M-polynomial and NM-polynomial indices of camptothecin–polymer conjugate IT-101 structure

Ling Teng; Lei Zhang; Muhammad Mohsin Abbas; Muhammad Naeem; Fairouz Tchier; Yong Wang

doi:10.1515/mgmc-2023-0020

Artikel Open Access

M-polynomial and NM-polynomial indices of camptothecin–polymer conjugate IT-101 structure

Ling Teng , Lei Zhang , Muhammad Mohsin Abbas , Muhammad Naeem , Fairouz Tchier und Yong Wang

Veröffentlicht/Copyright: 25. März 2025

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Abstract

Camptothecin is a naturally occurring alkaloid known for its significant and selective inhibition of the topoisomerase nuclear enzyme, a critical target in cancer treatment. IT-101, a polymeric drug conjugate of camptothecin, exhibits potent antiglioma activity in vitro, making it a highly promising candidate for cancer therapy. This conjugate not only slows the progression of various malignancies but also enhances the therapeutic efficacy of camptothecin. Topological indices, which are numerical values associated with the molecular structure of chemical compounds, serve as powerful tools for predicting physical properties and biological activities. Calculating these indices offers an efficient alternative to time-consuming and costly laboratory experiments. In this article, we first computed the M-polynomial and NM-polynomial of the camptothecin–polymer conjugate IT-101 structure. By applying various integration and differentiation formulas, we have computed different degree-based topological indices (TIs) for the camptothecin–polymer conjugate IT-101 structure. From an application perspective, we employed a linear regression model to estimate the physicochemical properties of 29 anticancer drugs using eccentricity-based TIs. The results demonstrate that two properties, namely, molecular weight and complexity, can be predicted with high accuracy using the first Zagreb eccentric index. The results may be useful to obtain insights into its molecular characteristics and potential applications in cancer treatment.

Keywords: camptothecin–polymer conjugate; IT-101; M-polynomial; topological indices; NM-polynomial

1 Introduction

Graph theory is utilized in a wide variety of scientific and engineering disciplines such as biology, chemistry, computer science, and mathematics [1,2]. In the study of mathematical chemistry, chemical graph theory has developed into a significant topic, as pioneered by Trinajstic [3,4,5], Graovac et al. [6], Gutman and Trinajstic [3], Randic [7], and Balaban [8]. For instance, one may conduct a mathematical study of a chemical network and attempt to improve techniques for computing indices, which have many applications in quantitative structure–activity relationship (QSAR) and quantitative structure–property relationship (QSPR). Chemical characteristics of graph theory are also helpful in stereochemistry and quantum chemistry [9,10]. Topological indices convert data contained in a chemical molecule into useful numbers. These indices are advantageous in the study of QSAR and QSPR.

In the literature, several articles exist where authors have developed quantitative structure–property relationship (QSPR) models linking physical properties to topological indices (TIs). Fatemeh Shafiei, for example, developed QSPR models for the thermodynamic properties of monocarboxylic acids based on three indices using linear regression [11]. Shirakol et al. proposed relationships between seven distance-based indices and eight physical properties of alkanes through linear regression [12]. Ahmad et al. studied the physical properties of anthracene and phenylene, identifying suitable quadratic curves for the Estrada and energy indices and demonstrating the inequality relation between exact and estimated values [13]. Khalid et al. determined vertex degree-based indices for two families of graphs, analyzing how the topology of a graph changes completely when an edge is added, otherwise remaining the same [14]. For further information about the relationship between physical properties, refer to the previous studies [15,16].

A topological descriptor/index is a graph invariant derived from a network that describes the structure and chemical characteristics of a molecule. Chemical graph theory, based on this concept, focuses on how the structural characteristics of certain molecules determine their behavior, aiding in understanding the mechanisms of chemical processes. Various properties such as melting point, vaporization temperatures, boiling point, and molar volume are investigated using TIs [17]. Moreover, these indices can characterize lipophilicity, toxicity, cell growth stimulation, nutritive behavior, and pH control, making them useful for identifying both physical characteristics and biological activities [18,19,20].

The M-polynomial represents a relatively new form of polynomial that promises to unveil novel chemical characteristics and provide fresh insights into the study of degree-based descriptors. Its main advantage lies in its ability to accurately generate degree-based indices [20,21]. This innovative polynomial is rapidly advancing. Recently, Kwun et al. calculated the M-polynomial indices for V-phenylene nanotubes and nanotori [22].

Let G be a connected graph, with E and V denoting its edge set and vertex set, respectively. For a vertex u ∈ V , let N ( u ) = { v ∈ V | u v ∈ E } denote the set containing all the neighbors of u . The degree of a vertex u , denoted as d u , is the number of elements in N ( u ) . The neighborhood degree of a vertex v is the sum of the degree of neighbors of v and is denoted by ℵ v . The eccentricity ec ( u ) of a vertex u ∈ V is the maximum distance of u and any vertex in G . We suggest that readers refer to West’s work on basic principles of graph theory [23].

The M-polynomial of a graph G is represented as follows [20]:

M ( G ) = ∑ i ≤ j ∣ N ( i , j ) ∣ x i y j

where ∣ N ( i , j ) ∣ is the number of edges having end vertices degree i and j and x and y are the variables.

The Wiener index [24] was the first topological introduced by Weiner, which is defined as follows:

W ( G ) = 1 2 ∑ u , v ∈ V ( P ) d ( u , v )

He used this index to estimate the boiling point of paraffin. We propose the interested readers to consult [25,26] for a comprehensive discussion of the Wiener index. The first and second Zagreb indices are defined as follows [3]:

M 1 ( G ) = ∑ u v ∈ E ( P ) ( d u + d v )

M 2 ( G ) = ∑ u v ∈ E ( P ) ( d u × d v )

More details on Zagreb indices can be found in the previous studies [27–29].

Nikolic et al. [4] proposed a new form of the second Zagreb index, which is called the modified second Zagreb index. It is denoted by M 2 m and is defined as follows:

M 2 m ( G ) = ∑ u v ∈ E ( G ) 1 d v d u

In 2011, Fath-Tabar [30] proposed the third Zagreb index, indicated by M 3 , which is defined as follows:

M 3 ( G ) = ∑ u v ∈ E ( G ) ∣ d v − d u ∣

The symmetric division index [6] is a topological index that is calculated as follows:

SDD ( G ) = ∑ u v ∈ E ( G ) max ( d v , d u ) min ( d v , d u ) + min ( d v , d u ) max ( d v , d u )

In 2010, Furtula et al. [6] developed the augmented Zagreb index, which is defined as follows:

AZ ( G ) = ∑ u v ∈ E ( G ) d v d u d v + d u − 2 3

Zhou and Trinajstić designed the sum-connectivity index ( S ) [31] as follows:

S ( G ) = ∑ u v ∈ E ( G ) 1 d v + d u

The graph invariant known as the inverse sum index has been examined as a key descriptor of surface area for isomers of octane. It is calculated as follows:

I ( G ) = ∑ u v ∈ E ( G ) d v d u d v + d u

In 2003, Caporossi et al. [32] established some fascinating and crucial mathematical characteristics. The Harmonic index [33] is defined as follows:

H ( G ) = ∑ u v ∈ E ( G ) 2 d v + d u

Munir et al. [21] proposed the M-polynomial idea in 2015. The role of M-polynomial is the same for degree-based indices as the Hosoya polynomial does for distance-based indices. For more information on the computation of M-Polynomials of various graphs, see the previous studies [6,22,34–36]. Let M ( G ; x , y ) = p ( x , y ) , and

D x = x ∂ p ( x , y ) ∂ x

D y = y ∂ p ( x , y ) ∂ y

I x = ∫ 0 x p ( t , y ) t d t

I y = ∫ 0 y p ( x , t ) t d t

J ( p ( x , y ) ) = p ( x , x )

Q α ( p ( x , y ) ) = x α p ( x , y )

One way to compute the degree-based TIs is by using M-polynomial. The mathematical formulas of the TIs in terms of M-polynomial are presented in Table 1. Recently, some new degree-based TIs were defined by different authors depending on the neighborhood degree. These neighborhood degree-based indices can be computed by using NM-polynomial. The mathematical formulas of neighborhood-based TIs in the form of NM-polynomila are given in Table 2.

