AYESHA Saeed Bhatti

(Department of Mathematics，Government College University，Lahore，Pakistan 54000)

Abstract: Hyaluronic Acid-Paclitaxel (HA-PTX) also known as Hyaluronic Acid (HA) is a compound that occurs naturally. It is a glycosaminoglycan polymer composed of a linear structure of units of D-glucuronic acid and N-acetyl-D-glucosamine, which are linked via alternating -1,3- and -1,4-glycosidic bonds. In chemical graph theory we associate a graph to a compound and compute topological indices that help us in guessing properties of understudy compound. So we concentrate to compute two sum connectivity indices such as 2-sum atomic bound connectivity index, 2-sum inverse Randic index, 2-sum geometric arithmetic index, 2-sum and first redefined Zagreb index, 2-sum and second redefined Zagreb index, 2-sum and third redefined Zagreb index for HA-PTX.

Key words: topological index；2-sum connectivity index；molecular graph；degree

0 Introduction

The description of an atomic structure by a graph where the vertices are represented by atoms and the edges are represented by the bonding, is a significant application of (connected undirected) graphs, studied in a field called chemical graph theory.

A molecular graphG=(V,E) is a simple graph havingn=|V| vertices andm=|E| edges. The verticesvi∈Vrepresent non-hydrogen atoms and the edges (vi,vj)∈Erepresent covalent bonds among the corresponding atoms. Hydrocarbons are made up of only carbon and hydrogen atoms and their molecular graphs are represented by the carbon skeleton of the molecule.

Topological indices possibly viewed as a huge variety of boundaries on a subatomic graph that is significant in theoretic physical science and pharmacology science[1-3]. There are some significant indices obsessed on the vertex degree, for example, Randic indices, harmonic index, Zagreb indices etc. The known Randic, has been utilized successfully as an atomic descriptor in QSPR and QSAR[4-5]. In this paper, we defined following 2-sum indices.

The 2-sum atomic bound connectivity index is defined as:

The 2-sum inverse Randic index is defined as:

The Redefined Zagreb indices are as follows:

1 Methodology

To figure our results, firstly we will construct the graph of involved molecular compounds and count the overall range of vertices and edges. Secondly, we tend to divide the edge set of involved graphs into totally different class supported the degrees of end vertices. By applying definitions of 2-sum Atomic Bound connectivity index, 2-sum Inverse Randic index, 2-sum First Redefined Zagreb index, 2-sum Second Redefined Zagreb index, 2-sum Third Redefined Zagreb index for some connectivity indices, we have a tendency to compute our desired results.

2 Hyaluronic Acid-Paclitaxel Conjugates

Threat is seen as one of the noteworthy explanations behind death on the planet and its end extent continues rising with the greater part of passing's achieved by chest, stomach, lung, and colon threatening growths. Regardless of the way that there have been unprecedented enhancements in threatening development science and medicines to treat tumors, challenges notwithstanding everything remain in fundamental and metastatic ailment treatment. Additionally, there are traps in current anticancer drugs which incite low identity and high destructiveness as such genuinely restricting their feasibility. Certain advances have happened in molecularly centered around sickness treatment over late years. Hyaluronic destructive (HA) is a fuel which occurs regularly. It is a glycosaminoglycan polymer made of a straight structure of units of D-glucuronic destructive and N-acetyl-D-glucosamine, which are associated by implies of pivoting -1,3-and-1,4-glycosidic bonds. Its fundamental structure, disaccharide, is energetically consistent. HA is a encouraging ailment sedate on account of its novel, biodegradable, bio-feasible, nontoxic, hydrophilic, likewise, non-immutable features; besides, HA receptors showed over-verbalization on various tumor cells. As a rapidly creating stage for zeroing in on CD44-over imparting cells these days, HA intends to improve anticancer therapies. HA is an ideal medicine carrier and prescription zeroing in on. Paclitaxel (PTX) is a convincing medicine, which is proposed for certain sorts of tumors, including ovarian, chest, lung, bladder, prostate, and esophageal tumors, etc. Whereas PTX association furthermore experience its own obstacles, for instance, its powerless dissolvability furthermore, appropriate indications, similarly as the excipients usually used in its arrangement. Ringsdorf at first suggested the procedure for making polymeric macromolecule-sedate structures, which was expected to pass on minimal hydrophobic drug particles to their objections of movement. The crucial good conditions of HA-PTX structures are the development of its water dissolvability and the protecting of its development, most significantly, it could be consumed as engaged medicine movement to strengthen unfriendly to tumor ampleness. The formation of hyaluronic acid-paclitaxel (HA-PTX) structures is represented in Figures 1-3.

