J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
221Ae241.
Calculating PI related indices and their polynomial of Hyaluronic Acid and Conjugates KANDAN1,2 .
SUBRAMANIAN2 and P.
RAJESH3
Assistant Professor.
PG and Research Department of Mathematics.
Government Arts College Chidambaram-608 102.
India.
/ kandan2k@gmail.
Department of Mathematics.
Annamalai University.
Annamalai Nagar- 608 002.
India.
/subramanian87@gmail.
Department of Physics.
School of Basic Sciences.
Vels Institute of Science Technology and Advance Studies.
Pallavaram.
Chennai-600 117.
India.
/rajesh.
ncc5oy@gmail.
Abstract.
A large number of medical experiments have confirmed that the features of drugs have a close correlation with their molecular structure.
Drug properties can be obtained by studying the molecular structure of corresponding drugs.
The calculation of the topological index of a chemical graph enables scientists to have a better understanding of the physical chemistry and biological characteristics of In this paper, we focus on Hyaluronic Acid-Curcumin/Paclitaxel conjugate which are widely used in the manufacture of anticancer drugs.
Therefore in this paper we have computed the Padmakar-Ivan related indices and its polynomial of the Hyaluronic Acid-curcumin/paclitaxel conjugate by using edge partitioning Key words and Phrases: Padmakar-Ivan index, weighted Padmakar-Ivan index.
Padmakar-Ivan related polynomial.
Hyaluronic Acid.
Hyaluronic AcidCurcumin/Paclitaxel conjugate.
Introduction and Preliminaries In a recent year, there is a rapid growth in pharmaceutical and chemical Biological activity, as well as physical, physicochemical, and chemical features of organic molecules, are all inherent in molecular structure, according to the primary paradigm of medical chemistry.
Crum-brown and Fraser pulished the first quantitative structure activity relationship in 1868 on the basis of this principle.
Despite many advances in the field of theoretical drug design, a large number of chemical experiments are needed to determine the pharmacological, chemical, and biological properties of these new compounds and drugs, which significantly increases the workload of pharmaceutical and chemical researchers.
The critical step 2020 Mathematics Subject Classification: 05-11, 05C05 Received: 25-04-2022, accepted: 31-07-2022.
KANDAN, et al in drug discovery remain the identification and optimization of lead compound in a rapid and cost effective way.
Computational tool techniques have advanced rapidly over the past decade and accordingly have played a major role in the development of a number of drugs now on the market or going through clinical trials.
These computational tool techniques are based on the use of a virtual world computer generated hypotheses, which are tested for practicality.
Developing structure-activity relationships for drug compounds using computational or theoretical methods, relies on appropriate representations of molecular Which predict the properties of drug without using any weight lab by analyzing the molecular structure of the relevant drug using a well-known tool of chemical graph theory known as topological index or molecular descriptor.
Topological indices are simply defined as numerical values associated with chemical graph which is used for correlation of molecular structure with numerous chacteristics such as chemical reactivity, pharmacological activity, and physical properties.
Topological indices play a vital role in quantitative structure property relationship (QSPR) and the quantitative structure-activity relationship (QSAR) in predicting different physicochemical properties and bioactivity that contribute in the discovery of drugs.
, .
In order to calculate topological indices, the structure of the drugs is represented as a graph known as chemical graph, where each vertex indicate an atom and each edge represents a chemical bond between the atoms.
Let G = (V.
E) be a molecular graph with vertex set V = V (G) and edge set E = E(G).
The number of vertices and edges in a graph G can be denoted by |V (G)| and |E(G)| respectively.
The degree of vertex uV (G) is denoted by degeG .
and is the number of vertices that are adjacent to u.
The edge eE(G) connecting the vertices u and v is denoted by e = uv.
We now recall the following quantities based on distances which will give the number of vertices, closer to the end vertices of e = uv by e .
= .
V (G) : dG .
, .
< dG .
, .
|, ne .
= .
V (G) : dG .
, .
< dG .
, .
|, where dG .
, .
is the distance between the vertex u and v.
That is nG e .
denote the number of vertices that are lying closer to the vertex u than the vertex v of e = uv.
Analogously, nG e .
denote the number of vertices that are lying closer to the vertex v than the vertex u of e = uv.
Vertex equidistant from both ends of the edge are not counted.
The concept of P topological index was first introduced by Wiener 1947 .
defined by W (G) = d.
, .
OIV (G) while working on the boiling points of alkanes.
Topological indices are classified into many classes such as degree-based, distance-based, and spectrum-based, etc.
Out of these distance based topological indices plays a vital role in the drug designs and theoretical chemistry.
Some important distance based indices are PadmakarIvan index.
Szeged index.
Mostar index and some degree-distance based indices are weighted Padmakar-Ivan index, weighted Szeged index, weighted Mostar index.
The Padmakar-Ivan index (PI) is one of the important topological indices widely applied to study the proertics of molecular structure, introduced by Khadikar et Calculating PI related indices and their polynomial of Hyaluronic Acid and Conjugates .
Padmakar-Ivan index defined as P I(G) = .
G e .
) e=uvE(G) The Padmakar-Ivan index is one of the highly correlated index with Wiener index and found many chemical applications see .
, .
Inspired by the extension of the Wiener index and in order to increase the bibartivity of graphs, illicU and Milosavljevic, .
, proposed the modifications in the vertex-PI index as the weighted version of vertex-PI index, which are defined as the AdditivelyP weighted vertex PI index is P IA (G) = .
egeG .
degeG .