Table 1

The formula of the degree-based indices in the form of M-polynomial

Topological index	TP	Degree based	Derivation
Second Zagreb	M 2	∑ u v ∈ E d u d v	( D x D y ) ⋅ ( M ( G ) ) ∣ x , y = 1
Second modified Zagreb	M 2 m	∑ u v ∈ E 1 d u d v	( I x I y ) ⋅ ( M ( G ) ) ∣ x , y = 1
First Zagreb	M 1	∑ u v ∈ E d u + d v	( D x + D y ) ⋅ ( M ( G ) ) ∣ x , y = 1
Augmented Zagreb	AZ	∑ u v ∈ E d u d v d u + d v ‒ 2 3	I x 3 Q ‒ 2 J D x 3 D y 3 ( M ( G ) ) ∣ x = 1
General Randic	R α	∑ u v ∈ E d u α d v α	D x α D y α ( M ( G ) ) ∣ x , y = 1
Harmonic	H	∑ u v ∈ E 2 d u + d v	2 I x J ( M ( G ) ) ∣ x = 1
Inverse-sum	I	∑ u v ∈ E d u d v d u + d v	I x J D x D y ( M ( G ) ) ∣ x = 1
Forgotten	F	∑ u v ∈ E d u 2 + d v 2	( D x 2 + D y 2 ) ⋅ ( M ( G ) ) ∣ x , y = 1
Symmetric division	SDD	∑ u v ∈ E d u 2 + d v 2 d u d v	( D x I y + I x D y ) ⋅ ( M ( G ) ) ∣ x , y = 1
Redefined third Zagreb	RZ 3	∑ u v ∈ E d u d v ( d u + d u )	D x ⋅ D y ( D x + D y ) ⋅ M ( G ) ∣ x , y = 1

Table 2

The formula of the neighborhood degree-based indices in the form of NM-polynomial

Topological index	TP	Nd degree sum	Derivation
Third version of Zagreb	M 1 ′	∑ u v ∈ E ℵ u + ℵ v	( D x + D y ) ⋅ ( NM ( G ) ) ∣ x , y =1
Neighborhood Harmonic	NH	∑ u v ∈ E 2 ℵ u + ℵ v	2 I x J ( NM ( G ) ) ∣ x = 1
Neighborhood inverse sum	NI	∑ u v ∈ E ℵ u ℵ v ℵ u + ℵ v	I x J D x D y ( N M ( G ) ) ∣ x = 1
Neighborhood second Zagreb	M 2 ⁎	∑ u v ∈ E ℵ u ℵ v	( D x D y ) ( NM ( G ) ) ∣ x , y =1
Third neighborhood	ND 3	∑ u v ∈ E ℵ u ℵ v ( ℵ u + ℵ u )	D x D y ( D x + D y ) ( NM ( G ) ) ∣ x , y =1
Modified neighborhood second
Zagreb	M 2 n m	∑ u v ∈ E 1 ℵ u ℵ v	( I x I y ) ( NM ( G ) ) ∣ x , y =1
Sanskruti	S	∑ u v ∈ E ℵ u ℵ v ℵ u + ℵ v ‒ 2 3	I x 3 Q ‒ 2 J D x 3 D y 3 ( NM ( G ) ) ∣ x =1
Neighborhood general Randic	NR α	∑ u v ∈ E ℵ u α ℵ v α	D x α D y α ( NM ( G ) ) ∣ x , y =1
Fifth neighborhood	ND 5	∑ u v ∈ E ℵ u 2 + ℵ v 2 ℵ u ℵ v	( D x I y + I x D y ) ( NM ( G ) ) ∣ x , y =1
Neighborhood forgotten	F N ⁎	∑ u v ∈ E ℵ u 2 + ℵ v 2	( D x 2 + D y 2 ) ( NM ( G ) ) ∣ x , y =1

2 Topological indices computation of camptothecin–polymer conjugate (IT-101)

A polymer–drug conjugate encompasses a wide range of components, from approved medicines to experimental substances like pegylated proteins. While all polymer–drug conjugates are now referred to as nanoscaled, only a select few qualify as “truly nanoparticles” [37]. Recent reviews on nanoparticle treatments in medicine have addressed this distinction [37]. One such example is IT-101, a camptothecin–polymer conjugate based on cyclodextrin [38], utilized as a cancer medication. Camptothecin, an alkaloid derived from a Chinese tree, profoundly affects the topoisomerase (I) nuclear enzyme, thereby impeding the development of several malignancies. Despite its efficacy, the systemic administration of camptothecin presents challenges due to its inherent toxicity. Nevertheless, its strong antiglioma activity in vitro makes it a promising candidate for polymer conjugation.

In the control model, patients receiving camptothecin polymers exhibited a median survival ranging from 19 to 120 days [39]. Interestingly, the initial peritumoral dosage of our medicine showed no significant effect on survival compared to the control group. When camptothecin was added to 50% polymers to detect 9L gliosarcoma in rats, a 69-day cure rate was observed, indicating efficacy and immunity in camptothecin-treated animals. Furthermore, under similar conditions, administering a high dose of cerebral camptothecin led to a substantial increase in longevity and a significant improvement in continuity [39].

In the next theorem, we compute the M-polynomial for the molecular structure of camptothecin–polymer conjugate IT-101. The molecular structure of camptothecin–polymer conjugate IT-101 is depicted in Figure 1.

Figure 1

Structure of camptothecin–polymer conjugate (IT-101).

Theorem 1

Let G be a molecular graph of camptothecin–polymer conjugate IT-101. Then, the M-polynomial of G is expressed as follows:

M ( G ; x , y ) = ( 4 x y 4 + x 2 y 4 + x 4 y 4 ) m n + ( 19 x y 2 + 25 x y 3 + 85 x y 4 + 8 x 2 y 3 + 60 x 2 y 4 + 35 x 3 y 3 + 20 x 3 y 4 + 43 x 4 y 4 ) m + x y 3 + 6 x y 4 + 2 x 3 y 4

Proof 1

Let G be a molecular graph of camptothecin–polymer conjugate IT-101 structure. The graph G has | V ( G ) | = 7 m n + 267 m + 10 vertices and | E ( G ) | = 6 m n + 295 m + 9 edges. To compute the M-polynomial of G , we need to find the edge partition of G based on the degree of end vertices of each edge. Let N ( i , j ) = { u v ∈ E ( G ) | d u = i , d v = j } . From the graph of G , it is easy to observe that the edge set of G can be partitioned in to eight sets. The edge partition is as follows: | N (1,2) | = 19 m , ∣ N ( 1 , 3 ) ∣ = 25 m + 1 , ∣ N ( 1 , 4 ) ∣ = 4 m n + 85 m + 6 , ∣ N (2,3) ∣ = 8 m , ∣ N ( 2 , 4 ) ∣ = m n + 60 m , ∣ N ( 3 , 3 ) ∣ = 35 m , ∣ N ( 3 , 4 ) ∣ = 20 m + 2 , and ∣ N ( 4 , 4 ) ∣ = m n + 43 m .□

Now, according to the definition of M-polynomial, we have

M ( G ) = ∑ i ≤ j ∣ N ( i , j ) ∣ x i y j = ∣ N ( 1 , 2 ) ∣ x 1 ⋅ y 2 + ∣ N ( 1 , 3 ) ∣ x 1 ⋅ y 3 + ∣ N ( 1 , 4 ) ∣ x 1 ⋅ y 4 + ∣ N ( 2 , 3 ) ∣ x 2 ⋅ y 3 + ∣ N ( 2 , 4 ) ∣ x 2 ⋅ y 4 + ∣ N ( 3 , 3 ) ∣ x 3 ⋅ y 3 + ∣ N ( 3 , 4 ) ∣ x 3 ⋅ y 4 + ∣ N ( 4 , 4 ) ∣ x 4 ⋅ y 4

Now, by using the values from the edge partition, we obtain

M ( G ) = 19 m x y 2 + ( 25 m + 1 ) x y 3 + ( 4 m n + 85 m + 6 ) x y 4 + 8 m x 2 y 3 + ( m n + 60 m ) x 2 y 4 + 35 m x 3 y 3 + ( 20 m + 2 ) x 3 y 4 + ( m n + 43 m ) x 4 y 4 = ( 4 x y 4 + x 2 y 4 + x 4 y 4 ) m n + ( 19 x y 2 + 25 x y 3 + 85 x y 4 + 8 x 2 y 3 + 35 x 3 y 3 + 60 x 2 y 4 + 20 x 3 y 4 + 43 x 4 y 4 ) m + x y 3 + 6 x y 4 + 2 x 3 y 4

Theorem 2

Let G be a molecular graph camptothecin polymer conjugate IT-101, then

M 1 ( G ) = 34 m n + 1676 m + 48

M 2 ( G ) = 40 m n + 2,224 m + 51

F ( G ) = 120 m n + 5 , 600 m + 162

RZ 3 ( G ) = 256 m n + 14 , 298 m + 300

R α ( G ) = ( 4 × 4 α + 2 3 α + 4 2 α ) m n + ( 19 × 2 α + 25 × 3 α + 85 × 4 α + 8 × 6 α + 60 × 2 3 α + 35 × 3 2 α + 20 × 12 α + 43 × 4 2 α ) m + 3 α + 6 × 4 α + 2 × 12 α

