CoM-polynomial and topological coindices of hyaluronic acid conjugates

1 Introduction

The emergence of new diseases and advancements in medical technology have fueled research into more contemporary drug discovery techniques in recent years (Azam, 2021; Schneider, 2018). Thus, investigations into the chemical, pharmacological and biopharmaceutical features of the novel drugs should be undertaken, which will need the use of appropriate reagents and equipment, as well as the participation of qualified researchers (Ahmed, 2016; Mustafa et al., 2017). The researchers established a satisfactory connection between the chemical, pharmacodynamic, and molecular structures of medications using a graph theory approach called topological index. In other words, topological index computing methods can aid in obtaining accessible biological and medical information on novel pharmaceuticals without requiring chemical testing, making them appropriate for usage in disadvantaged areas of the globe (Liu et al., 2016; Sarfraz et al., 2016). (See Figs. 1-2.).

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Fig. 1

The structure of hyaluronic acid conjugates.

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Fig. 2

Molecular graphs HA-conjugates. $\mathrm{f}\mathrm{o}\mathrm{r}n=1\mathrm{a}\mathrm{n}\mathrm{d}n=3.$

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A molecular graph is created by transforming a chemical molecule into a graph with vertices and edges represented by atoms and bonds. For molecular graph $G=\left(V,E\right)$ , $V\left(G\right)$ is vertex set and $E(G)$ is edge set. $\left|V(G)\right|$ and $\left|E(G)\right|$ are used to represent the number of vertices and edges in a graph G. The number of vertices adjacent to $u\in V\left(G\right)$ is called degree of vertex and denoted by $d\left(u\right)$ . The simple graph ( $\overline{G}$ ) having vertex set $V\left(G\right)$ such that any two vertices $u\mathrm{a}\mathrm{n}\mathrm{d}v$ are adjacent in $\overline{G}$ if and only if $u\mathrm{a}\mathrm{n}\mathrm{d}v$ are not adjacent in $G$ , is called complement of a graph G (Trinajstic, 2018).

Wiener was the first to apply topological index in QSPR, demonstrating that his index, dubbed the Wiener index after him, was closely aligned with alkane boiling points.(Wiener, 1947) The Wiener index has been used to describe a range of chemical and physical properties of molecules, as well as to associate a molecule's structure to its biological activity (Dobrynin et al., 2001). Since the introduction of Wiener index, there have been hundreds of topological indices constructed nonetheless, in literature, the Wiener index, Randic index, Zagreb indices and variant, are the most extensively utilized (Wiener, 1947; Gutman et al., 2018; Gutman and Furtula, 2008; Li and Shi, 2008; Randic, 1975; Li et al., 2006; Gutman and Trinajstić, 1972; Miličević et al., 2004; Gutman et al., 2018; Nikolić et al., 2003; Gutman and Furtula, 2010; Gutman and Das, 2004). Symmetric division index (SDD), Inverse sum index, Harmonic index, F-Index, augmented Zagreb index and redefined versions of Zagreb indices are among the indices regularly encountered in the literature (Gutman and Trinajstić, 1972; Gupta et al., 2016; Vukičević and Gašperov, 2010; Fajtlowicz, 1987; Furtula and Gutman, 2015; Furtula et al., 2010; Ranjini et al., 2013; Fang et al., 2022; Chu, 2021; Rafiullah et al., 2021).

The contributions of adjacent vertex pairs are taken into account by the majority of degree-based topological indices. However, over time, researchers have begun to incorporate non-adjacent pairing of vertices into consideration when computing some topological features of graphs, resulting in degree-based topological indices known as coindices (TCI).

With the hopes of improving our capacity to assess the effect of non-adjacent vertices on a variety of molecular properties, Doslic (Došlić, 2008) formalized a set of new invariants, of graph G as follows.

1st. ${\mathrm{Z}\mathrm{a}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{b}\mathrm{c}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}=\overline{M}}_1\left(G\right)={\sum }_{\mathit{uv}\notin E(G)}\left(d\left(u\right)+d\left(v\right)\right)$

and.