G e .
) and e=uvE(G) MultiplicativelyPweighted vertex-P I index is P IM (G) = .
egeG .
degeG .
G e .
) e=uvE(G) Recently the weighted vertex-P I index of the tensor and strong products of graphs have been studied in .
Extensively more applications on weighted version of PI index has been studied, see(.
, 28, .
Numerous graph polynomials have been developed for measuring structural information of molecular graphs.
Graph polynomials found applications in chemistry in connection with the molecular orbital theory of unsaturated compounds and also an important source of structural descriptors used in developing structure property models .
, 14, .
Distance-based and degree-based graph polynomials are useful because they contain a wealth of information about topological indices, vertex-P I polynomial of a graph G is.
P I(G, .
= x.
) .
e=uvE(G) Various topological indices can be derived from polynomials by taking their value at some point directly, or by taking integrals or derivatives.
For instance, the first derivative of the vertex PI polynomial evaluated at x = 1 equals to the PI index.
In light of the preceding condition and the newly proposed weighted version of the vertex-PI polynomial, motivates us to define the Additively weighted vertex-PI polynomial of a graphs G is P IA (G, .
= x.
) e=uvE(G) the Multiplicatively weighted vertex-P I polynomal of a grpah G is P IM (G, .
= x.
) e=uvE(G) The computation of topological indices for special molecular drug structures has raised considerable interest among medical and pharmaceutical researchers as it is useful to make up the medicinal and chemical experimental defects.
Recently Mondal et al.
several topological indices of COVID-19 based drugs such as remdesivir, chloroquine, hydroxychloroquine and theaflavin have been investigated.
Wang et al.
, .
computed topological indices of Hyaluronic Acid, which is thoroughly investigated in cancer therapy because of its excellent physical features.
Moreover Ali et al.
, .
, .
have theoretically analyzed and obtained several degree-based topological indices and indeg polynomial of Hyaluronic Acid-curcumin KANDAN, et al Likewise.
Zheng et al.
have determined many degree-based topological indices and polynomials for the Hyaluronic Acid-Paclitaxel conjugate, widely used in the therapeutics of cancer.
Motivation Many pharmaceutical enterprises have made great efforts to develop many drugs to treat heart disease one such is Hyaluronic Acid(HA).
Hyaluronic Acid is contributes most important to chemical graph multiplication and relocation, which is nontoxic, bio-compatible, biodegradable, and nonimmunogenic.
Hyaluronic Acid (HA) is frequently used in anticancer drug delivery, visco-supplementation theraphy, eye surgery, corneal transplantation and even more medical fields.
Since ancient times, curcumin is known to be endowed with remarkable medicinal properties against a range of ailing conditions such as inflammation, cancer, diabetes, neurodegenerative disorders, cardiovascular diseases and asthma, even curcumin very tolerated in human body, due to poor aqueous-solubility of curcumin leads to inadequate bioavailability in biological system.
To overcome this conjugates of curcumin with Hyaluronic Acid(HA) has received appreciable attention in recent times not only for increasing bioavailability but also to target tumor cells and tumor metastases for the treatment of numerous cancer.
Therefore, conjugates of curcumin with HA is regarded as promising pharmaceutical strategy to improve water solubility, prolong curcumin release at the target site, improve tissue distribution and potentiate therapeutic outcomes.
, .
, computed many degree-based topological indices and polynomials.
Paclitaxel is another compound pharamatical departments which is unique antiproliferative mechanism makes it an efficient anticancer drug .
, 34, 35, 36, .
It is an important medication that is prescribed in various forms of cancer in spite of its limitations, such as low solubility and relevant adverse effects.
Galer et .
, 42, 41, 43, .
synthesized the HA-Paclitaxel conjugate (HA-PTX), in order to reduce the toxicity of taxanes and improve the antitumor activity.
Therefore, different antitumor drugs can be covalently bonded to HA, forming HAdrugs conjugates.
Owing to having pharmaceutical to developed of anticancer durgs many topological indices has been studied.
For such as molecular structure, from mathematical perspective.
Recently many articles has investigates degree based topological indices and its polynomial of HA and HA-based conjugates see .
, 3, 4, 5, 6, 8, 9, 15, 17, 18, 20, .
Graph polynomials have many applications and is an important source of structural descriptors see .
Since the distance based indices has been shown to have greater predictive ability, the current research on Hyaluronic Acid.
Hyaluronic Acid-Curcumin/Paclitaxel conjugates.
Computing distance based indices to the family of graph is more complicated then computing degree based indices, in this work we use the edge partition technique to calculate the distance based indices to the Hyaluronic Acid based molecular graphs.
Calculating PI related indices and their polynomial of Hyaluronic Acid and Conjugates Figure 1.
The molecular structure of Hyaluronic Acid HAd .
Figure 2.
The chemical graph of Hyaluronic HA2 .
Main Result and Proof Hyaluronic Acid.
In this section, we computing various bond additive topological indices like the Padmakar-Ivan index.
Additively Padmakar-Ivan index and Multiplicatively Padmakar-Ivan index and its polynomial of Hyaluronic Acid.
Figure 1.
illustrate the molecular structure of Hyaluronic Acid.
Let H denote the chemical graph of Hyaluronic Acid HAd with the linear iteration d units.
Figure represented the chemical graph of Hyaluronic Acid for d = 2 units.