M 2 m ( G ) = 19 16 m n + 8 , 087 144 m + 2

SDD ( G ) = 43 2 m n + 10 , 285 m + 33

H ( G ) = 131 60 m n + 42 , 909 420 m + 243 60

I ( G ) = 98 15 m n + 151 , 957 420 m + 1 , 257 140

AZ ( G ) = 11 , 704 27 m n + 7 , 592 , 921 , 805 216 , 000 m + 1 , 221 , 621 27 , 000

Proof 2

The formula of the M-polynomial of G is expressed as follows:

p ( x , y ) = ( 4 x y 4 + x 2 y 4 + x 4 y 4 ) m n + ( 19 x y 4 + 25 x y 3 + 85 x y 4 + 8 x 2 y 3 + 35 x 3 y 3 + 60 x 2 y 4 + 20 x 3 y 4 + 43 x 4 y 4 ) m + x y 3 + 6 x y 4 + 2 x 3 y 4

Then the degree-based TIs can be calculated as follows:

( D x + D y ) p ( x , y ) = ( 20 x y 4 + 6 x 2 y 4 + 8 x 4 y 4 ) m n + ( 57 x y 2 + 100 x y 3 + 425 x y 4 + 40 x 2 y 3 + 360 x 2 y 4 + 210 x 3 y 3 + 140 x 3 y 4 + 344 x 4 y 4 ) m + 4 x y 3 + 30 x y 4 + 14 x 3 y 4 M 1 ( G ) = 34 m n + 1676 m + 48

D x D y ( p ( x , y ) ) = ( 16 x y 4 + 8 x 2 y 4 + 16 x 4 y 4 ) m n + ( 38 x y 2 + 75 x y 3 + 340 x y 4 + 48 x 2 y 3 + 480 x 2 y 4 + 315 x 3 y 3 + 240 x 3 y 4 + 688 x 4 y 4 ) m + 3 x y 3 + 24 x y 4 + 24 x 3 y 4 M 2 ( G ) = 40 m n + 2224 m + 51

( D x 2 + D y 2 ) p ( x , y ) = ( 68 x y 4 + 20 x 2 y 4 + 32 x 4 y 4 ) m n + ( 95 x y 2 + 250 x y 3 + 1,445 x y 4 + 104 x 2 y 3 + 1 , 200 x 2 y 4 + 630 x 3 y 3 + 500 x 3 y 4 + 1,376 x 4 y 4 ) m + 10 x y 3 + 102 x 2 y 4 + 50 x 3 y 4 F ( G ) = 120 m n + 5 , 600 m + 162

D x D y ( D x + D y ) p ( x , y ) = ( 80 x y 4 + 48 x 2 y 4 + 128 x 4 y 4 ) m n + ( 104 x y 2 + 300 x y 3 + 1 , 700 x y 4 + 240 x 2 y 3 + 2,880 x 2 y 4 + 1,890 x 3 y 3 + 1 , 680 x 3 y 4 + 5,504 x 4 y 4 ) m + 12 x y 3 + 120 x y 4 + 168 x 3 y 4 RZ 3 ( G ) = 256 m n + 14 , 298 m + 300

D x α D y α ( p ( x , y ) ) = ( 4 × 4 α x y 4 + 2 3 α x 2 y 4 + 4 2 α x 4 y 4 ) m n + ( 19 × 2 α x y 2 + 25 × 3 α x y 3 + 85 × 4 α x y 4 + 8 × 6 α x 2 y 3 + 60 × 8 α x 2 y 4 + 35 × 9 α x 3 y 3 + 20 × 12 α x 3 y 4 + 43 × 16 α x 4 y 4 ) m + 3 α x y 3 + 6 × 4 α x y 4 + 2 × 12 α x 3 y 4 R α ( G ) = ( 4 × 4 α + 2 3 α + 4 2 α ) m n + ( 19 × 2 α + 25 × 3 α + 85 × 4 α + 8 × 6 α + 60 × 8 α + 35 × 9 α + 20 × 12 α + 43 × 16 α ) m + 3 α + 6 × 4 α + 2 × 12 α

I x I y ( p ( x , y ) ) = x y 4 + 1 8 x 2 y 4 + 1 16 x 4 y 4 m n + 19 2 x y 2 + 25 3 x y 3 + 85 4 x y 4 + 4 3 x 2 y 3 + 15 2 x 2 y 4 + 35 9 x 3 y 3 + 5 3 x 3 y 4 + 43 16 x 4 y 4 m + 1 3 x y 3 + 3 2 x y 4 + 1 6 x 3 y 4 M 2 m ( G ) = 19 16 m n + 8 , 087 144 m + 2

( D x I y + I x D y ) p ( x , y ) = 17 x y 4 + 5 2 x 2 y 4 + 2 x 4 y 4 m n + 95 2 x y 2 + 250 3 x y 3 + 1,445 4 x y 4 + 52 3 x 2 y 3 + 150 x 2 y 4 + 70 x 3 y 3 + 125 3 x 3 y 4 + 86 x 4 y 4 m + 10 3 x y 3 + 51 2 x y 4 + 25 6 x 3 y 4 SDD ( G ) = 43 2 m n + 10 , 285 m + 33 2 I x J ( p ( x , y ) ) = 8 5 x 5 + 1 3 x 6 + 1 4 x 8 m n + 38 3 x 3 + 25 2 x 4 + 34 x 5 + 16 5 x 5 + 20 x 6 + 35 3 x 6 + 40 7 x 7 + 43 4 x 8 m + 1 2 x 4 + 12 5 x 5 + 4 7 x 7 H ( G ) = 131 60 m n + 42 , 909 420 m + 243 60

I x J D x D y ( p ( x , y ) ) = 16 5 x 5 + 4 3 x 6 + 2 x 8 m n + 38 3 x 3 + 75 4 x 4 + 48 5 x 5 + 315 6 x 6 + 240 7 x 7 + 68 x 5 + 80 x 6 + 86 x 8 m + 3 4 x 4 + 24 5 x 5 + 24 7 x 7 I ( G ) = 98 15 m n + 151 , 957 420 + 1 , 257 140

I x 3 Q − 2 J D x 3 D y 3 ( p ( x , y ) ) = 10 , 976 27 x 5 + 512 27 x 8 + 8 x 6 m n + 32 , 832 x 3 + 675 8 x 4 + 5 , 440 27 x 5 + 64 x 5 + 480 x 6 + 25 , 515 64 x 6 + 34 , 560 125 x 7 + 176 , 128 216 x 8 m + 27 8 x 4 + 128 9 x 5 + 3 , 456 125 x 7

AZ ( G ) = 11 , 704 27 m n + 7 , 592 , 921 , 805 216 , 000 m + 1 , 221 , 621 27 , 000

We have plotted the 3D graph of each of the computed topological index. These plots are depicted in Figures 2–6. The 3D plots help to understand the behavior of these TIs. We have also computed the numerical values of these TIs for different values of m and n . These values are presented in Tables 3 and 4. It is easy to observe that the value of each topological index increases with the increase in the value of m and n .

$Figure 2 3D plot of (a) M 1 ( G ) {M}_{1}(G) and (b) M 2 ( G ) {M}_{2}(G) .$

Figure 2

3D plot of (a) M 1 ( G ) and (b) M 2 ( G ) .

$Figure 3 3D plot of (a) F ( G ) F(G) and (b) RZ 3 ( G ) {\text{RZ}}_{\text{3}}(G) .$

Figure 3

3D plot of (a) F ( G ) and (b) RZ 3 ( G ) .

$Figure 4 3D plot of (a) R α ( G ) {R}_{\alpha }(G) and (b) M 2 m ( G ) {}^{m}M_{2}(G) .$

Figure 4

3D plot of (a) R α ( G ) and (b) M 2 m ( G ) .

$Figure 5 3D plot of (a) SDD( G ) \text{SDD(}G\text{)} and (b) H ( G ) H(G) .$

Figure 5

3D plot of (a) SDD( G ) and (b) H ( G ) .

$Figure 6 3D plot of (a) I ( G ) I(G) and (b) AZ( G ) \text{AZ(}G\text{)} .$

Figure 6

3D plot of (a) I ( G ) and (b) AZ( G ) .

Table 3

Numerical values of degree-based TIs for different values of m and n

[m, n]	M ₁(G)	M ₂(G)	F(G)	RZ ₃(G)	R _a(G)
[1,1]	1,758	2,315	5,882	14,854	2,315
[2,2]	3,536	4,659	11,842	29,920	35,449
[3,3]	5,382	7,083	18,042	45,498	674,817
[4,4]	7,296	9,587	24,482	61,588	13,370,349
[5,5]	9,278	12,171	31,162	78,190	262,249,257
[6,6]	11,328	14,835	38,082	95,304	5,043,315,389
[7,7]	13,446	17,579	45,242	112,930	95,143,703,849
[8,8]	15,632	20,403	52,642	131,068	1,765,429,617,469
[9,9]	17,886	23,307	60,282	149,718	32,306,904,629,769
[10,10]	20,208	26,291	68,162	168,880	584,398,181,015,037

Table 4

Numerical values of degree-based TIs for different values of m and n .