2nd. ${\mathrm{Z}\mathrm{a}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{b}\mathrm{c}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}=\overline{M}}_2={\sum }_{\mathit{uv}\notin E(G)}d\left(u\right)d\left(v\right)$

It's worth noting that Zagreb coindices of G and Zagreb indices of $\overline{G}$ are not same; The degrees are with regard to G, whereas the defining sums are over. $E(\overline{G})$

Xu et al. (Xu et al., 2013) defined multiplicative versions of Zagreb coindices described as below. $${\overline{\prod }}_1(G)={\prod }_{\mathit{uv}\notin E(G)}\left(d\left(u\right)+d\left(v\right)\right)$$

and. $${\overline{\prod }}_2(G)={\prod }_{\mathit{uv}\notin E(G)}d\left(u\right)d\left(v\right)$$

respectively.

De and colleagues (De et al., 2016) demonstrated that the values of octane isomers F-coindex and Log P, where P is the octanol–water partition coefficient, show a high correlation (r = 0.96), leading to the introducing the F-coindex $(\overline{F})$ of G as follows. $$\overline{F}\left(G\right)=\sum _{\mathit{uv}\notin E(G)}\left(d^2\left(u\right)+d^2\left(v\right)\right)$$

Recently Berhe et al. (Berhe and Wang, 2019) computed various coindices of C₄C₈(S) nanotubes, nanotorus & graphene sheets. For more literature on the topological coindices readers might refer to (Gutman et al., 2015; Basavanagoud et al., 2015; Basavanagoud and Patil, 2016; Hua et al., 2012; Ashrafi et al., 2010; Ashrafi, et al., 2011).

Other degree-based topological coindices mentioned in this article can be defined in the same way as their classical degree counterparts:

Second modified Zagreb coindex:. ${m\overline{M}}_2\left(G\right)={\sum }_{\mathit{uv}\notin E(G)}\frac{1}{d\left(u\right)d(v)}$

Redefined third Zagreb coindex:. $\overline{\mathit{ReZ}{G}_3}\left(G\right)={\sum }_{\mathit{uv}\notin E(G)}d\left(u\right)d(v)\left(d\left(u\right)+d\left(v\right)\right)$

Randić coindex:. ${\overline{R}}_k\left(G\right)={\sum }_{\mathit{uv}\notin E(G)}{(d\left(u\right)d\left(v\right))}^k$

Inverse Randić coindex:. ${\overline{\mathit{RR}}}_k\left(G\right)={\sum }_{\mathit{uv}\notin E(G)}\frac{1}{{(d\left(u\right)d\left(v\right))}^k}$

Symmetric Division coindex:. $\overline{\mathit{SDD}}(G)={\sum }_{\mathit{uv}\notin E(G)}\frac{d^2\left(u\right)+d^2\left(v\right)}{d\left(u\right)d(v)}$

Harmonic Coindex:. $\overline{H}\left(G\right)={\sum }_{\mathit{uv}\notin E(G)}\frac{2}{d\left(u\right)+d(v)}$

Inverse sum indeg Coindex:. $\overline{I}\left(G\right)={\sum }_{\mathit{uv}\notin E(G)}\frac{d\left(u\right)d(v)}{\left(d\left(u\right)+d\left(v\right)\right)}$

Augmented Zagreb Coindex:. $\overline{A}\left(G\right)={\sum }_{\mathit{uv}\notin E(G)}{\left\{\frac{d\left(u\right)d(v)}{d\left(u\right)+d\left(v\right)-2}\right\}}^3$

Several distance-based graph polynomials have been proposed in the literature to speed up the computation of certain graph indices (Hosoya, 1988; Khadikar, 2000; Schultz, 1989; Klavžar and Gutman, 1997). Similarly, Deutsch and Klavzar introduced the M−polynomial in 2015, a degree-based polynomial that is used to compute various degree-based-TI (Deutsch and Klavžar, 2020). Because of its broad adaptability, it has been utilized in a number of articles to obtain topological indices (Ali et al., 2018; Basavanagoud and Barangi, 2019; Kwun, 2018; Kwun et al., 2018; Munir and Nazeer, 2016; Munir et al., 2016; Raza and Sukaiti, 2020; Yang et al., 2019). As the M−polynomial takes into account the contributions of pairs of adjacent vertices, a polynomial that takes into account non-adjacent pairs of vertices is necessary to speed up the time-consuming process of computing various types of graph coindices. This encourages us to cultivate polynomial based on non-adjacent pairs of vertices of chemical compounds.