For our convenience here we assume H[Hi ] be the induced sub graph of H, where Hi is the ith units of H.
In order to obtain the results, let start our discussion with partitioning the edge set of H.
By chemical graph structure analysis on the basic of degree of vertices, the edge set of H can be divided into five groups which are summarized as follow:
E1 (H) = e = uvE(H) : degeH .
= 1, degeH .
= 2 , |E1 (H)| = d E2 (H) = e = uvE(H) : degeH .
= 1, degeH .
= 3 , |E2 (H)| = 7d 1 E3 (H) = e = uvE(H) : degeH .
= 2, degeH .
= 3 , |E3 (H)| = 11d Oe 2 E4 (H) = e = uvE(H) : degeH .
= 3, degeH .
= 3 , |E4 (H)| = 9d Oe 1 and E5 (H) = e = uvE(H) : degeH .
= 2, degeH .
= 2 , |E5 (H)| = 1.
So that the order of vertex set and edge set of Hyaluronic Acid is |V (H)| = 26d and |E(H)| = 28d Oe 1.
The main results discussed in this section are as follows:
Theorem 3.
The Padmakar-Ivan index of Hyaluronic Acid HAd is given by P I(HAd ) = .
d Oe .
Proof.
By graph structure analysis to the Hyaluronic Acid and applying the definition of Padmakar-Ivan index, we get P I(HAd ) = .
HA .
nHA .
) e=uvE(HAd ) .
H e .
) i=1 e=uvE(Hi ) dOe1 .
H e .
), e=xyE(Hi OHi 1 ) where Hi O Hi 1 is the link of Hi and Hi 1 dOe1 P I(Hi ) (|V (H)| Oe |V (Hi )|) |V (H)| i=1 e=uvE(Hi ) KANDAN, et al P I(Hi ) |E(Hi )|(|V (H)| Oe |V (Hi )|) .
Oe .
|V (H)| Since P I(Hi ) = |E(Hi )||V (Hi )| is true for every i, so we have P I(HAd ) = |E(Hi )||V (Hi )| |E(Hi )|(|V (H)| Oe |V (Hi )|) .
Oe .
|V (H)| |E(Hi )||V (Hi )| |E(Hi )||V (H)| Oe |E(Hi )||V (Hi )| .
Oe .
|V (H)| |E(Hi )||V (H)| .
Oe .
|V (H)|, since |E(Hi )| = |E(H)| Oe .
Oe .
= (|E(H)| Oe .
Oe .
)|V (H)| .
Oe .
|V (H)| = |V (H)||E(H)| = .
d Oe .
26d, since |V (H)| = 26d Hence the theorem.
Theorem 3.
The Additively weighted Padmakar-Ivan index of Hyaluronic Acid HAd is given by P IA (HAd ) = .
d Oe .
Proof.
From the graph structure of Hyaluronic Acid and by the definition of Additively weighted Padmakar-Ivan index, we get P IA (HAd ) = .
egeHAd .
degeHAd .
HA .
nHA .
) e=uvE(HAd ) e=uvE1 (H) e=uvE2 (H) e=uvE3 (H) e=uvE4 (H) e=uvE5 (H) .
egeH .
) P dege .
H .
) e .
) e=uvE2 (H) e=uvE1 (H) .
H .
H e .
) e .
) e=uvE4 (H) e=uvE3 (H) .
H e .
) e=uvE4 (H) = d.
|V (H)| .
|V (H)| .
d Oe .
|V (H)| .
d Oe .
|V (H)| .
|V (H)| = 3.
V (H)| 4.
|V (H)| 5.
d Oe .
|V (H)| 6.
d Oe .
|V (H)| .
V (H)| = .
d 28d 4 4 55d Oe 10 54d Oe .
|V (H)|, since |V (H)| = 26d = .
d Oe .
26d Hence the theorem.
Theorem 3.
The Multiplicatively weighted Padmakar-Ivan index of Hyaluronic Acid HAd is given by P IM (HAd ) = .
d Oe .
Proof.
By graph structure analysis to the Hyaluronic Acid and from the definition of Multiplicatively weighted Padmakar-Ivan index, we get .
) .
nHA P IM (HAd ) = .
eguHAd .
degeHAd .
HA e=uvE(HAd ) e=uvE1 (H) e=uvE2 (H) e=uvE3 (H) e=uvE4 (H) ).
eguH .
degvH .
H .
) e=uvE5 (H) Calculating PI related indices and their polynomial of Hyaluronic Acid and Conjugates .
H e .
) e=uvE1 (H) e=uvE2 (H) e=uvE3 (H) .
H e .
) e=uvE4 (H) .
H e .
) .
H e .
) .
H e .
) e=uvE5 (H) = d.
|V (H)| .
|V (H)| .
d Oe .
|V (H)| .
d Oe .
|V (H)| .
|V (H)| = .
d 21d 3 4 66d Oe 12 81d Oe .
|V (H)| = .
d Oe .
26d, since |V (H)| = 26d Hence the theorem.
Theorem 3.
The Padmakar-Ivan polynomial of Hyaluronic Acid HAd is given by P I(HAd , .
= .
d Oe .
Proof.
From the graph structure analysis to the Hyaluronic Acid and applying the definition of Padmakar-Ivan ploynomial, we get HAd HAd P I(HAd , .
= x.
) e=uvE(HAd ) dOe1 ) i=1 e=uvE(Hi ) x.