[m, n]	^m M ₂(G)	SDD(G)	H(G)	I(G)	AZ(G)
[1,1]	59.347	10,339.5	108.39762	377.31429	35,631.14247
[2,2]	119.07	20,689.0	217.11190	758.71667	72,084.00268
[3,3]	181.17	31,081.5	330.19286	1,153.18571	109,403.82585
[4,4]	245.64	41,517.0	447.64048	1,560.72143	147,590.61198
[5,5]	312.49	51,995.5	569.45476	1,981.32381	186,644.36108
[6,6]	381.71	62,517.0	695.63571	2,414.99286	226,565.07314
[7,7]	453.31	73,081.5	826.18333	2,861.72857	267,352.74816
[8,8]	527.28	83,689.0	961.09762	3,321.53095	309,007.38615
[9,9]	603.63	94,339.5	1,100.37857	3,794.40000	351,528.98709
[10,10]	682.35	105,033.0	1,244.02619	4,280.33571	394,917.55101

In the next theorem, we give an explicit expression for the NM-polynomial of camptothecin–polymer conjugate IT-101.

Theorem 3

Let G be a molecular graph of camptothecin-polymer conjugate IT-101 structure. Then, the NM-polynomial of G is expressed as follows:

NM ( G ; x , y ) = ( 4 x 4 y 8 + 2 x 8 y 8 ) m n + ( 19 x 2 y 5 + 18 x 3 y 7 + 6 x 3 y 8 + x 3 y 9 + 10 x 4 y 7 + 30 x 4 y 8 + 11 x 4 y 9 + 4 x 4 y 10 + 30 x 4 y 11 + 4 x 6 y 8 + 14 x 7 y 7 + 5 x 5 y 8 + 14 x 5 y 11 + 4 x 7 y 12 + 12 x 8 y 8 + 21 x 8 y 9 + 26 x 8 y 11 + 4 x 7 y 8 + 6 x 7 y 9 + 10 x 7 y 10 + 2 x 8 y 10 + 2 x 9 y 10 + 2 x 10 y 10 + 4 x 10 y 8 + x 9 y 6 + 8 x 9 y 11 + 2 x 10 y 7 + 4 x 10 y 12 + 21 x 11 y 11 ) m + x 3 y 9 + 6 x 4 y 6 + x 9 y 6 + x 9 y 9

Proof 3

Let G be a molecular graph of camptothecin polymer conjugate IT-101 structure. The graph G has ∣ V ( G ) ∣ = 7 m n + 267 m + 10 vertices and ∣ E ( G ) ∣ = 6 m n + 295 m + 9 edges. To compute the M-polynomial of G , we need to find the edge partition of G based on the neighborhood degree of end vertices of each edge. Let N ( i , j ) ⁎ = { u v ∈ E ( G ) | ℵ u = i , ℵ v = j } . From the graph of G , it is easy to observe that the edge set of G can be partitioned into 29 sets. The edge partition is as follows: ∣ N ( 2 , 5 ) ⁎ ∣ = 19 m , ∣ N ( 3 , 7 ) ⁎ ∣ = 18 m , ∣ N ( 3 , 8 ) ⁎ ∣ = 6 m , ∣ N ( 3 , 9 ) ⁎ ∣ = m + 1 , ∣ N ( 4 , 6 ) ⁎ ∣ = 6 , ∣ N ( 4 , 7 ) ⁎ ∣ = 10 m , ∣ N ( 4 , 8 ) ⁎ ∣ = 4 m n + 30 m , ∣ N ( 4 , 9 ) ⁎ ∣ = 11 m , ∣ N ( 4 , 10 ) ⁎ ∣ = 4 m , ∣ N ( 4 , 11 ) ⁎ ∣ = 30 m , ∣ N ( 6 , 8 ) ⁎ ∣ = 4 m , ∣ N ( 7 , 7 ) ⁎ ∣ = 14 m , ∣ N ( 5 , 8 ) ⁎ ∣ = 5 m , ∣ N (5,11) ⁎ ∣ = 14 m , ∣ N ( 7 , 12 ) ⁎ ∣ = 4 m , ∣ N ( 8 , 8 ) ⁎ ∣ = 2 m n + 12 m , ∣ N ( 8 , 9 ) ⁎ ∣ = 21 m , ∣ N ( 8 , 10 ) ⁎ ∣ = 6 m , ∣ N ( 8 , 11 ) ⁎ ∣ = 26 m , ∣ N ( 7 , 8 ) ⁎ ∣ = 4 m , ∣ N ( 7 , 9 ) ⁎ ∣ = 6 m , ∣ N ( 7 , 10 ) ⁎ ∣ = 12 m , ∣ N ( 6 , 9 ) ⁎ ∣ = m + 1 , ∣ N ( 9 , 9 ) ⁎ ∣ = 1 , ∣ N ( 9 , 10 ) ⁎ ∣ = 2 m , ∣ N ( 9 , 11 ) ⁎ ∣ = 8 m , ∣ N ( 10 , 10 ) ⁎ ∣ = 2 m , ∣ N ( 10 , 12 ) ⁎ ∣ = 4 m , ∣ N ( 11 , 11 ) ⁎ ∣ = 21 m .□

Now, by using the definition of NM-polynomial, we have

NM ( G ) = ∑ i ≤ j ∣ N ( i , j ) ⁎ ∣ x i y j = ∣ N ( 2 , 5 ) ⁎ ∣ x 2 ⋅ y 5 + ∣ N ( 3 , 7 ) ⁎ ∣ x 3 ⋅ y 7 + ∣ N ( 3 , 8 ) ⁎ ∣ x 3 ⋅ y 8 + ∣ N ( 3 , 9 ) ⁎ ∣ x 3 ⋅ y 9 + ∣ N ( 4 , 6 ) ⁎ ∣ x 4 ⋅ y 6 + ∣ N ( 4 , 7 ) ⁎ ∣ x 4 ⋅ y 7 + ∣ N ( 4 , 8 ) ⁎ ∣ x 4 ⋅ y 8 + ∣ N ( 4 , 9 ) ⁎ ∣ x 4 ⋅ y 9 + ∣ N ( 4 , 10 ) ⁎ ∣ x 4 ⋅ y 10 + ∣ N ( 4 , 11 ) ⁎ ∣ x 4 ⋅ y 11 + ∣ N (6,8) ⁎ ∣ x 6 ⋅ y 8 + ∣ N (5,8) ⁎ ∣ x 5 ⋅ y 8 + ∣ N (5,11) ⁎ ∣ x 5 ⋅ y 11 + ∣ N (7,7) ⁎ ∣ x 7 ⋅ y 7 + ∣ N (7,12) ⁎ ∣ x 7 ⋅ y 12 + ∣ N (8,8) ⁎ ∣ x 8 ⋅ y 8 + ∣ N (8,9) ⁎ ∣ x 8 ⋅ y 9 + ∣ N (8,10) ⁎ ∣ x 8 ⋅ y 10 + ∣ N (8,11) ⁎ ∣ x 8 ⋅ y 11 + ∣ N (7,8) ⁎ ∣ x 7 ⋅ y 8 + ∣ N (7,9) ⁎ ∣ x 7 ⋅ y 9 + ∣ N (7,10) ⁎ ∣ x 7 ⋅ y 10 + ∣ N (6,9) ⁎ ∣ x 6 ⋅ y 9 + ∣ N (9,9) ⁎ ∣ x 9 ⋅ y 9 + ∣ N (9,10) ⁎ ∣ x 9 ⋅ y 10 + ∣ N (9,19) ⁎ ∣ x 9 ⋅ y 11 + ∣ N (10,10) ⁎ ∣ x 10 ⋅ y 10 + ∣ N (10,12) ⁎ ∣ x 10 ⋅ y 12 + ∣ N (11,11) ⁎ ∣ x 11 ⋅ y 11

By using the values of ∣ N ( i , j ) ⁎ ∣ , we obtain

NM ( G ; x , y ) = 19 m x 2 y 5 + 18 m x 3 y 7 + 6 m x 3 y 8 + ( m + 1 ) x 3 y 9 + 6 x 4 y 6 + 10 m x 4 y 7 + (4 m n + 30 m ) x 4 y 8 + 11 m x 4 y 9 + 4 m x 4 y 10 + 30 m x 4 y 11 + 4 m x 6 y 8 + 14 m x 7 y 7 + 5 m x 5 y 8 + 14 m x 5 y 11 + 4 m x 7 y 12 + (2 m n + 12 m ) x 8 y 8 + 21 m x 8 y 9 + 6 m x 8 y 10 + 26 m x 8 y 11 + 4 m x 7 y 8 + 6 m x 7 y 9 + 12 m x 7 y 10 + ( m + 1 ) x 6 y 9 + x 9 y 9 + 2 m x 9 y 10 + 8 m x 9 y 11 + 2 m x 10 y 10 + 4 m x 10 y 12 + 21 m x 11 y 11