2 Motivation

The cancer treatment regimen relies heavily on methotrexate, also known as 2,4-diaminopropyl-N-10-methyl-beta-glutamylethylamide, which has demonstrated success in many different types of cancer, including head and neck cancer, breast cancer, acute lymphocytic leukemia, non-lymphoma, choriocarcinoma, and Hodgkin's osteosarcoma (Jolivet et al., 1983). However, because methotrexate is distributed indiscriminately throughout the body following systemic injection, approximately 50% of patients experience significant side effects such as headaches, nausea, hair loss, and skin discoloration, limiting its long-term usage and acceptability. Furthermore, methotrexate is also reported to cause liver fibrosis, pneumonitis, and myelosuppression (Abolmaali et al., 2013). Therefore, methotrexate has been conjugated with hyaluronic acid as a targeted therapy approach in order to avoid indiscriminate distribution in the body (Shin, 2014).

Paclitaxel, a tricyclic diterpenoid molecule found naturally in the bark and needles of Taxus brevifolia, is well recognized for its unique anticancer action and has been hailed as one of the most effective and extensively used natural anticancer agents. It increases tubulin assembly into microtubules and inhibits microtubule dissociation, therefore slowing cell cycle progression, halting mitosis, and limiting cancer cell development. Additionally, it is utilized to treat coronary heart disease, renal and hepatic fibrosis, inflammation, axon regeneration, and skin problems, and clinical studies for degenerative brain illnesses are now underway (Zhu and Chen, 2019). However, paclitaxel's therapeutic utility is harmed by a distribution difficulty induced by the drug's poor pharmacokinetic and physicochemical characteristics. Due to the inclusion of cremophor in the paclitaxel formulation, several adverse events have been documented, including nephrotoxicity, hypersensitivity reactions, and neurotoxicity. Significant research has been undertaken to create an alternate formulation strategy for paclitaxel that increases its water solubility without the use of cremophor, therefore reducing its toxicity. In this context, it has been revealed that a compound of paclitaxel and hyaluronic acid (HA) offers numerous benefits over conventional pharmaceutical formulations (Singh and Dash, 2009). Galer et al. developed the HA-paclitaxel conjugate to improve anticancer activity while reducing taxanes' toxicity. (Galer, 2011).

Curcumin, a naturally occurring chemical has been found to have a number of pharmacological actions, including antioxidant, anti-inflammatory, anti-proliferative, and chemo-sensitizing, among others. Cancer prevention and therapy are among its most well-known applications, and it is particularly well-known as a chemopreventive agent (Giordano and Tommonaro, 2019). Curcumin's limited water solubility, along with its poor absorption in biological systems, may make it unsuitable for use in clinical practice (Kharat and McClements, 2019). Curcumin conjugates with HA are viewed as a potentially beneficial therapeutic approach for extending curcumin release at the site of action, optimizing tissue dispersion, and improving clinical outcomes. Additionally, it has garnered great attention for its potential to improve bioavailability while also targeting tumor cells and metastases in the treatment of many kinds of cancer (Saravanakumar et al., 2014; Wang, 2018).

Because of the high pharmaceutical interest in HA-drugs conjugates, Zheng et al. have obtained several topological indices of Hyaluronic Acid-Paclitaxel conjugates (Zheng et al., 2019), however their results were modified (Kirmani and Ali, 2023). Ali et al. (Ali and Kirmani, 2020) recently focused on a variety of topological indices and polynomials for curcumin conjugate with HA. Kirmani et al. (2021) obtained various results on ve-degree/ev-degree for conjugates of curcumin/paclitaxel with Hyaluronic acid. Rauf et al. (Rauf et al., 2021) derived the quantitative closed formulations for the Zagreb, Randic, geometric-arithmetic and atom-bond connectivity (ABC) indices for the HA-curcumin molecular graph based on ev-degrees and ve-degrees.

Recently Jun Yang et al. consider topological coindices of Hydroxyethyl Starch Conjugated with Hydroxychloroquine (Yang, 2021). Very recently various coindices for the set of antiviral drugs were considered and it is established that the topological coindices have a substantial relationship with the physicochemical properties of antiviral drugs (Kirmani et al., 2022). It reveals that topological coindices are very useful tool for future drug research and QSPR analysis. In this note, we define CoM-polynomial for HA-curcumin/paclitaxel/Methotrexate conjugates and derives various coindices for these conjugates.

3 Preliminaries

Let us use the following notation for the rest of paper.