) , where Hi O Hi 1
e=xyE(Hi OHi 1 ) denotes the link of graphs dOe1
P |V (H)|
|E(Hi ).
V (H)| 27.
V (H)| .
Oe .
V (H)| = .
Oe .
V (H)| = .
d Oe .
x26d , since |V (H)| = 26d Hence the theorem.
Theorem 3.
The Additively weighted Padmakar-Ivan polynomial Hyaluronic Acid HAd is given by P IA (HAd , .
= dx78d .
d Oe .
d Oe .
Proof.
From the graph structure by the Hyaluronic Acid and by the definition of Additively weighted Padmakar-Ivan polynomial, we have HAd HAd HAd HAd P IA (HAd , .
= x.
) e=uvE(HAd ) e=uvE1 (H) e=uvE2 (H) e=uvE3 (H) e=uvE1 (H) x.
) e=uvE3 (H) e=uvE5 (H) e=uvE4 (H) e=uvE5 (H) e .
) x.
deg x.
) e=uvE2 (H) .
H e .
) e=uvE4 (H) x.
) x.
) KANDAN, et al = dx.
V (H)| .
V (H)| .
d Oe .
V (H) | .
d Oe .
V (H)| .
V (H)| = dx.
V (H)| .
V (H)| .
d Oe .
V (H)| .
d Oe .
V (H)| = dx3.
d Oe .
d Oe .
= dx78d .
d Oe .
d Oe .
x156d , since |V (H)| = 26d Hence the theorem.
Theorem 3.
The Multiplicatively weighted Padmakar- Ivan polynomial of Hyaluronic Acid HAd is given by P IM (HAd , .
= dx52d .
d Oe .
d Oe .
x234d x104d .
Proof.
By graph structure analysis to the Hyaluronic Acid and applying the definition of Multiplicatively weighted Padmakar-Ivan polynomial, we have HAd HAd HAd HAd P IM (HAd , .
= x.
) e=uvE(HAd ) e=uvE1 (H) e=uvE2 (H) e=uvE3 (H) x.
) e=uvE4 (H) )x.
) e=uvE5 (H) e=uvE1 (H) e=uvE2 (H) .
H e .
) e=uvE3 (H) x.
) x.
) e=uvE4 (H) .
H e .
) e=uvE5 (H) .
V (H)| = dx .
V (H)| .
d Oe .
V (H)| .
d Oe .
V (H)| x.
V (H)| 2.
= dx .
d Oe .
d Oe .
= dx .
d Oe .
d Oe .
x234d x104d ,since |V (H)| = 26d Hence the theorem.
Note that similar work had been done for computing spectrum based indices like Estrada indices see.
The following corollaries shows the relationship between Padmakar-Ivan index.
Additively Padmakar-Ivan index and Multiplicatively Padmakar-Ivan index for Hyaluronic Acid.
Corollary 3.
P IA (H) = 5P I(H) Oe .
V (H)|.
Corollary 3.
P IM (H) = 6P I(H) .
d Oe .
|V (H)|.
Corollary 3.
P I(H) < P IA (H) < P IM (H).
In this connection one may generalize the above corollaries for a families of Hyaluronic Acid-Curcumin conjugates In this section, we computing various bond additive topological indices like the Padmakar-Ivan index.
Additively Padmakar-Ivan index and Multiplicatively Padmakar-Ivan index and its polynomial of Hyaluronic Acid-Curcumin conjugate.
Calculating PI related indices and their polynomial of Hyaluronic Acid and Conjugates Figure 3.
illustrate the molecular structure of Hyaluronic Acid-Curcumin conjugate.
Let J denote the chemical graph of Hyaluronic Acid-Curcumin conjugates HACd with the linear iteration d units.
Figure 4.
represented the chemical graph of Hyaluronic-Acid Curcumin conjugates for d = 1 unit.
For our convenience here we assume J[Ji ] be the induced sub-graph of J, where Ji is the ith units of J.
that the order of vertex set and edge set of Hyaluronic Acid is |V (H)| = 26d and |E(H)| = 28d Oe 1.
The main results discussed in this section are as follows:
In order to obtain the results, let start our discussion with partitioning the edge set of J.
By chemical graph structure analysis on the basic of degree of vertices the edge sets of J can be divided into five groups which are summarized as follows:
E1 (J) = e = uvE(J) : degeJ .
= 1, degeJ .
= 2 , |E1 (J)| = 3d E2 (J) = e = uvE(J) : degeJ .
= 1, degeJ .
= 3 , |E2 (J)| = 9d 1 E3 (J) = e = uvE(J) : degeJ .
= 2, degeJ .
= 2 , |E3 (J)| = 4d 1 E4 (J) = e = uvE(J) : degeJ .
= 2, degeJ .
= 3 , |E4 (J)| = 29d Oe 2 and E5 (J) = e = uvE(J) : degeJ .
= 3, degeJ .
= 3 , |E5 (J)| = 11d Oe 1.
So that the order of vertex set and edge set of Hyaluronic Acid-Curcumin conjugate is |V (J)| = 52d and |E(J)| = 56d Oe 1.
The main results discussed in this section are as follows:
Theorem 4.
The Padmakar-Ivan index of Hyaluronic Acid-Curcumin conjugate HACd is given by P I(HACd ) = .
d Oe .
Proof.
From the graph structure of the Hyaluronic Acid-Curcumin conjugate and by the definition of P Padmakar-Ivan index, we have P I(HACd ) = .
eHACd .
nHAC .
