} xy3 + 2x2y2 + x2y3 + x3y3 X26 {v3v4, v5v6, v7v8, v9v10, v11v13} xy3 + 2x2y2 + x2y3 + x3y3 Thus, for each case of maximum matching we classify the independent edges and ob- tain the M-polynomial as given in the third column. Finally, we add all M-polynomials in the third column to get the maximum independent bond set polynomials. MIBSP (HAspirin) = 46xy3 + 38x2y2 + 38x2y3 + 8x3y3. Corollary 1. Let HAspirin be a graph of Aspirin drug structure. Then, (i) MM1(HAspirin) = 574. (ii) MM2(HAspirin) = 590. (iii) MRγ(HAspirin) = 46(3γ) + 38(4γ) + 38(6γ) + 8(9γ). (iv) MSSD(HAspirin) = 983 3 . (v) MH(HAspirin) = 898 15 . (vi) MI(HAspirin) = 1301 10 . J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 12 of 22 The graph derived from the chemical structure of Paracetamol has order |V (HParacetamol)| = 11 and size |E(HParacetamol)| = 11. In addition the vertices labeled as vi, i = 1, 2, ..., 11. Figure 3: Chemical structure of Paracetamol drug, adapted from [42]. Figure 4: Graph of Paracetamol drug structure. The following theorem determines the maximum independent bond set polynomials of the graph representing the structure of the Paracetamol drug. Theorem 3. Let HParacetamol be a graph of Paracetamol drug structure. Then, the maxi- mum independent bond set polynomials is: MIBSP (HParacetamol) = 4xy3+4x2y2+2x2y3. Proof. Let HParacetamol be a graph of Paracetamol drug structure. Then, the matching number β(HParacetamol) = 5, this means the maximum independent bond set contains 5 edges and we have 2 maximum matching which are: X1 = {v1v2, v3v4, v6v7, v5v8, v9v10} and X2 = {v1v2, v3v4, v6v7, v5v8, v9v11}. Since, M(X1;x, y) = M(X2;x, y) = 2xy3 + 2x2y2 + x2y3. Hence, MIBSP (HParacetamol) = M(X1;x, y) +M(X2;x, y) = 4xy3 + 4x2y2 + 2x2y3 Corollary 2. Let HParacetamol be a graph of Paracetamol drug structure. Then, (i) MM1(HParacetamol) = 42. (ii) MM2(HParacetamol) = 40. (iii) MRγ(HParacetamol) = 4(3γ) + 4(4γ) + 2(6γ). (iv) MSSD(HParacetamol) = 77 3 . (v) MH(HParacetamol) = 24 5 . (vi) MI(HParacetamol) = 47 5 . J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 13 of 22 The graph derived from the chemical structure of Caffeine has order |V (HCaffeine)| = 14 and size |E(HCaffeine)| = 15. In addition the vertices labeled as vi, i = 1, 2, ..., 14. Figure 5: Chemical structure of Caffeine drug, adapted from [43]. Figure 6: Graph of Caffeine drug structure. The following theorem determines the maximum independent bond set polynomials of the graph representing the structure of the Caffeine drug. Theorem 4. Let HCaffeine be a graph of Caffeine drug structure. Then, the maximum independent bond set polynomials is: MIBSP (HCaffeine) = 5xy3 + x2y2 + x3y3. Proof. Let HCaffeine be a graph of Caffeine drug structure. Then, the matching num- ber β(HCaffeine) = 7, this means the maximum independent bond set contains 7 edges and