$n_i=\left|V_i\right|$ for. $V_i=\left\{v\in V\left(G\right)\right|d\left(v\right)=i\}$

$m_{\mathit{ij}}=\left|E_{\mathit{ij}}\right|$ for. $E_{\mathit{ij}}=\left\{uv\in E\left(G\right)\right|d\left(u\right)=i\mathrm{a}\mathrm{n}\mathrm{d}d\left(v\right)=j\}$

${\overline{m}}_{\mathit{ij}}=\left|{\overline{E}}_{\mathit{ij}}\right|$ for. ${\overline{E}}_{\mathit{ij}}=\left\{uv\in E(\overline{G})\right|d\left(u\right)=i\mathrm{a}\mathrm{n}\mathrm{d}d\left(v\right)=j\}$

With the help above notation, we extend the concept of M−polynomial for non-adjacent pair of vertices and define CoM-polynomial as follows. $$CoM\left(G;x,y\right)=\overline{M}\left(G;x,y\right)=\sum _{i\le j}{\overline{m}}_{\mathit{ij}}\left(G\right)x^i{y}^j$$

The relationship between several degree-based TCI and the CoM-polynomial is shown in Table 1.

The following result is due to Berhe (Berhe and Wang, 2019).

where $n_i$ , $m_{\mathit{ij}}$ and ${\overline{m}}_{\mathit{ij}}$ are defined above.

4 Main results and discussion

Chemical structures of Hyaluronic Acid (HA) conjugated with curcumin, paclitaxel and methotrexate are shown in Figure-1 while Figure-2 displays their corresponding molecular graphs.

Moreover, Fig. 3, Fig. 4 and Fig. 5 exhibit plots of CoM-polynomials of various HA-conjugates.

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Fig. 3

Plot of CoM-polynomial for HACC for n = 1.

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Fig. 4

Plot of CoM-polynomial for HAPTX for n = 1.

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Fig. 5

Plot of CoM-polynomial for HAM for n = 1.

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Let HACC be the nth iteration of the equivalent molecular graphs of HA-curcumin conjugate. Figure-2(a) displays graphs of HACC for n = 1 and n = 3.

We begin by determining the expression of the CoM-polynomial for HACC.

Proof: From the Figure-2(a), it is straightforward to deduce that $\left|E(HACC)\right|=56n,\left|V(HACC)\right|=52n+1$ . The $\left|E(HACC)\right|$ can be classified into five classes based on the vertex degrees: $$E_{12}=\left\{\mathit{uv}\in E(HACC)|d\left(u\right)=1,d\left(v\right)=2\right\},E_{13}=\left\{\mathit{uv}\in E(HACC)|d\left(u\right)=1,d\left(v\right)=3\right\},$$

$E_{22}=\left\{uv\in E\left(HACC\right)|d\left(u\right)=2,d\left(v\right)=2\right\},{E_{23}=\left\{uv\in E\left(HACC\right)|d\left(u\right)=2,d\left(v\right)=3\right\},E}_{33}=\left\{\mathit{uv}\in E(HACC)|d\left(u\right)=3,d\left(v\right)=3\right\}$ such that. $$\left|E_{12}\right|=3n+1,\left|E_{13}\right|=9n,\left|E_{22}\right|=4n+1,\left|E_{23}\right|=29n-1,\left|E_{33}\right|=11n-1$$

Similarly, HACC 's vertex set can be classified into three classes based on their degrees. $$n_1=\left|V_1\right|=12n+1,n_2=\left|V_2\right|=20n+1,n_3=\left|V_3\right|=20n-1$$

Using lemma 1, we have. $${\overline{m}}_{12}=\left(12n+1\right)\left(20n+1\right)-\left(3n+1\right)$$ $$=240n^2+29n$$ $${\overline{m}}_{22}=\frac{\left(20n+1\right)\left(20n+1-1\right)}{2}-\left(4n+1\right)$$ $$=200n^2+6n-1$$

Similarly, ${\overline{m}}_{13}=\left|{\overline{E}}_{13}\right|=240n^2-n-1$ , ${\overline{m}}_{23}=\left|{\overline{E}}_{23}\right|=400n^2-29n$ ,. ${\overline{m}}_{33}=\left|{\overline{E}}_{33}\right|=200n^2-41n+2$