NM ( G ; x , y ) = ( 4 x 4 y 8 + 2 x 8 y 8 ) m n + ( 19 x 2 y 5 + 18 x 3 y 7 + 6 x 3 y 8 + x 3 y 9 + 10 x 4 y 7 + 30 x 4 y 8 + 11 x 4 y 9 + 4 x 4 y 10 + 30 x 4 y 11 + 4 x 6 y 8 + 14 x 7 y 7 + 5 x 5 y 8 + 14 x 5 y 11 + 4 x 7 y 12 + 12 x 8 y 8 + 21 x 8 y 9 + 26 x 8 y 11 + 4 x 7 y 8 + 6 x 7 y 9 + 12 x 7 y 10 + x 6 y 9 + 2 x 9 y 10 + 8 x 9 y 11 + 2 x 10 y 10 + x 9 y 6 + 4 x 10 y 12 + 21 x 11 y 11 ) m + x 3 y 9 + 6 x 4 y 6 + x 9 y 6 + x 9 y 9

Theorem 4

Let G be a molecular graph of camptothecin–polymer conjugate IT-101 structure, then

M 1 ′ = 80 m n + 4 , 441 m + 105

M 2 ⁎ = 256 m n + 16 , 776 m + 306

F N ⁎ = 576 m n + 37 , 795 m + 681

S = 883 , 949 , 568 2 , 744 , 000 m n + 5 , 960 , 284 , 677 250 , 000 m + 1 , 765 , 118 , 848 , 000 , 000 4 , 607 , 442 , 944 , 000

ND 3 = 3 , 584 m n + 286 , 726 m + 4 , 032

ND 5 = 14 m n + 4 , 913 , 119 6 , 930 m + 41 2

NH = 11 12 m n + 137 , 624 , 243 3 , 233 , 230 m + 29 18

NI = 56 3 m n + 3 , 335 , 165 , 313 3 , 233 , 230 m + 99 4

M 2 n m ⁎ = 5 32 m n + 8 , 450 , 851 1 , 306 , 800 m + 103 324

NR α = ( 4 × 2 5 α + 2 × 2 6 α ) m n + ( 19 × 10 α + 18 × 21 α + 6 × 24 α + 10 × 28 α + 30 × 2 5 α + 27 α + 11 × 6 2 α + 4 × 40 α + 30 × 44 α + 4 × 48 α + 14 × 7 2 α + 5 × 40 α + 14 × 55 α + 4 × 84 α + 12 × 2 6 α + 21 × 72 α + 26 × 88 α + 4 × 56 α + 6 × 63 α + 10 × 70 α + 2 × 80 α + 2 × 90 α + 2 × 10 2 α + 4 × 80 α + 54 α + 8 × 99 α + 2 × 70 α + 4 × 120 α + 21 × 11 2 α ) m + 3 3 α + 6 × 24 α + 54 α + 9 2 α

Proof 4

The formula of the NM-polynomial of G is given as follows:

Then the degree based TIs can be calculated as follows:

( D x + D y ) ( NM ( G ) ) = ( 48 x 4 y 8 + 32 x 8 y 8 ) m n + ( 133 x 2 y 5 + 180 x 3 y 7 + 66 x 3 y 8 + 12 x 3 y 9 + 110 x 4 y 7 + 360 x 4 y 8 + 143 x 4 y 9 + 56 x 4 y 10 + 450 x 4 y 11 + 56 x 6 y 8 + 196 x 7 y 7 + 65 x 5 y 8 + 224 x 5 y 11 + 76 x 7 y 12 + 192 x 8 y 8 + 357 x 8 y 9 + 494 x 8 y 11 + 60 x 7 y 8 + 96 x 7 y 9 + 170 x 7 y 10 + 36 x 8 y 10 + 38 x 9 y 10 + 40 x 10 y 10 + 72 x 10 y 8 + 15 x 9 y 6 + 160 x 9 y 11 + 34 x 10 y 7 + 88 x 10 y 12 + 462 x 11 y 11 ) m + 12 x 3 y 9 + 60 x 4 y 6 + 15 x 9 y 6 + 18 x 9 y 9 M 1 ′ ( G ) = 80 m n + 4 , 441 m + 105

D x D y ( NM ( G ) ) = ( 128 x 4 y 8 + 128 x 8 y 8 ) m n + ( 190 x 2 y 5 + 378 x 3 y 7 + 144 x 3 y 8 + 27 x 3 y 9 + 280 x 4 y 7 + 960 x 4 y 8 + 396 x 4 y 9 + 160 x 4 y 10 + 1 , 320 x 4 y 11 + 192 x 6 y 8 + 686 x 7 y 7 + 200 x 5 y 8 + 770 x 5 y 11 + 336 x 7 y 12 + 768 x 8 y 8 + 1 , 512 x 8 y 9 + 2,288 x 8 y 11 + 224 x 7 y 8 + 378 x 7 y 9 + 700 x 7 y 10 + 160 x 8 y 10 + 180 x 9 y 10 + 200 x 10 y 10 + 320 x 10 y 8 + 54 x 9 y 6 + 792 x 9 y 11 + 140 x 10 y 7 + 480 x 10 y 12 + 2 , 541 x 11 y 11 ) m + 27 x 3 y 9 + 144 x 4 y 6 + 54 x 9 y 6 + 81 x 9 y 9 M 2 ⁎ ( G ) = 256 m n + 16 , 776 m + 306

( D x 2 + D y 2 ) ( NM ( G ) ) = ( 320 x 4 y 8 + 256 x 8 y 8 ) m n + ( 551 x 2 y 5 + 1,044 x 3 y 7 + 438 x 3 y 8 + 90 x 3 y 9 + 650 x 4 y 7 + 2 , 400 x 4 y 8 + 1,067 x 4 y 9 + 464 x 4 y 10 + 4 , 110 x 4 y 11 + 400 x 6 y 8 + 1,372 x 7 y 7 + 445 x 5 y 8 + 2 , 044 x 5 y 11 + 772 x 7 y 12 + 1,536 x 8 y 8 + 3,045 x 8 y 9 + 4 , 810 x 8 y 11 + 452 x 7 y 8 + 780 x 7 y 9 + 1,490 x 7 y 10 + 328 x 8 y 10 + 362 x 9 y 10 + 400 x 10 y 10 + 656 x 10 y 8 + 117 x 9 y 6 + 1 , 616 x 9 y 11 + 298 x 10 y 7 + 976 x 10 y 12 + 5 , 082 x 11 y 11 ) m + 90 x 3 y 9 + 312 x 4 y 6 + 117 x 9 y 6 + 162 x 9 y 9 F N ⁎ ( G ) = 576 m n + 37 , 795 m + 681

I x 3 Q − 2 J D x 3 D y 3 ( NM ( G ) ) = 131 , 072 1 , 000 x 10 + 524 , 288 2 , 744 x 14 m n + 19 , 000 125 x 5 + 166 , 698 512 x 8 + 82 , 944 729 x 9 + 19 , 683 1 , 000 x 10 + 219 , 520 729 x 9 + 983 , 040 1 , 000 x 10 + 513 , 216 1 , 331 x 11 + 256 , 000 1 , 728 x 12 + 2 , 555 , 520 2 , 197 x 13 + 442 , 368 1 , 728 x 12 + 1 , 647 , 086 1 , 728 x 12 + 320 , 000 1 , 331 x 11 + 2 , 329 , 250 2 , 744 x 14 + 2 , 370 , 816 4 , 913 x 17 + 3 , 145 , 728 2 , 744 x 14 + 7 , 838 , 208 3 , 375 x 15 + 17 , 718 , 272 4 , 913 x 17 + 702 , 464 2 , 197 x 13 + 1 , 500 , 282 2 , 744 x 14 + 3 , 430 , 000 3 , 375 x 15 + 1 , 024 , 000 4 , 096 x 16 + 1 , 458 , 000 4 , 913 x 17 + 2 , 000 , 000 5 , 832 x 18 + 2 , 048 , 000 4 , 096 x 16 + 157 , 464 2 , 197 x 13 + 7 , 762 , 392 5 , 832 x 18 + 686 , 000 3 , 375 x 15 + 6 , 912 , 000 8 , 000 x 20 + 37 , 202 , 781 8 , 000 x 20 m + 19 , 683 1 , 000 x 10 + 82 , 944 512 x 8 + 157 , 464 2 , 197 x 13 + 531 , 441 4 , 096 x 16

S ( G ) = 883 , 949 , 568 274 , 4000 m n + 5 , 960 , 284 , 677 250 , 000 m + 1 , 765 , 118 , 848 , 000 , 000 4 , 607 , 442 , 944 , 000