) e=uvE(HACd ) dOe1 .
Je .
nJe .
) .
Je .
nJe .
), e=xyE(Ji OJi 1 ) i=1 e=uvE(Ji ) where Ji O Ji 1 is the link of Ji and Ji 1 (|V (J)| Oe |V (Ji )|) .
Oe .
|V (J)| P I(Ji ) P I(Ji ) i=1 eE(Ji ) |E(Ji )|(|V (J)| Oe |V (Ji )|) .
Oe .
|V (J)| Since P I(Ji ) = |E(Ji )||V (Ji )| is true for every i, so we have P I(HACd ) = |E(Ji )||V (Ji )| |E(Ji )|(|V (J)| Oe |V (Ji )|) .
Oe .
|V (J)| (|E(Ji )||V (Ji )| |E(Ji )||V (J)| Oe |E(Ji )||V (Ji )) .
Oe .
|V (J)| = (|E(J)| Oe .
Oe .
)|V (J)| .
Oe .
|V (J)| = |V (J)||E(J)| = .
d Oe .
52d Hence the theorem.
KANDAN, et al Figure 3.
The molecular structure of Hyaluronic Acid-Curcumin conjugates HACd .
Figure 4.
The chemical graph of Hyaluronic Acid-Curcumin conjugates HAC1 .
Theorem 4.
The Additively weighted Padmakar-Ivan index of Hyaluronic AcidCurcumin conjugate HACd is given by P IA (HACd ) = .
d Oe .
Proof.
By graph structure analysis to the Hyaluronic Acid-Curcumin conjugate and applying the definition of Additively weighted Padmakar-Ivan index, we have .
nHAC .
P IA (HACd ) = .
egeHACd .
degeHACd .
HAC e=uvE(HACd ) e=uvE1 (J) e=uvE2 (J) e=uvE3 (J) e=uvE4 (J) ).
egeJ .
degeJ .
Je .
nJe .
) e=uvE5 (J) .
Je .
nJe .
) .
Je .
nJe .
) e=uvE1 (J) e=uvE2 (J) .
Je .
nJe .
) .
Je .
nJe .
) e=uvE4 (J) e=uvE3 (J) .
Je .
nJe .
) e=uvE5 (J) = 3d.
|V (J)| .
|V (J)| .
|V (J)| .
d Oe .
|V (J)| .
d Oe .
|V (J)| = .
d 36d 4 16d 4 145d Oe 10 66d Oe .
|V (J)| as |V (J)| = 52d = .
d Oe .
52d Hence the theorem.
Theorem 4.
The Multiplicatively weighted Padmakar-Ivan index of Hyaluronic Acid-Curcumin conjugate HACd is given by P IM (HACd ) = .
d Oe .
Proof.
From the graph structure of Hyaluronic Acid- Curcumin conjugate and from the definition of Multiplicatively weighted Padmakar-Ivan index, we have .
nHAC .
P IM (HACd ) = .
egeHACd .
degeHACd .
HAC e=uvE(HACd ) e=uvE1 (J) e=uvE2 (J) e=uvE3 (J) e=uvE4 (J) ).
egeJ .
degeJ .
Je .
nJe .
) e=uvE5 (J) .
Je .
nJe .
) .
Je .
nJe .
) e=uvE1 (J) e=uvE2 (J) Calculating PI related indices and their polynomial of Hyaluronic Acid and Conjugates .
Je .
nJe .
) e=uvE3 (J) .
Je .
nJe .
) e=uvE4 (J) .
Je .
nJe .
) e=uvE5 (J) = .
d Oe .
d Oe .
) |V (J)| = |V (J)|.
d 27d 3 16d 4 174d Oe 12 99d Oe .
, since |V (G)| = 52d = .
d Oe .
52d Hence the theorem.
Theorem 4.
The Padmakar-Ivan polynomial of Hyaluronic Acid-Curcumin conjugate (HACd ) is given by P I(HACd , .
= .
d Oe .
x52d Proof.
By graph structure analysis to the Hyaluronic Acid-Curcumin conjugate and applying the definition of Padmakar-Ivan polynomial, we have HACd HACd .
) P I(HACd , .
= x.
e e=uvE(HACd ) dOe1 x.
) i=1 e=uvE(Ji ) x.
) , where Ji O Ji 1 de- e=uvE(Ji OJi 1 ) noted the link of graphs |E(Ji ).
V (J)| .
Oe .
V (J)| = 55d.
V (J)| .
Oe .
V (J)| = .
d d Oe .
V (J)| , since |V (J)| = 52d = .
d Oe .
x52d Hence the theorem.
Theorem 4.
The Additively weighted Padmakar-Ivan polynomial of Hyaluronic Acid-Curcumin conjugate.
HACd is given by P IA (HACd , .
= 3dx156d .
d Oe .
d Oe .
Proof.
From the graph structure of the Hyaluronic Acid-Curcumin conjugate and by the definition of Additively weighted Padmakar-Ivan polynomial, we have HACd HACd HACd HACd .
) P IA (HACd , .
= x.
ege e=uvE(HACd ) e=uvE1 (J) e=uvE2 (J) e=uvE3 (J) e=uvE4 (J) )x.
) e=uvE5 (J) x.
) e=uvE1 (J) e=uvE5 (J) .
V (J)| x.
) e=uvE2 (J) .