By definition of CoM-polynomial. $$CoM\left(G;x,y\right)=\sum _{i\le j}{\overline{m}}_{\mathit{ij}}\left(G\right)x^i{y}^j$$ $$CoM\left(HACC;x,y\right)=\sum _{1\le 2}{\overline{m}}_{12}x{y}^2+\sum _{1\le 3}{\overline{m}}_{13}x{y}^3+\sum _{2\le 2}{\overline{m}}_{22}{x}^2{y}^2+\sum _{2\le 3}{\overline{m}}_{23}{x}^2{y}^3+\sum _{3\le 3}{\overline{m}}_{33}{x}^3{y}^3=\left|{\overline{E}}_{12}\right|x{y}^2+\left|{\overline{E}}_{13}\right|x{y}^3+\left|{\overline{E}}_{22}\right|x^2{y}^2+\left|{\overline{E}}_{23}\right|x^2{y}^3+\left|{\overline{E}}_{33}\right|x^3{y}^3=\left(240n^2+29n\right)x{y}^2+\left(240n^2-n-1\right)x{y}^3+\left(200n^2+6n-1\right)x^2{y}^2+\left(400n^2-29n\right)x^2{y}^3+\left(200n^2-41n+2\right)x^3{y}^3$$

Hence the result.

With the help of Theorem 1, we can now recover some degree-based topological coindices of the HACC in the next proposition.

then.

• $D_{x}f\left(x,y\right)=\left(240n^2+29n\right)x{y}^2+\left(240n^2-n-1\right)x{y}^3+2(200n^2+6n-1)x^2{y}^2+2(400n^2-29n)x^2{y}^3+3(200n^2-41n+2)x^3{y}^3$

• $D_{y}f\left(x,y\right)=2\left(240n^2+29n\right)x{y}^2+3\left(240n^2-n-1\right)x{y}^3+2(200n^2+6n-1)x^2{y}^2+3(400n^2-29n)x^2{y}^3+3(200n^2-41n+2)x^3{y}^3.$

• ${{(D}_x+D}_y)(f\left(x,y\right))=3\left(240n^2+29n\right)x{y}^2+4\left(240n^2-n-1\right)x{y}^3+4(200n^2+6n-1)x^2{y}^2+5(400n^2-29n)x^2{y}^3+6(200n^2-41n+2)x^3{y}^3.$

• ${D_{y}D}_x(f\left(x,y\right))=2\left(240n^2+29n\right)x{y}^2+3\left(240n^2-n-1\right)x{y}^3+4(200n^2+6n-1)x^2{y}^2+6(400n^2-29n)x^2{y}^3+9(200n^2-41n+2)x^3{y}^3.$

• $\left(D_x^2+D_y^2\right)\left(f\left(x,y\right)\right)=5\left(240n^2+29n\right)x{y}^2+10\left(240n^2-n-1\right)x{y}^3+8\left(200n^2+6n-1\right)x^2{y}^2+13\left(400n^2-29n\right)x^2{y}^3+18\left(200n^2-41n+2\right)x^3{y}^3.$

• ${D_x}^k{D_y}^k(f\left(x,y\right))=2^k\left(240n^2+29n\right)x{y}^2+3^k\left(240n^2-n-1\right)x{y}^3+2^k{2}^k(200n^2+6n-1)x^2{y}^2+2^k{3}^k(400n^2-29n)x^2{y}^3+3^k{3}^k(200n^2-41n+2)x^3{y}^3.$

• ${D_{x}D}_y{{(D}_x+D}_y)(f\left(x,y\right)=2\left(720n^2+87n\right)x{y}^2+3\left(960n^2-4n-4\right)x{y}^3+4(800n^2+24n-4)x^2{y}^2+6(2000n^2-145n)x^2{y}^3+9(1200n^2-246n+12)x^3{y}^3.$

• ${S_{x}S}_y\left(f\left(x,y\right)\right)=\frac{\left(240n^2+29n\right)}{2}x{y}^2+\frac{\left(240n^2-n-1\right)}{3}x{y}^3+\frac{\left(200n^2+6n-1\right)}{4}{x}^2{y}^2+\frac{\left(400n^2-29n\right)}{6}{x}^2{y}^3+\frac{\left(200n^2-41n+2\right)}{9}{x}^3{y}^3.$

• ${S_x}^k{S_y}^k(f\left(x,y\right))=\frac{\left(240n^2+29n\right)}{2^k}x{y}^2+\frac{\left(240n^2-n-1\right)}{3^k}x{y}^3+\frac{\left(200n^2+6n-1\right)}{4^k}{x}^2{y}^2+\frac{\left(400n^2-29n\right)}{6^k}{x}^2{y}^3+\frac{\left(200n^2-41n+2\right)}{9^k}{x}^3{y}^3.$