D x D y ( D x + D y ) ( NM ( G ) ) = ( 1 , 536 x 4 y 8 + 2,048 x 8 y 8 ) m n + ( 1 , 330 x 2 y 5 + 3,780 x 3 y 7 + 1 , 584 x 3 y 8 + 324 x 3 y 9 + 3,080 x 4 y 7 + 11,520 x 4 y 8 + 5 , 148 x 4 y 9 + 2,240 x 4 y 10 + 19,800 x 4 y 11 + 2 , 688 x 6 y 8 + 9 , 604 x 7 y 7 + 2,600 x 5 y 8 + 12,320 x 5 y 11 + 6,384 x 7 y 12 + 12 , 288 x 8 y 8 + 25,704 x 8 y 9 + 43,472 x 8 y 11 + 3 , 360 x 7 y 8 + 6 , 048 x 7 y 9 + 11,900 x 7 y 10 + 2,880 x 8 y 10 + 3 , 420 x 9 y 10 + 4 , 000 x 10 y 10 + 5,760 x 10 y 8 + 810 x 9 y 6 + 15,840 x 9 y 11 + 2 , 380 x 10 y 7 + 10,560 x 10 y 12 + 55,902 x 11 y 11 ) m + 324 x 3 y 9 + 1,440 x 4 y 6 + 810 x 9 y 6 + 1,458 x 9 y 9 ND 3 ( G ) = 3 , 584 m n + 286 , 726 m + 4 , 032

( D x I y + I x D y ) ( NM ( G ) ) = ( 10 x 4 y 8 + 4 x 8 y 8 ) m n + 551 10 x 2 y 5 + 348 7 x 3 y 7 + 73 4 x 3 y 8 + 10 3 x 3 y 9 + 325 14 x 4 y 7 + 75 x 4 y 8 + 1 , 067 36 x 4 y 9 + 58 5 x 4 y 10 + 2 , 055 22 x 4 y 11 + 25 3 x 6 y 9 + 28 x 7 y 7 + 89 8 x 5 y 8 + 2 , 044 55 x 5 y 11 + 193 21 x 7 y 12 + 24 x 8 y 8 + 1 , 015 24 x 8 y 9 + 2 , 405 44 x 8 y 11 + 113 14 x 7 y 8 + 260 21 x 7 y 9 + 149 7 x 7 y 10 + 41 10 x 8 y 10 + 181 45 x 9 y 10 + 4 x 10 y 10 + 41 5 x 10 y 8 + 13 6 x 9 y 6 + 1 , 616 99 x 9 y 11 + 149 35 x 10 y 7 + 122 15 x 10 y 12 + 42 x 11 y 11 ) m + 10 3 x 3 y 9 + 13 x 4 y 6 + 13 6 x 9 y 6 + 2 x 9 y 9 ND 5 ( G ) = 14 m n + 4 , 913 , 119 6 , 930 m + 41 2

2 I x J ( NM ( G ) ) = 2 3 x 12 + 1 4 x 16 m n + 38 7 x 7 + 18 5 x 10 + 12 11 x 11 + 1 6 x 12 + 20 11 x 11 + 5 x 12 + 22 13 x 13 + 4 7 x 14 + 4 x 15 + 4 7 x 14 + 2 x 14 + 10 13 x 13 + 7 4 x 16 + 8 19 x 19 + 6 4 x 16 + 42 17 x 17 + 52 19 x 19 + 8 15 x 15 + 3 4 x 16 + 20 17 x 17 + 2 9 x 18

+ 4 19 x 19 + 1 5 x 20 + 4 9 x 18 + 2 15 x 15 + 4 5 x 20 + 4 17 x 17 + 4 11 x 22 + 21 11 x 22 m + 1 6 x 12 + 5 6 x 10 + 2 15 x 15 + 1 9 x 18 NH ( G ) = 11 12 m n + 137 , 624 , 243 323 , 3230 m + 29 18

I x J D x D y ( NM ( G ) ) = 128 12 x 12 + 128 16 x 16 m n + 190 7 x 7 + 378 10 x 10 + 144 11 x 11 + 27 12 x 12 + 280 11 x 11 + 960 12 x 12 + 396 13 x 13 + 160 14 x 14 + 88 x 15 + 96 7 x 14 + 49 x 14 + 200 13 x 13 + 385 8 x 16 + 336 19 x 19 + 48 x 16 + 1 , 512 17 x 17 + 2 , 288 19 x 19 + 224 15 x 15 + 189 8 x 16 + 700 17 x 17 + 80 9 x 18 + 180 19 x 19 + 200 20 x 20 + 160 9 x 18 + 54 15 x 15 + 198 5 x 20 + 140 17 x 17 + 240 11 x 22 + 231 2 x 22 m + 27 12 x 12 + 72 5 x 10 + 54 15 x 15 + 81 18 x 18 NI ( G ) = 56 3 m n + 3 , 335 , 165 , 313 3 , 233 , 230 m + 99 4

I x I y ( NM ( G ) ) = 1 8 x 4 y 8 + 1 32 x 8 y 8 m n + 19 10 x 2 y 5 + 6 7 x 3 y 7 + 1 4 x 3 y 8 + 1 27 x 3 y 9 + 5 14 x 4 y 7 + 15 16 x 4 y 8 + 11 36 x 4 y 9 + 1 10 x 4 y 10 + 15 22 x 4 y 11 + 1 12 x 6 y 8 + 2 7 x 7 y 7 + 1 8 x 5 y 8 + 14 55 x 5 y 11 + 1 21 x 7 y 12 + 3 16 x 8 y 8 + 7 24 x 8 y 9 + 13 44 x 8 y 11 + 1 14 x 7 y 8 + 2 21 x 7 y 9 + 1 7 x 7 y 10 + 1 40 x 8 y 10 + 1 145 x 9 y 10 + 1 50 x 10 y 10 + 1 20 x 10 y 8 + 1 54 x 9 y 6 + 8 99 x 9 y 11 + 1 35 x 10 y 7 + 1 30 x 10 y 12 + 21 121 x 11 y 11 m + 1 27 x 3 y 9 + 1 4 x 4 y 6 + 1 54 x 9 y 6 + 1 81 x 9 y 9 M 2 n m ⁎ ( G ) = 5 32 m n + 8 , 450 , 851 1 , 306 , 800 m + 103 324

D x α D y α ( NM ( G ) ) = ( 4 × 2 5 α x 4 y 8 + 2 × 2 6 α x 8 y 8 ) m n + ( 19 × 10 α x 2 y 5 + 18 × 21 α x 3 y 7 + 6 × 24 α x 3 y 8 + 10 × 28 α x 4 y 7 + 30 × 2 5 α x 4 y 8 + 27 α x 3 y 9 + 11 × 6 2 α x 4 y 9 + 4 × 40 α x 4 y 10 + 30 × 44 α x 4 y 11 + 4 × 48 α x 6 y 8 + 14 × 7 2 α x 7 y 7 + 5 × 40 α x 5 y 8 + 14 × 55 α x 5 y 11 + 4 × 84 α x 7 y 12 + 12 × 2 6 α x 8 y 8 + 21 × 72 α x 8 y 9

+ 26 × 88 α x 8 y 11 + 4 × 56 α x 7 y 8 + 6 × 63 α x 7 y 9 + 10 × 70 α x 7 y 10 + 2 × 80 α x 8 y 10 + 2 × 90 α x 9 y 10 + 2 × 10 2 α x 10 y 10 + 4 × 80 α x 10 y 8 + 54 α x 9 y 6 + 8 × 99 α x 9 y 11 + 2 × 70 α x 10 y 7 + 4 × 120 α x 10 y 12 + 21 × 11 2 α x 11 y 11 ) m + 3 3 α x 3 y 9 + 6 × 24 α x 4 y 6 + 54 α x 9 y 6 + 9 2 α x 9 y 9 NR α ( G ) = ( 4 × 2 5 α + 2 × 2 6 α ) m n + ( 19 × 10 α + 18 × 21 α + 6 × 24 α + 10 × 28 α + 30 × 2 5 α + 27 α + 11 × 6 2 α + 4 × 40 α + 30 × 44 α + 4 × 48 α + 14 × 7 2 α + 5 × 40 α + 14 × 55 α + 4 × 84 α + 12 × 2 6 α + 21 × 72 α + 26 × 88 α + 4 × 56 α + 6 × 63 α + 10 × 70 α + 2 × 80 α + 2 × 90 α + 2 × 10 2 α + 4 × 80 α + 54 α + 8 × 99 α + 2 × 70 α + 4 × 120 α + 21 × 11 2 α ) m + 3 3 α + 6 × 24 α + 54 α + 9 2 α

We have plotted the 3D graph of each of the neighborhood degree-based topological index. These plots are depicted in Figures 7–11. The 3D plots helps to understand the behavior of TIs. We have also computed the numerical values of these TIs for different values of m and n . These values are given in Tables 5 and 6. It is easy to observe that the value of each topological index increases with the increase in the value of m and n .