we have only one maximum matching which is: X = {v4v10, v3v11, v2v12, v1v13, v7v14, v5v6, v8v9}. This matching is called a perfect match- ing in HCaffeine because every vertex in V (HCaffeine) is incident to exactly one edge in X. Since, we have only one maximum matching. Therefore, MIBSP (HCaffeine) = M(X;x, y) = 5xy3 + x2y2 + x3y3 Corollary 3. Let HCaffeine be a graph of Caffeine drug structure. Then, (i) MM1(HCaffeine) = 30. J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 14 of 22 (ii) MM2(HCaffeine) = 28. (iii) MRγ(HCaffeine) = 5(3γ) + (4γ) + 9γ. (iv) MSSD(HCaffeine) = 62 3 . (v) MH(HCaffeine) = 10 3 . (vi) MI(HCaffeine) = 75 12 . The graph derived from the chemical structure of Ibuprofen has order |V (HIbuprofen)| = 15 and size |E(HIbuprofen)| = 15. In addition the vertices labeled as vi, i = 1, 2, ..., 15. Figure 7: Chemical structure of Ibuprofen drug, adapted from [41]. Figure 8: Graph of Ibuprofen drug structure. The following theorem determines the maximum independent bond set polynomials of the graph representing the structure of the Ibuprofen drug. Theorem 5. Let HIbuprofen be a graph of Ibuprofen drug structure. Then, the maximum independent bond set polynomials is: MIBSP (HIbuprofen) = 76xy3 + 36x2y2 + 52x2y3 + 4x3y3. Proof. Let HIbuprofen be a graph of Ibuprofen drug structure. Then, the matching number β(HIbuprofen) = 6, this means the maximum independent bond set contains 6 edges and we have 28 maximum matching which are given in the next Table 3. J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 15 of 22 Table 3: Maximum matching Xi, i = 1, 2, ..., 28 of HIbuprofen and their M-polynomials Xi Maximum matching M(Xi;x, y) X1 {v1v3, v4v5, v6v7, v8v9, v10v11, v12v13} 2xy3 + x2y2 + 3x2y3 X2 {v2v3, v4v5, v6v7, v8v9, v10v11, v12v13} 2xy3 + x2y2 + 3x2y3 X3 {v1v3, v4v5, v6v7, v8v9, v10v11, v13v14} 3xy3 + x2y2 + 2x2y3 X4 {v2v3, v4v5, v6v7, v8v9, v10v11, v13v14} 3xy3 + x2y2 + 2x2y3 X5 {v1v3, v4v5, v6v7, v8v9, v10v11, v13v15} 3xy3 + x2y2 + 2x2y3 X6 {v2v3, v4v5, v6v7, v8v9, v10v11, v13v15} 3xy3 + x2y2 + 2x2y3 X7 {v1v3, v4v5, v6v11, v9v10, v7v8, v12v13} 2xy3 + x2y2 + 3x2y3 X8 {v2v3, v4v5, v6v11, v9v10, v7v8, v12v13} 2xy3 + x2y2 + 3x2y3 X9 {v1v3, v4v5, v6v11, v9v10, v7v8, v13v14} 3xy3 + x2y2 + 2x2y3 X10 {v2v3, v4v5, v6v11, v9v10, v7v8, v13v14} 3xy3 + x2y2 + 2x2y3 X11 {v1v3, v4v5, v6v11, v9v10, v7v8, v13v15} 3xy3 + x2y2 + 2x2y3 X12 {v2v3, v4v5, v6v11, v9v10, v7v8, v13v15} 3xy3 + x2y2 + 2x2y3 X13 {v1v3, v4v6, v7v8, v10v11, v9v12, v13v15} 2xy3 + 2x2y2 + x2y3 + x3y3 X14 {v2v3, v4v6, v7v8, v10v11, v9v12, v13v15} 2xy3 + 2x2y2 + x2y3 + x3y3 X15 {v1v3, v4v6, v7v8, v10v11, v9v12, v13v14} 2xy3 + 2x2y2 + x2y3 + x3y3 X16 {v2v3, v4v6, v7v8, v10v11, v9v12, v13v14} 2xy3 + 2x2y2 + x2y3 + x3y3 X17 {v1v3, v4v5, v7v8, v10v11, v9v12, v13v15} 3xy3 + 2x2y2 + x2y3 X18 {v2v3, v4v5, v7v8, v10v11, v9v12, v13v15} 