J e .
) e=uvE3 (J) x.
) e=uvE4 (J) .
J e .
) = 3dx .
V (J)| .
V (J)| .
dOe.
V (J)| .
dOe.
V (J)| 3.
= 3dx .
dOe.
dOe.
since |V (J)| = 52d KANDAN, et al = 3dx156d .
d Oe .
d Oe .
x312d Hence the theorem.
Theorem 4.
The Multiplicatively weighted Padmakar-Ivan polynomial of Hyaluronic Acid-Curcumin conjugate HACd , is given by P IM (HACd , .
= 3dx104d .
d Oe .
d Oe .
Proof.
By graph structure analysis to Hyaluronic Acid-Curcumin conjugate and applying the definition of Multiplicatively weighted Padmakar-Ivan index polynomial, we have HACd HACd HACd HACd .
) P IM (HACd , .
= x.
ege e=uvE(HACd ) e=uvE1 (J) e=uvE2 (J) e=uvE3 (J) e=uvE4 (J) )x.
) e=uvE5 (J) x.
) e=uvE1 (J) x.
) e=uvE2 (J) .
J e .
) e=uvE3 (J) x.
) e=uvE4 (J) x.
) e=uvE5 (J) = 3dx.
V (J)| .
V (J)| .
V (J)| .
dOe.
V (J)| .
dOe.
V (J)| = 3dx2.
d Oe .
d Oe .
since |V (J)| = 52d = 3dx104d .
d Oe .
d Oe .
x468d Hence the theorem.
The following corollaries shows the relationship between Padmakar-Ivan index.
Additively Padmakar-Ivan index and Multiplicatively Padmakar-Ivan index for Hyaluronic Acid-Curcumin conjugate.
Corollary 4.
P IA (J) = 5P I(J) Oe .
|V (J)|.
Corollary 4.
P IM (J) = 5P I(J) Oe .
d Oe .
|V (J)|.
Corollary 4.
P I(J) < P IA (J) < P IM (J).
Hyaluronic Acid-Paclitaxel conjugates In this section, we calculate the exact value of the some bond additive topological indices and its polynomial as discussed in the previous sections for the Hyaluronic Acid-Paclitaxel Conjugates Figure 5.
illustrate the molecular structure of Hyaluronic Acid-Paclitaxel conjugates.
Let K denote the chemical graph of Hyaluronic Acid-Paclitaxel conjugates HAPd with the linear iteration d units.
Figure 6.
represented the chemical graph of Hyaluronic Acid-Paclitaxel conjugates d = 1 unit.
For our convenience here we assume K[Ki ] be the induced sub-graph of K, where Ki is the ith units of K.
In order to obtain the results, let start our discussion with partitioning the edge set K.
By chemical graph structure analysis Calculating PI related indices and their polynomial of Hyaluronic Acid and Conjugates Figure 5.
The molecular structure of Hyaluronic Acid-Paclitaxel conjugates HAPd.
Figure 6.
The chemical graph of Hyaluronic Acid-Paclitaxel conjugates HAP1 .
on the basic of degree of vertices the edge set of K can be divided into five groups which aresummarized as follows:
E1 (K) = e = uvE(K) : degeK .
= 1, degeK .
= 2 , |E1 (K)| = d E2 (K) = e = uvE(K) : degeK .
= 1, degeK .
= 3 , |E2 (K)| = 16d 1 E3 (K) = e = uvE(K) : degeK .
= 2, degeK .
= 2 , |E3 (K)| = 13d 1 E4 (K) = e = uvE(K) : degeK .
= 1, degeK .
= 4 , |E4 (K)| = 4d E5 (K) = e = uvE(K) : degeK .
= 2, degeK .
= 3 , |E5 (K)| = 32d Oe 2 E6 (K) = e = uvE(K) : degeK .
= 2, degeK .
= 4 , |E6 (K)| = 3d E7 (K) = e = uvE(K) : degeK .
= 3, degeK .
= 3 , |E7 (K)| = 19d Oe 1 E8 (K) = e = uvE(K) : degeK .
= 3, degeK .
= 4 , |E8 (K)| = 7d and E9 (K) = e = uvE(K) : degeK .
= 4, degeK .
= 4 , |E9 (K)| = d.
So that the order of vertex set and edge set of Hyaluronic Acid-Paclitaxel conjugates is |V (K)| = 87d and |E(K)| = 96d Oe 1.
The main results discussion in this section are as follows:
Theorem 5.
The Padmakar-Ivan index of Hyaluronic Acid-Paclitaxel conjugate HAPd is given by P I(HAPd ) = .
d Oe .
Proof.
From the graph structure of Hyaluronic Acid-Paclitaxel conjugate and by the definition of Padmakar-Ivan index we get P I(HAPd ) = .
HAP .
nHAP .
) e=uvE(HAPd ) .
K e .
) i=1 e=uvE(Ki ) dOe1 .
K e .
), e=xyE(Ki OKi 1 ) where Ki O Ki 1 is the link of Ki and Ki 1 P I(Ki ) (|V (K)| Oe |V (Ki )|) .
Oe .
|V (K)| i=1 e=uvE(Ki ) KANDAN, et al Since P I(Ki ) = |E(Ki )||V (Ki )| is true for every i, so we have P I(HAPd ) = |E(Ki )||V (Ki )| |E(Ki )|(|V (K)| Oe |V (Ki )|) .