$Figure 7 3D plot of (a) M 1 ′ ( G ) {M}_{1}^{^{\prime} }(G) and (b) M 2 ⁎ ( G ) {M}_{2}^{\ast }(G) .$

Figure 7

3D plot of (a) M 1 ′ ( G ) and (b) M 2 ⁎ ( G ) .

$Figure 8 3D plot of (a) F N ⁎ ( G ) {F}_{N}^{\ast }(G) and (b) S ( G ) S(G) .$

Figure 8

3D plot of (a) F N ⁎ ( G ) and (b) S ( G ) .

$Figure 9 3D plot of (a) ND 3 ( G ) {\text{ND}}_{\text{3}}(G) and (b) ND 5 ( G ) {\text{ND}}_{\text{5}}(G) .$

Figure 9

3D plot of (a) ND 3 ( G ) and (b) ND 5 ( G ) .

$Figure 10 3D plot of (a) NH ( G ) \text{NH}(G) and (b) NI ( G ) \text{NI}(G) .$

Figure 10

3D plot of (a) NH ( G ) and (b) NI ( G ) .

$Figure 11 3D plot of (a) M 2 n m ⁎ ( G ) {}^{nm}M_{2}\ast (G) and (b) NR α ( G ) {\text{NR}}_{\alpha }(G) .$

Figure 11

3D plot of (a) M 2 n m ⁎ ( G ) and (b) NR α ( G ) .

Table 5

Numerical values of neighborhood degree-based TIs for different values of m and n .

[m, n]	M ₁′(G)	M ₂ ^⁎(G)	F _N ^⁎(G)	S(G)	ND₃(G)
[1,1]	4,626	17,338	39,052	24,546.37938	294,342
[2,2]	9,307	34,882	78,575	49,353.93526	591,820
[3,3]	14,148	52,938	119,250	74,805.76924	896,466
[4,4]	19,149	71,506	161,077	100,901.88134	1,208,280
[5,5]	24,310	90,586	204,056	127,642.27154	1,527,262
[6,6]	29,631	110,178	248,187	155,026.93986	1,853,412
[7,7]	35,112	130,282	293,470	183,055.88629	2,186,730
[8,8]	40,753	150,898	339,905	211,729.11083	2,527,216
[9,9]	46,554	172,026	387,492	241,046.61348	2,874,870
[10,10]	52,515	193,666	436,231	271,008.39424	3,229,692

Table 6

Numerical values of neighborhood degree-based TIs for different values of m and n .

[m, n]	ND₅(G)	NH(G)	NI(G)	ⁿ M ₂⁎(G)	NR_a(G)
[1,1]	743.46	45.093	1,074.9	6.9408	17,338
[2,2]	1,494.4	90.409	2,162.5	13.876	2.5193 × 10 06
[3,3]	2,273.4	137.56	3,287.3	21.124	3.2322 × 10 08
[4,4]	3,080.4	186.54	4,449.5	28.684	4.0893 × 10 10
[5,5]	3,915.3	237.36	5,649.1	36.557	5.2005 × 10 12
[6,6]	4,778.3	290.00	6,885.9	44.743	6.6621 × 10 14
[7,7]	5,669.2	344.49	8,160.1	53.241	8.5835 × 10 16
[8,8]	6,588.2	400.80	9,471.6	62.051	1.1099 × 10 19
[9,9]	7,535.2	458.95	10,820	71.174	1.4375 × 10 21
[10,10]	8,510.1	518.93	12,207	80.609	1.8620 × 10 23

3 Topological descriptors of different anticancer drugs

Cancer drugs, also known as anticancer or antineoplastic agents, are medications designed to treat various types of cancer by targeting and killing rapidly dividing cells. These drugs include chemotherapy agents like alkylating agents, antimetabolites, and mitotic inhibitors, as well as newer targeted therapies and immunotherapies. Chemotherapy disrupts cell division or damages DNA, while targeted therapies focus on specific molecular pathways driving cancer growth. Immunotherapies harness the immune system to fight cancer more effectively. Although highly effective, these treatments can cause side effects such as fatigue, nausea, and weakened immunity due to their impact on normal cells. Cancer drugs remain a cornerstone of modern oncology, often used in combination to maximize efficacy.

In this section, we check the chemical applicability of TIs. We have considered the eccentricity-based TIs to estimate the physical properties of 29 anticancer drugs. The eccentricity-based TIs that we have considered are eccentric connectivity index, total eccentricity index, average eccentricity index, eccentricity-based geometric–arithmetic index, eccentric ABC index, and first Zagreb eccentricity index. The mathematical formula of these TIs is presented in Table 7.

Table 7

Formula of eccentricity based TIs

Topological index	Mathematical formula
Eccentric connectivity index [40]	∑ v ∈ V d v ec ( v )
Total eccentricity index [41]	∑ v ∈ V ec ( v )
Average eccentricity index [42]	1 \| V \| ∑ v ∈ V ec ( v )
Eccentricity-based geometric–arithmetic index [43]	∑ v v ∈ E 2 ec ( u ) ec ( v ) ec ( u ) + ec ( v )
Eccentric ABC index [44]	∑ v v ∈ E ec ( u ) + ec ( v ) ‒ 2 ec ( u ) ec ( v )
First Zagreb eccentricity index [45]	∑ u v ∈ V ec ( v ) + ec ( u )

We use a linear regression model P = a T I + b to estimate the physicochemical properties of 29 anticancer drugs, where TIs are considered independent variables, and the physicochemical properties are considered dependent variables. The 29 anticancer drugs under consideration are as follows: altretamine, bendamustine, busulfan, carboquone, carmustine, chlorambucil, chlormethine, chlorozotocin, cyclophosphamide, dacarbazine, fotemustine, ifosfamide, lomustine, melphalan, melphalan flufenamide, mitobronitol, nimustine, nitrosoureas, pipobroman, ranimustine, semustine, streptozotocin, temozolomide, thiotepa, treosulfan, triaziquone, triethylenemelamine, and trofosfamide.

We estimate four physicochemical properties – molecular weight, complexity, molecular refractivity, and molar volume – for these anticancer drugs. The experimental values of these properties were obtained from ChemSpider. The chemical formulas, molecular structures, and physicochemical property values of these drugs are presented in Table 8.

Table 8

Chemical formula and values of physicochemical properties of anticancer drugs

Drug	Molecular formula	Molecular weight	Complexity	Molecular refractivity	Molar volume
Altretamine	C₉H₁₈N₆	210.28	148.0	2.7	48.4
Bendamustine	C₁₆H₂₁Cl₂N₃O₂	358.3	380.0	2.9	58.4
Busulfan	C₆H₁₄O₆S₂	246.3	293.0	−0.5	104.0
Carboquone	C₁₅H₁₉N₃O₅	321.33	643.0	−0.2	102.0
Carmustine	C₅H₉Cl₂N₃O₂	214.05	156.0	1.5	61.8
Chlorambucil	C₁₄H₁₉Cl₂NO₂	304.2	250.0	1.7	40.5
Chlormethine	C₅H₁₁Cl₂N	156.05	43.7	0.9	3.2
Chlorozotocin	C₉H₁₆ClN₃O₇	313.69	333.0	−2.6	160.0
Cyclophosphamide	C₇H₁₅Cl₂N₂O₂P	261.08	212.0	0.6	41.6
Dacarbazine	C₆H₁₀N₆O	182.18	215.0	−0.6	99.7
Fotemustine	C₉H₁₉ClN₃O₅P	315.69	334.0	0.9	97.3
Ifosfamide	C₇H₁₅Cl₂N₂O₂P	261.08	218.0	0.9	41.6
Lomustine	C₉H₁₆ClN₃O₂	233.69	219.0	2.8	61.8
Melphalan	C₁₃H₁₈Cl₂N₂O₂	305.2	265.0	−0.5	66.6
Melphalan flufenamide	C₂₄H₃₀Cl₂FN₃O₃	498.4	579.0	3.2	84.7
Mitobronitol	C₆H₁₂Br₂O₄	307.96	110.0	−0.3	80.9
Nimustine	C₉H₁₃ClN₆O₂	272.69	292.0	0.4	114.0
Nitrosoureas	CH₃N₃O₂	89.05	69.2	−0.8	84.6
Pipobroman	C₁₀H₁₆Br₂N₂O₂	356.05	227.0	0.4	40.6
Ranimustine	C₁₀H₁₈ClN₃O₇	327.72	362.0	−1.7	141.0
Semustine	C₁₀H₁₈ClN₃O₂	247.72	242.0	3.3	61.8
Streptozotocin	C₈H₁₅N₃O₇	265.22	315.0	−1.4	152.0
Temozolomide	C₆H₆N₆O₂	194.15	315.0	−1.1	106.0
Thiotepa	C₆H₁₂N₃PS	189.22	194.0	0.5	41.1
Treosulfan	C₆H₁₄O₈S₂	278.3	345.0	−2.2	144.0
Triaziquone	C₁₂H₁₃N₃O₂	231.25	494.0	−0.1	43.2
Triethylenemelamine	C₉H₁₂N₆	204.23	205.0	0.7	47.7
Trofosfamide	C₉H₁₈Cl₃N₂O₂P	323.6	265.0	1.8	32.8

To compute the eccentricity-based TIs, we developed a MAPLE-based algorithm, the pseudocode of which is provided in Algorithm 1 (see supplementary file). The values of the TIs for the 29 anticancer drugs were computed using this algorithm and are presented in Table 9. Linear regression models were developed using SPSS software, and regression parameters, namely, the correlation coefficient (R), R ², and adjusted R ², were calculated for each case. The values of these regression parameters for the physicochemical properties, including molecular weight, complexity, molecular refractivity, and molar volume, are presented in Tables 10–13.