3xy3 + 2x2y2 + x2y3 X19 {v1v3, v4v5, v7v8, v10v11, v9v12, v13v14} 3xy3 + 2x2y2 + x2y3 X20 {v2v3, v4v5, v7v8, v10v11, v9v12, v13v14} 3xy3 + 2x2y2 + x2y3 X21 {v1v3, v4v5, v6v7, v10v11, v9v12, v13v15} 3xy3 + x2y2 + 2x2y3 X22 {v2v3, v4v5, v6v7, v10v11, v9v12, v13v15} 3xy3 + x2y2 + 2x2y3 X23 {v1v3, v4v5, v6v7, v10v11, v9v12, v13v14} 3xy3 + x2y2 + 2x2y3 X24 {v2v3, v4v5, v6v7, v10v11, v9v12, v13v14} 3xy3 + x2y2 + 2x2y3 X25 {v1v3, v4v5, v7v8, v6v11, v9v12, v13v15} 3xy3 + x2y2 + 2x2y3 X26 {v2v3, v4v5, v7v8, v6v11, v9v12, v13v15} 3xy3 + x2y2 + 2x2y3 X27 {v1v3, v4v5, v7v8, v6v11, v9v12, v13v14} 3xy3 + x2y2 + 2x2y3 X28 {v2v3, v4v5, v7v8, v6v11, v9v12, v13v14} 3xy3 + x2y2 + 2x2y3 Thus, for each case of maximum matching we classify the independent edges and ob- tain the M-polynomial as given in the third column. Finally, we add all M-polynomials in the third column to get the maximum independent bond set polynomials. MIBSP (HIbuprofen) = 76xy3 + 36x2y2 + 52x2y3 + 4x3y3. Corollary 4. Let HIbuprofen be a graph of Ibuprofen drug structure. Then, (i) MM1(HIbuprofen) = 732. (ii) MM2(HIbuprofen) = 720. J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 16 of 22 (iii) MRγ(HIbuprofen) = 76(3γ) + 36(4γ) + 52(6γ) + 4(9γ). (iv) MSSD(HIbuprofen) = 446. (v) MH(HIbuprofen) = 1172 15 . (vi) MI(HIbuprofen) = 807 5 . The graph derived from the chemical structure of Phenacetin has the same order and size |V (HPhenacetin)| = |E(HPhenacetin)| = 13. In addition the vertices labeled as vi, i = 1, 2, ..., 13. Figure 9: Chemical structure of Phenacetin drug, adapted from [42]. Figure 10: Graph of Phenacetin drug structure. The following theorem determines the maximum independent bond set polynomials of the graph representing the structure of the Phenacetin drug. Theorem 6. Let HPhenacetin be a graph of Phenacetin drug structure. Then, the maximum independent bond set polynomials is: MIBSP (HPhenacetin) = 2xy2+2xy3+4x2y2+4x2y3. Proof. Let HPhenacetin be a graph of Phenacetin drug structure. Then, the matching number β(HPhenacetin) = 6, this means the maximum independent bond set contains 6 edges and we have 2 maximum matching which are: X1 = {v1v2, v3v4, v5v6, v8v9, v7v10, v11v12} and X2 = {v1v2, v3v4, v5v6, v8v9, v7v10, v11v13}. Since, M(X1;x, y) = M(X2;x, y) = xy2 + xy3 + 2x2y2 + 2x2y3. Therefore, MIBSP (HPhenacetin) = M(X1;x, y) + M(X2;x, y) = 2xy2 + 2xy3 + 4x2y2 + 4x2y3. J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 17 of 22 Corollary 5. Let HPhenacetin be a graph of Phenacetin drug structure. Then, (i) MM1(HPhenacetin) = 50. (ii) MM2(HPhenacetin) = 50. (iii) MRγ(HPhenacetin) = 2(2γ) + 2(3γ) + 4(4γ) + 4(6γ). (iv) MSSD(HPhenacetin) = 85 3 . (v) MH(HPhenacetin) = 89 15 . (vi) MI(HPhenacetin) = 349 30 . The graph derived from the chemical structure of Salicylic acid also has the same order and size |V (HSalicylicacid)| = |E(HSalicylicacid)| = 10. In addition the vertices labeled as vi, i = 1, 2, ..., 10. Figure 11: Chemical structure of Salicylic acid drug, adapted from [44]. Figure 12: Graph of Salicylic acid drug structure. J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 18 of 22 The following theorem determines the maximum independent bond set polynomials of the graph representing the structure of the Salicylic acid drug. Theorem 7. Let HSalicylicacid be a graph of Salicylic acid drug structure. Then, the maximum independent bond set polynomials is: MIBSP (HSalicylicacid) = 17xy3+16x2y2+ 8x2y3 + 3x3y3. Proof. Let HSalicylicacid be a graph of Salicylic acid drug structure. Then, the matching number β(HSalicylicacid) = 4, this means the maximum independent bond set contains 4 edges and we have 11 maximum matching which are given in the next Table 4. Table 4: Maximum matching Xi, i = 1, 2, ..., 11 of HSalicylicacid and their M-polynomials Xi Maximum matching M(Xi;x, y) X1 {v1v3, v4v5, v6v7, v8v9} xy3 + 2x2y2 + x3y3 X2 {v2v3, v4v5, v6v7, v8v9} xy3 + 2x2y2 + x3y3 X3 {v1v3, v5v6, v7v8, v4v9} xy3 + x2y2 + 2x2y3 X4 {v2v3, v5v6, v7v8, v4v9} xy3 + x2y2 + 2x2y3 X5 {v1v3, v4v9, v7v8, v5v10} 2xy3 + x2y2 + x2y3 X6 {v2v3, v4v9, v7v8, v5v10} 2xy3 + x2y2 + x2y3 X7 {v1v3, v4v9, v6v7, v5v10} 2xy3 + x2y2 + x2y3 X8 {v2v3, v4v9, v6v7, v5v10} 2xy3 + x2y2 + x2y3 X9 {v1v3, v8v9, v6v7, v5v10} 2xy3 + 2x2y2 X10 {v2v3, v8v9, v6v7, v5v10} 2xy3 + 2x2y2 X11 {v3v4, v8v9, v6v7, v5v10} xy3 + 2x2y2 + x3y3 Thus, for each case of maximum matching we classify the independent edges and ob- tain the M-polynomial as given in the third column. Finally, we add all M-polynomials in the third column to get the maximum independent bond set polynomials. MIBSP (HSalicylicacid) = 17xy3 + 16x2y2 + 8x2y3 + 3x3y3. Corollary 6. Let HSalicylicacid be a graph of Salicylic acid drug structure. Then, (i) MM1(HSalicylicacid) = 190. (ii) MM2(HSalicylicacid) = 190. (iii) MRγ(HSalicylicacid) = 17(3γ) + 16(4γ) + 8(6γ) + 3(9γ). (iv) MSSD(HSalicylicacid) = 112. (v) MH(HSalicylicacid) = 207 10 . (vi) MI(HSalicylicacid) = 857 20 . J. J. Ali, D. A. Ali / Eur. J. Pure Appl. Math, 18 (2) (2025), 5992 19 of 22 4. Conclusions Unlike distance-based resolvers (Wiener [18]), MIBSP specializes in electronic struc- ture analysis through degree-weighted matchings. This approach enables the detection of stability variations in conjugated systems such as the perfect matching in Caffeine that remain undetectable via path-length analysis. In this study, we introduced the concept of the maximum independent bond set polynomial MIBSP (H;x, y) and applied it to derive degree-based topological indices for the chemical graphs of several painkiller molecules, including Aspirin, Paracetamol, Caffeine, Ibuprofen, Phenacetin, and Salicylic acid. 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