Oe .
|V (K)| (|E(Ki )||V (Ki )| |E(Ki )||V (K)| Oe |E(Ki )||V (Ki )|) .
Oe .
|V (K)| |E(Ki )|(|V (K)|) .
Oe .
|V (K)| = (|E(K)| Oe .
Oe .
)|V (K)| .
Oe .
|V (K)| = |V (K)||E(K)| = .
d Oe .
87d Hence the theorem.
Another interesting degree based descriptor called Sombor index has been investigated for some extremal graphs .
and the above result may extend for extremal graphs.
Theorem 5.
The Additively weighted Padmakar-Ivan index of Hyaluronic AcidPaclitaxel conjugate (HAPd ) is given by P IA (HAPd ) = .
d Oe .
Proof.
By graph structure analysis to the Hyaluronic Acid-Paclitaxel conjugate and from the definition ofPAdditively weighted Padmakar-Ivan index, we have .
nHAP .
) P IA (HAPd ) = .
egeHAPd .
degeHAPd .
HAP e=uvE(HAPd ) e=uvE1 (K) e=uvE2 (K) e=uvE3 (K) e=uvE4 (K) e=uvE5 (K) e=uvE6 (K) .
egeK .
degeK .
K e .
) e=uvE7 (K) e=uvE8 (K) e=uvE9 (K) .
K .
K e .
) e .
) e=uvE1 (K) e=uvE2 (K) .
K .
K e .
) e .
) e=uvE4 (K) e=uvE3 (K) .
K .
K e .
) e .
) e=uvE6 (K) e=uvE5 (K) .
K .
K e .
) e .
) e=uvE7 (K) e=uvE8 (K) .
K e .
) e=uvE9 (K) = 3.
V (K)| 4.
|V (K)| 4.
|V (K)| 5.
|V (K)| 5.
dOe.
|V (K)| 6.
|V (K)| 6.
d Oe .
|V (K)| 7.
|V (K)| 8.
V (K)| = .
d 64d 4 52d 4 20d 150d 10d Oe 10 18d 114d Oe 6 49d 8.
|V (K)| = .
d Oe .
Since |V (K)| = 87d Hence the theorem.
Theorem 5.
The Multiplicatively weighted Padmakar-Ivan index of the Hyaluronic Acid-Paclitaxel conjugate HAPd is given by P IM (HAPd ) = .
d Oe .
Proof.
By graph structure analysis to the Hyaluronic Acid-Paclitaxel conjugate and applying the definition of Multiplicatively weighted Padmakar-Ivan index, we get Calculating PI related indices and their polynomial of Hyaluronic Acid and Conjugates .
egeHAPd .
degeHAPd .
HAP .
nHAP .
e=uvE(HAPd ) P IM (HAPd ) = ( e=uvE1 (K) e=uvE2 (K) e=uvE3 (K) e=uvE4 (K) e=uvE5 (K) ).
egeK .
degeK .
K e .
) e=uvE6 (K) e=uvE7 (K) e=uvE8 (K) e=uvE9 (K) .
K .
K e .
) e .
) e=uvE1 (K) e=uvE2 (K) .
K .
K e .
) e .
) e=uvE3 (K) e=uvE4 (K) .
K .
K e .
) e .
) e=uvE5 (K) e=uvE6 (K) .
K .
K e .
) e .
) e=uvE7 (K) e=uvE8 (K) .
K e .
) P IM (HAPd ) = e=uvE9 (K) = 2.
V (K)| 3.
|V (K)| 4.
|V (K)| 4.
|V (K)| 6.
dOe.
|V (K)| 8.
|V (K)| 9.
d Oe .
|V (K)| 12.
|V (K)| 16.
V (K)| = .
d 48d 3 52d 4 16d 180d 12d Oe 12 24d 171d Oe 9 84d 16.
|V (K)| = .
d Oe .
87d, since |V (K)| = 87d Hence the theorem.
Theorem 5.
The Padmakar-Ivan polynomial of Hyaluronic Acid-Paclitaxel conjugate HAPd is given by P I(HAPd , .
= .
d Oe .
x87d Proof.
From the graph structure analysis to the Hyaluronic Acid-Paclitaxel conjugate and by the definition of Padmakar-Ivan polynomial, we have HAPd HAPd .
) P I(HAPd , .
= x.
e e=uvE(HAPd ) i=1 e=uvE(Ki ) dOe1 x.
) x.
) , where Ki O Ki 1 e=xyE(Ki OKi 1 ) denotes the link of graphs |E(Ki ).
V (K)| .
Oe .
V (K)| = 95d.
V (K)| .
Oe .
V (K)| = .
V (K)| .
d d Oe .
, since |V (K)| = 87d = .
d Oe .
x87d Hence the theorem.
Theorem 5.
The Additively weighted Padmakar-Ivan polynomial of Hyaluronic Acid-Paclitaxel conjugate HAPd is given by P IA (HAPd , .
= dx261d .
d Oe .
)x435d .
d Oe .
x522d 7dx609d dx696d .
Proof.
By graph structure analysis to the Hyaluronic Acid-Paclitaxel conjugate and applying the definition of Additively weighted Padmakar-Ivan polynomial, we have KANDAN, et al HAPd x.
ege P IA (HAPd , .
= HAPd .
dege HAPd .
e HAPd .
) e=uvE(HAPd ) .
egeK .
degeK .
K e .
) e=uvEi (K) i=1,2,.
) e=uvE1 (K) x.