Figure 1 Unit Structure Figure 2 Molecular Graph

For Hyaluronic Acid-Paclitaxel conjugates (HA-PTX[n]), we have

|p{1,2,3}|=n+1,|p{1,3,3}|=12n,|p{1,3,2}|=10n+1,|p{1,4,4}|=3n,

|p{1,4,3}|=6n,|p{1,3,4}|=2n,|p{1,4,1}|=n,|p{1,4,2}|=n,|p{1,3,1}|=3n,

|p{2,2,2}|=9n,|p{2,3,2}|=8n,|p{2,4,2}|=n,|p{2,4,3}|=3n,|p{2,4,4}|=n,

|p{2,3,4}|=4n,|p{2,3,3}|=29n,|p{3,3,3}|=11n,|p{3,4,4}|=2n,

|p{3,4,3}|=3n,|p{3,2,3}|=10n-1,|p{3,2,2}|=9n,|p{4,3,4}|=n,

|p{4,2,3}|=2n,|p{4,3,3}|=6n.

Theorem1LetGbe the molecular graph of Hyaluronic Acid-Paclitaxel conjugates (HA-PTX), then the 2-sum atomic bound connectivity index is

ABC2(G)= 87.327 1n+1.055 7.

ProofBy the definition of the 2-sum atomic bound connectivity index, we have

=0.816 5n+0.816 5+8.944 3n+8.165 0n+0.816 5+1.984 3n+4.242 6n

+1.414 2n+n+0.790 6n+3n+6.364 0n+5.164 0n+0.162 4n+1.620 2n

+0.5n+2.160 2n+16.743 1n+5.601 0n+0.866 0n+1.414 2n+5.773 5n

-0.577 4+5.809 5n+0.433 0n+1.080 1n+2.828 4n

=87.327 1n+1.055 7.

Theorem2LetGbe the molecular graph of Hyaluronic Acid-Paclitaxel conjugates (HA-PTX), then the 2-sum inverse Randic index is

R2(G)= 34.274 2n+0.580 7.

ProofBy the definition of the 2-sum inverse Randic index, we have

=0.408 2n+0.408 2+4n+0.625n+0.408 2+0.75n+1.732 1n+0.577 4n+0.5n

+0.353 6n+1.732 1n+3.182 0n+2.309 4n+0.25n+0.612 4n+0.176 8n+0.816 5n

+6.835 4n+2.117 0n+0.288 7n+0.5n+2.357 0n-0.235 7+2.598 1n+0.144 3n

+0.408 2n+n=34.274 2n+0.580 7.

Theorem3LetGbe the molecular graph of Hyaluronic Acid-Paclitaxel conjugates (HA-PTX), then the 2-sum first redefined Zagreb index is

ProofBy the definition of the 2-sum first redefined Zagreb index, we have

=n+1+9.333 3n+10n+1+1.687 5n+4n+1.333 3n+1.5n+0.875n+5n+6.75n

+4.666 7n+0.5n+1.125n+0.312 5n+1.5n+12.888 9n+3.666 7n+0.458 3n

+0.833 3n+4.444 4n-0.444 4+5.25n+0.229 2n+0.75n+1.666 7n

=79.770 8n+1.555 6.

Theorem4LetGbe the molecular graph of Hyaluronic Acid-Paclitaxel conjugates (HA-PTX), then the 2-sum second redefined Zagreb Index is

ProofBy the definition of the 2-sum second redefined Zagreb index, we have

=n+1+15.428 6n+10n+1+5.333 3n+9n+3n+0.666 7n+1.142 9n+1.8n

+12n+3.714 3n+2n+8n+3.2n+10.666 7n+65.25n+33n+8.727 3n+10.8n

+22.5n-2.25+15.428 6n+4.363 6n+5.333 3n+21.6n

=283.955 3n-0.25.

Theorem5LetGbe the molecular graph of Hyaluronic Acid-Paclitaxel conjugates (HA-PTX), then the 2-sum third redefined Zagreb index is

ProofBy definition of the 2-sum third redefined Zagreb index, we have

=36n+36+756n+360n+36+432n+576n+192n+24n+56n+45n+432n+672n

+128n+648n+320n+864n+4 176n+2 673n+1 056n+1 080n+1 440n-144+756n

+528n+432n+2 160n

=19 842n-72.

3 Conclusions

Topological indices can help to solve many problems of chemistry with using lab. In this paper, we introduced concept of 2-sum connectivity indices. We defined 2-sum inverse Randic index, 2-sum atomic bound connectivity index and 2-sum redefined first, second and third Zagreb indices. We also computed aforementioned indices for the molecular graph of Hyaluronic Acid-Paclitaxel conjugates (HA-PTX).