Table 9

Values of eccentricity-based TIs

Medicine	Eccentric connectivity index	Total eccentricity index	Average eccentricity index	Eccentricity-based geometric–arithmetic index	Eccentric ABC index	First Zagreb eccentricity index
Altretamine	147.0	78.0	5.2	14.92	23.95	147.0
Bendamustine	460.0	230.0	10.0	23.97	42.84	460.0
Busulfan	190.0	106.0	7.57	12.97	22.36	190.0
Carboquone	356.0	169.0	7.35	24.94	42.88	356.0
Carmustine	129.0	74.0	6.17	10.96	18.19	129.0
Chlorambucil	343.0	178.0	9.37	18.97	33.66	343.0
Chlormethine	61.0	37.0	4.63	6.95	10.75	61.0
Chlorozotocin	312.0	171.0	8.55	18.97	33.31	312.0
Cyclophosphamide	149.0	78.0	5.57	13.94	22.69	149.0
Dacarbazine	143.0	75.0	5.77	12.95	21.22	143.0
Fotemustine	274.0	151.0	7.95	17.96	31.21	274.0
Ifosfamide	162.0	85.0	6.07	13.94	23.1	162.0
Lomustine	206.0	106.0	7.07	14.96	25.57	206.0
Melphalan	317.0	165.0	8.68	18.97	33.31	317.0
Melphalan flufenamide	932.0	461.0	13.97	33.97	62.92	932.0
Mitobronitol	114.0	66.0	5.5	10.95	17.73	114.0
Nimustine	301.0	155.0	8.61	17.97	31.64	301.0
Nitrosoureas	31.0	20.0	3.33	4.93	6.78	31.0
Pipobroman	254.0	132.0	8.25	15.97	27.88	254.0
Ranimustine	360.0	186.0	8.86	20.96	37.04	360.0
Semustine	240.0	124.0	7.75	15.96	27.68	240.0
Streptozotocin	262.0	137.0	7.61	17.95	30.99	262.0
Temozolomide	153.0	75.0	5.36	14.95	24.03	153.0
Thiotepa	86.0	38.0	3.45	12.86	18.06	86.0
Treosulfan	212.0	118.0	7.38	14.96	25.65	212.0
Triaziquone	217.0	94.0	5.53	19.93	32.51	217.0
Triethylenemelamine	183.0	78.0	5.2	17.92	28.95	183.0
Trofosfamide	201.0	106.0	6.24	16.93	28.18	201.0

Table 10

Regression parameters for molecular weight

Topological index	Correlation coefficient (R)	R ²	Adjusted R ²
Eccentric connectivity index	0.853	0.728	0.718
Total eccentricity index	0.876	0.767	0.749
Average eccentricity index	0.882	0.778	0.751
Eccentricity-based geometric–arithmetic index	0.911	0.831	0.802
ABC index	0.913	0.834	0.797
First Zagreb eccentricity index	0.916	0.839	0.796

Table 11

Regression parameters for complexity

Topological index	Correlation coefficient (R)	R ²	Adjusted R ²
Eccentric connectivity index	0.723	0.522	0.505
Total eccentricity index	0.770	0.593	0.561
Average eccentricity index	0.774	0.599	0.551
Eccentricity-based geometric–arithmetic index	0.878	0.772	0.734
ABC index	0.888	0.789	0.744
First Zagreb eccentricity index	0.891	0.795	0.739

Table 12

Regression parameters for molecular refractivity

Topological index	Correlation coefficient (R)	R ²	Adjusted R ²
Eccentric connectivity index	0.278	0.077	0.043
Total eccentricity index	0.332	0.110	0.042
Average eccentricity index	0.336	0.113	0.007
Eccentricity-based geometric–arithmetic index	0.417	0.174	0.036
ABC index	0.427	0.182	0.005
First Zagreb eccentricity index	0.461	0.212	−0.003

Table 13

Regression parameters for molar volume

Topological index	Correlation coefficient (R)	R ²	Adjusted R ²
Eccentric connectivity index	0.230	0.053	0.018
Total eccentricity index	0.444	0.197	0.135
Average eccentricity index	0.445	0.198	0.101
Eccentricity-based geometric–arithmetic index	0.539	0.290	0.172
ABC index	0.561	0.314	0.165
First Zagreb eccentricity index	0.588	0.346	0.168

4 Discussion

Linear regression is a statistical method used to model the relationship between one dependent variable and one or more independent variables. It predicts the dependent variable based on the values of the independent variables by fitting a straight line, known as the regression line, to the data points. The performance and quality of a linear regression model are often evaluated using metrics such as the correlation coefficient (R), R ², and adjusted R ². This value of R measures the strength and the direction of the linear relationship between the dependent and independent variables. It ranges from −1 to 1, where values close to 1 or −1 indicate a strong linear relationship (positive or negative), and values near 0 suggest no linear relationship. This metric R ² quantifies the proportion of variance in the dependent variable that is explained by the independent variables in the model. It ranges from 0 to 1, where higher values indicate a better fit of the model to the data. For example, an R ² value of 0.8 means 80% of the variability in the dependent variable is explained by the independent variables. While R ² increases with the addition of more independent variables, adjusted R ² accounts for the number of predictors and adjusts the value to prevent overfitting. It provides a more accurate measure of model quality by penalizing the inclusion of variables that do not significantly improve the model’s explanatory power. Together, these metrics help evaluate the effectiveness and reliability of a linear regression model, guiding analysts in building accurate and interpretable predictive models.

From Tables 10–13, it can be observed that the eccentricity-based TIs exhibit a strong correlation with the physicochemical properties, particularly molecular weight and complexity. For molecular weight, the correlation coefficient ( R ) is 0.916, the R 2 value is 0.839, and the adjusted R 2 value is 0.796 for the first Zagreb eccentricity index. This indicates that 83.9% of the variability in molecular weight is explained by the First Zagreb eccentricity index. Similarly, for complexity, the R value is 0.891, the R 2 value is 0.795, and the adjusted R 2 value is 0.739, meaning that 79.5% of the variability in complexity is accounted for by the first Zagreb eccentricity index. For the remaining two properties, the regression parameter values are not statistically significant. Therefore, these properties cannot be reliably estimated using the considered TIs.

5 Conclusion

This study focuses on the computation of M- and NM-polynomials for the camptothecin–polymer conjugate IT-101. From these polynomials, we derived various degree-based and neighborhood degree-based TIs for the molecule. The computation of these TIs offers valuable insights into the physical properties and chemical behavior of camptothecin–polymer conjugate IT-101. By utilizing these indices, it becomes possible to predict important molecular properties, thereby contributing to the drug design and development process.

In addition, we applied eccentricity-based TIs to estimate the physical properties of 29 anticancer drugs. The results demonstrate that molecular weight and complexity can be accurately predicted using the first Zagreb eccentricity index, with high correlation and predictive reliability. These findings underscore the utility of TIs as a tool for understanding molecular properties and optimizing drug discovery efforts.

Funding information: This work was supported by Researchers Supporting Project number (RSP2025R401), King Saud University, Riyadh, Saudi Arabia.
Author contributions: LT: Conceptualization, Methodology. LZ: Software, Validation. MMA: Data curation, Writing – Original draft preparation. MN: Visualization, Investigation. FT: Visualization, Writing – Reviewing and Editing. YW: Writing – Reviewing and Editing.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2023-07-07

Accepted: 2025-01-18

Published Online: 2025-03-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

Supplementary material

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https://doi.org/10.1515/mgmc-2023-0020

Schlagwörter für diesen Artikel

camptothecin–polymer conjugate; IT-101; M-polynomial; topological indices; NM-polynomial

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BY 4.0

M-polynomial and NM-polynomial indices of camptothecin–polymer conjugate IT-101 structure

Artikel

Abstract

1 Introduction

2 Topological indices computation of camptothecin–polymer conjugate (IT-101)

Theorem 1

Proof 1

Theorem 2

Proof 2

Theorem 3

Proof 3

Theorem 4

Proof 4

3 Topological descriptors of different anticancer drugs

4 Discussion

5 Conclusion

References

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