• ${{(S}_{y}D}_x+{S_{x}D}_y)(f\left(x,y\right))=\frac{5\left(240n^2+29n\right)}{2}x{y}^2+\frac{10\left(240n^2-n-1\right)}{3}x{y}^3+2\left(200n^2+6n-1\right)x^2{y}^2+\frac{13\left(400n^2-29n\right)}{6}{x}^2{y}^3+2\left(200n^2-41n+2\right)x^3{y}^3.$

• $S_{x}J\left(f\left(x,y\right)\right)=\frac{2\left(240n^2+29n\right)}{3}{x}^3+\frac{2\left(240n^2-n-1\right)}{4}{x}^4+\frac{2\left(200n^2+6n-1\right)}{4}{x}^4+\frac{2\left(400n^2-29n\right)}{5}{x}^5+\frac{2\left(200n^2-41n+2\right)}{6}{x}^6.$

• $S_{x}J{D_{y}D}_x(f\left(x,y\right))=\frac{2\left(240n^2+29n\right)}{3}{x}^3+\frac{3\left(240n^2-n-1\right)}{4}{x}^4+\frac{4\left(200n^2+6n-1\right)}{4}{x}^4+\frac{6\left(400n^2-29n\right)}{5}{x}^5+\frac{9\left(200n^2-41n+2\right)}{6}{x}^6.$

• ${S_x}^3{{Q_{-2}JD}_x}^3{D_y}^3\left(f\left(x,y\right)\right)=\frac{8\left(240n^2+29n\right)}{1}x+\frac{27\left(240n^2-n-1\right)}{8}{x}^2+\frac{8\left(200n^2+6n-1\right)}{1}{x}^2+\frac{8\left(400n^2-29n\right)}{1}{x}^3+\frac{729\left(200n^2-41n+2\right)}{64}{x}^4.$

With the help of Table 1 we get. $$1.{\overline{M}}_1\left(HACC\right)={\left.\left(D_x+D_y\right)f\left(x,y\right)\right|}_{x=y=1}=5680n^2-284n+4$$ $$2.{\overline{M}}_2\left(HACC\right)={\left.D_x{D}_{y}f\left(x,y\right)\right|}_{x=y=1}=6200n^2-464n+11$$ $$3.{m\overline{M}}_2\left(HACC\right)={\left.S_x{S}_{y}f\left(x,y\right)\right|}_{x=y=1}=\frac{3050}{9}{n}^2+\frac{113}{18}n-\frac{13}{36}$$ $$4..\overline{\mathit{ReZ}{G}_3}\left(HACC\right)={\left.{D_{x}D}_y{{(D}_x+D}_y)f\left(x,y\right)\right|}_{x=y=1}=30320n^2-2826n+80.$$ $$5.\overline{F}\left(HACC\right)={\left.(D_x^2+D_y^2)(f\left(x,y\right))\right|}_{x=y=1}=14000n^2-932n+18.$$ $$6.{6.\overline{R}}_k\left(HACC\right)={\left.{D_x}^k{D_y}^{k}f\left(x,y\right)\right|}_{x=y=1}=\left({{240.2}^k+240.3^k+200.2}^{2k}+{400.2}^k{3}^k+200.3^{2k}\right)n^2+\left({{29.2}^k-3^k+6.2}^{2k}-29{.2}^k{3}^k-41.3^{2k}\right)n-\left(3^k+2^{2k}-{2.3}^{2k}\right)$$ $$7.{\overline{\mathit{RR}}}_k(HACC)={{{S_x}^k{S}_y}^{k}f(x,y)|}_{x=y=1}=(\frac{240}{2^k}+\frac{240}{3^k}+\frac{200}{2^{2k}}+\frac{400}{2^k{3}^k}+\frac{200}{3^{2k}})n^2+(\frac{29}{2^k}-\frac{1}{3^k}+\frac{6}{2^{2k}}-\frac{29}{2^k{3}^k}-\frac{41}{3^{2k}})n-(\frac{1}{3^k}+\frac{1}{2^{2k}}-\frac{2}{3^{2k}})$$ $$8.\overline{\mathit{SDD}}\left(HACC\right)={\left.{(S}_y{D}_x+S_x{D}_y)f\left(x,y\right)\right|}_{x=y=1}=\frac{9200}{3}{n}^2-\frac{191}{3}n-\frac{4}{3}.$$ $$9.\overline{H}\left(HACC\right)={\left.{2S}_{x}Jf\left(x,y\right)\right|}_{x=y=1}=\frac{1820}{3}{n}^2-\frac{103}{30}n-\frac{1}{3}$$ $$10.\overline{I}\left(HACC\right)={\left.S_{x}J{D}_x{D}_{y}f\left(x,y\right)\right|}_{x=y=1}=1320n^2-\frac{4303}{60}n+\frac{5}{4}.$$ $$11.\overline{A}\left(HACC\right)={\left.{S_x}^3{Q}_{-2}J{D_x}^3{D_y}^{3}f\left(x,y\right)\right|}_{x=y=1}=\frac{78465}{8}{n}^2-\frac{27033}{64}n+\frac{365}{32}.$$