) e=uvE6 (K) x.
) e=uvE7 (K) x.
) e=uvE4 (K) x.
) e=uvE5 (K) e=uvE2 (K) .
K e .
) e=uvE3 (K) x.
) x.
) e=uvE8 (K) x.
) e=uvE9 (K) .
V (K)| = dx .
V (K)| .
V (K)| .
V (K)| .
dOe.
V (K)| .
V (K)| .
d Oe .
V (K)| .
V (K)| dx.
V (K)| , since |V (K)| = 87d 3.
= dx .
dOe.
)x5.
dOe.
= dx .
d Oe .
d Oe .
x522d 7dx609d dx696d Hence the theorem.
Theorem 5.
The Multiplicatively weighted Padmakar-Ivan polynomial of Hyaluronic Acid-Paclitaxel conjugate HAPd is given by P IM (HAPd , .
= dx174d .
d Oe .
d Oe .
x1044d Proof.
By graph structure analysis to the Hyaluronic Acid-Paclitaxel conjugate and from the definition of Multiplicatively weighted Padmakar-Ivan polynomial, we have HAPd HAPd HAPd HAPd .
) P IM (HAPd , .
= x.
ege e=uvE(HAPd ) .
egeK .
degeK .
K e .
) e=uvEi (K) i=1,2,.
) e=uvE1 (K) e=uvE9 (K) .
V (K)| x.
) x.
) e=uvE6 (K) .
K e .
) e=uvE7 (K) e=uvE4 (K) x.
) e=uvE5 (K) x.
) e=uvE2 (K) x.
) e=uvE3 (K) x.
) e=uvE8 (K) .
K e .
) = dx .
V (K)| .
V (K)| .
V (K)| .
dOe.
V (K)| .
V (K)| .
d Oe .
V (K)| .
V (K)| dx.
V (K)| = dx .
dOe.
dOe.
x763d Calculating PI related indices and their polynomial of Hyaluronic Acid and Conjugates Figure 7.
Comparison of P I.
P IA and P IM indices for Hyaluronic Acid HAd Figure 8.
Comparison of P I.
P IA and P IM indices for Hyaluronic Acid-Curcumin conjugate HACd .
x1044d dx1392d .
Since |V (K)| = 87d Hence the theorem.
The following corollaries shows the relationship between Padmakar-Ivan index.
Additively Padmakar-Ivan index and Multiplicatively Padmakar-Ivan index for Hyaluronic Acid-Paclitaxel conjugate.
Corollary 5.
P IA (K) = 5P I(K) .
d Oe .
|V (K)| Corollary 5.
P IM (K) = 6P I(K) .
d Oe .
|V (K)|.
Corollary 5.
P I(K) < P IA (K) < P IM (K)
Comparisons and discussions The calculation of the molecular invariants of a chemical graphs enables scientists to have a better understanding of the physical chemistry and biological characteristics of drugs.
Since drug tests show strong inner relationships between the biomedical and pharmacological characteristics of the drugs and their molecular Hence we studied the exact value of chemical graph P I.
P IA and P IM for HAd .
HACd .
HAPd and compared it.
It was found that these indices increases with increasing of d units .
ee Fig.
7, 8, .
and from Fig.
10 show that PI index has strong correlation for HAPd .
As these results are helpful in drug discovery as well as chemical science point of view, in this regard, the mathematical properties of Hyaluronic Acid-Curcumin/Paclitaxel conjugate are worth to investigate for future KANDAN, et al Figure 9.
Comparison of P I.
P IA and P IM indices for Hyaluronic Acid-Paclitaxel conjugate HAPd .
Figure 10.
Comparison of P I index for HAd .
HACd and HAPd .
Conclusion Studying the physicochemical properties of the molecular structure is not only time consuming but also a costly process.
Therefore, the reearchers in mathematical chemistry shown interest on finding topological indices for various chemical As the number of vertices of a chemical graph increases, it is difficult to obtain data and make calculations on it.
In this case, computational and statistical methods become important for analyzing the behavior of the data.
In this way, the desired values can become predictable without time-consuming experiments.
Therefore, it has been considered necessary to calculate the molecular invariant called PI related indices and its polynomial for molecular sturcture in order to examine the physicochemical and biological properties of these molecules.
Hyaluronic Acid.
Hyaluronic Acid Curcumin/Paclitaxel conjugate, are popular lately, especially in drug manufacturing.
In this work, we have calculated some distance, degree-distance based topological indices of Hyaluronic Acid.
Hyaluronic Acid-Curcumin/Paclitaxel conjugate such as PI index, weighted Additively PI index, weighted Multiplicatively PI index.
In addition, associated polynomials of the above topological indices for Hyaluronic Acid.
Hyaluronic Acid-Curcumin/Paclitaxel conjugate were also computed.
The results obtained in this study can be incorporated in various QSAR and QSPR models for its applications in the chemical and pharmaceutical science.
We extended this work to obtain Szeged and Mostar indices.
The computation of Calculating PI related indices and their polynomial of Hyaluronic Acid and Conjugates other molecular invariants and polynomial for Hyaluronic Acid.
Hyaluronic AcidCurcumin/Paclitaxel conjugate may be another interesting topic for future studies.
Here we pose a problem.
Problem.
Characterize the maximal value of PI index, weighted Additively PI index, weighted Multiplicatively PI index among various conjugation to the Hyaluronic Acid.
REFERENCES