Following that, Figure-1(b) depicts the chemical structure of Hyaluronic acid conjugated with paclitaxel and let HAPTX represent the nth iteration of equivalent molecular graphs illustrated in Figure-2(b) for n = 1 and n = 3.

Now we obtain the expression of CoM-polynomial for HAPTX.

Proof: From the Figure-2(b), we have, number of vertices =  $87n+1$ and number of edges = $96n$ . Structure analysis reveals that the HAPTX edge set may be classified into nine classes and HAPTX vertex set can be put into three groups: $$\left|E_{12}\right|=n+1,\left|E_{13}\right|=16n,\left|E_{22}\right|=13n+1,\left|E_{14}\right|=4n,\left|E_{23}\right|=32n-1,\left|E_{24}\right|=3n,$$ $$\left|E_{33}\right|=19n-1,\left|E_{34}\right|=7n\left|E_{44}\right|=n$$ $$n_1=\left|V_1\right|=21n+1n_2=\left|V_2\right|=31n+1n_3=\left|V_3\right|=31n-1,n_4=\left|V_4\right|=4n$$

Using lemma 1, we have. $${\overline{m}}_{12}=651n^2+51n$$ $${\overline{m}}_{22}=961n^2+18n-1$$

Similarly. $${\overline{m}}_{13}=\left|{\overline{E}}_{13}\right|=651n^2-6n-1{\overline{m}}_{14}=\left|{\overline{E}}_{14}\right|=84n^2{\overline{m}}_{23}=\left|{\overline{E}}_{23}\right|=961n^2-32n{\overline{m}}_{24}=\left|{\overline{E}}_{24}\right|=124n^2+n{\overline{m}}_{33}=\left|{\overline{E}}_{33}\right|=961n^2-112n+3{\overline{m}}_{34}=\left|{\overline{E}}_{34}\right|=124n^2-11n{\overline{m}}_{44}=\left|{\overline{E}}_{44}\right|=16n^2-5n$$

By definition of CoM-polynomial. $$CoM\left(HAPTX;x,y\right)=\sum _{i\le j}{\overline{m}}_{\mathit{ij}}\left(HAPTX\right)x^i{y}^j$$ $$CoM\left(HAPTX;x,y\right)=\sum _{1\le 2}{\overline{m}}_{12}x{y}^2+\sum _{1\le 3}{\overline{m}}_{13}x{y}^3+\sum _{1\le 4}{\overline{m}}_{14}x{y}^4+\sum _{2\le 2}{\overline{m}}_{22}{x}^2{y}^2+\sum _{2\le 3}{\overline{m}}_{23}{x}^2{y}^3+\sum _{2\le 4}{\overline{m}}_{24}{x}^2{y}^4+\sum _{3\le 3}{\overline{m}}_{33}{x}^3{y}^3+\sum _{3\le 4}{\overline{m}}_{34}{x}^3{y}^4+\sum _{4\le 4}{\overline{m}}_{44}{x}^4{y}^4$$ $$=\left(651n^2+51n\right)x{y}^2+\left(651n^2-6n-1\right)x{y}^3+84n^{2}x{y}^4+\left(961n^2+18n-1\right)x^2{y}^2+\left(961n^2-32n\right)x^2{y}^3+\left(124n^2+n\right)x^2{y}^4+\left(961n^2-112n+3\right)x^3{y}^3+\left(1{24n}^2-11n\right)x^3{y}^4+\left(16n^2-5n\right)x^4{y}^4$$

This completes the proof of the theorem.

With the help of Theorem 2, we derive various degree-based TCI for the HAPTX in next result.