Structural descriptors of anthracene using topological coindices through CoM-polynomial

Abstract

Topological indices and coindices are numerical invariants that relate to quantitative structure property/activity connections. The purpose of topological indices and coindices were introduced to draw the data related to chemical graphs with respect to adjacent & non adjacent pairs of vertex degrees respectively. These indices equip the researchers with a lot of information related to the properties and structure of the chemical compound. In this article, CoM-polynomials for molecular graph of linear and multiple Anthracene are computed from which eleven degree based topological coindices are derived.

Keywords

CoM-polynomial topological coindex anthracene

1 Introduction

An impressive branch of science that talks about the structure and the properties of chemical compound without using any apparatus, is the mathematical chemistry. As the tools applied in this science are all numerical, it is not time and cost consuming. This fascinating branch of science uses topological indices as one of its tools that helps in forseeing the structure and property studies. In computing these indices, generally hydrogen atoms are overlooked.

In this study, a tricyclic aromatic hydrocarbon known as anthracene is studied. An aromatic compound with three fused benzene rings having 14 carbon atoms and 10 hydrogen atoms is anthracene. It is one of the polycyclic aromatic hydrocarbons that finds its applications in the production of dyes, scintillation crystals and in organic semiconductor research. Anthracene is used in insecticides, coating materials, wood preservatives, and UV tracer in conformal coatings of printed wiring boards. It is used in identifying high energy part of photons, electrons, and alpha particles. The emission spectrum of anthracene lies between 400nm and 440nm as it used in radiation therapy dosimetry. Anthracene and its derivatives having hydroxyl group, are used in pharmacology, the science which mainly focusses on the drug reaction. This refers to the derivatives of anthracene being pharmacologically active.

Anthracene is the major source of coal tar which occupies 1.5% in coal tar. A tricyclic aromatic hydrocarbon is on the EPA’s (Environmental protection agents) high priority list of the pollutants. It is found on the surface of drinking water, exhaust emissions, cigarette and cigar smoke, ambient air, and in edible aquatic organisms [1, 2]. Humans are exposed to anthracene through tobacco smoke and intake of contaminated food containing combustion products, as these polycyclic aromatic hydrocarbons are generated by combustion processes. Studies specify that anthracene is noncarcinogenic. It has been found that it is negative in various in vitro and in vivo genotoxicity tests. It is biodegradable in soil in the presence of light.

Fritzache studied on solid hydrocarbons and discovered anthracene in the distillation process of coal tar in the year 1866. He found from the studies that, saturated solutions of anthracene when exposed to sunlight gave crystals which were colorless and it regenerated to anthracene when melted them. The main ingredient for dye is anthroquinone that is obtained from anthracene. Alizarin is a red dye that is produced using anthracene. Even though anthracene is colorless it exhibits florescence blue when exposed to ultraviolet light. Anthracene melts at 218°C and boils at 354°C. It is not soluble in water, but it is soluble in alcohol, benzene, hydro naphthalene, chloroform and disulphide.

A chemical reaction where the electrons of a compound are lost refers to oxidation. Oxidation of anthracene takes place when strong oxidants such as chromic acid or hydrogen peroxide comes into contact with anthracene. In this process, anthraquinone is obtained as by product. Anthraquinone finds its applications in many fields such as, dyes, additive in paper industry to aid the digestion of paper pulp in alkaline processes and in flow batteries as electrolyte. To decontaminate the water containing hydrocarbons, anthracene oxidation is used.

The pairs of adjacent vertices contribute to degree based topological indices. However, non-adjacent pairs of vertices refer to degree based topological coindices for computing few topological properties of graphs. In this work, CoM-polynomials of anthracene are established from which eleven degree based topological coindices are derived. In graph theory, a molecular structure of the chemical compound is assumed as a graph in which the joins of the bonds refer to the vertices and the bonds to the edges. Numerical quantities obtained scientifically from the atomic structure is a topological index/coindex. They refer to the constitution showing the relationship of the structure with various physical and chemical properties.

2 Applications of anthracene

Anthracence has several applications in drug delivery having significant biological activities against tumor cells. It is very effective in skin related ailments. The structural system of the anthracene nucleus has overlapping with the deoxyribonucleic acid (DNA) base pairs.

Many compounds of anthracene are formed by re-arrangement of molecules of methyl group. 2,7,12-trimethylbenzanthracene was synthesized by Newman et.al. which does not exhibit cancer causing activity, also called carcinogenic activity. As the 1,2,3 and 5 position of 7,12-dimethylbenzanthracene are substituted with methyl group, this compound does not exhibit carcinogenic activity. The substitution of methyl group in different positions yields some compounds that are carcinogenic anthracene-based such as 7-bromomethylbenzanthracene, 7-bromomethyl-12-methylbenzanthracene and 4-chloro-7-bromomethylbenzanthracene. These are tested on mice and found that there was an increase in the risk of tumour development [3].

DNA photo-cleavage is a reaction that slices one of the linkages of covalent sugar-phosphate between nucleotides that forms the sugar phosphate backbone of DNA. This process may be catalyzed chemically, enzymatically or by radiation. Cleavage may be formed by removing the end nucleotide, called exonucleolytic or by splitting the strand, called endonucleolytic. Among many compounds, the derivatives of acridine and anthracene are found to be good DNA photo cleavers while naphthalene and benzene are inactive. Acridine showed copper (II) enhanced photo cleaving at micromolar concentrations while anthracene derivatives of 0.25μm was enough to cleave DNA completely.

DNA in living organisms has very extensive functions and also advantageous in preventing and curing diseases. The studies on binding of small molecules with DNA are significant in the design of new drugs targeted to DNA. The combination of two words reduction and oxidation gives rise to the word redox means reaction with oxygen to form an oxide. Anthracene appended with cyclam rings show high redox activity. It was found that, anthracene is reversibly oxidised and was reported for chemo sensors. Anthracenes are used as antibiotics in the treatment of cancer chemotherapy in the form of anthracyclines. They inhibit DNA and Ribonucleic acid (RNA) synthesis between base pairs of the DNA/RNA strand which retards further growth of cancer cells. Anthracenediones are used in the drugs of anti-tumour [4, 5].

3 Motivation

A mathematical model of the structure of a drug is expressed as a graph such that, every vertex represents an atom and each edge indicates a chemical bond between these atoms. Assuming G to be a simple graph of a drug, where atom set and chemical bond set are represented by V (G) and E (G) respectively [6, 7].

A topological index is a function that maps molecular structure of a drug to real number. Since four decades, lot of studies have taken place to introduce many important indices and coindices viz., Wiener index, Zagreb indices & coindices, Randic indices & coindices, PI index, eccentric indices and many more to measure various characteristics of drug molecules through Quantitative Structure-Activity/Property/Toxicity Relationship (QSAR/QSPR/QSTR) studies [8 –10].

Topological indices and coindices contribute to the pharmaceutical, chemical and nano materials engineering. Topological indices play a major role in contributing to the biological and chemical characteristics of novel drugs to help the medical scientists [11 –15].

In this study, Anthracene is focused as it as numerous applications in drugs, DNA cleavage, DNA binding, anti-carcinogenic, chemo sensors and redox activity. It is found that, the presence of Anthracene in antibiotics have warded cancer cells. It is effective in dermatological ailments.

Various work on Anthracene using topological indices based on adjacent pair of vertex degrees have been carried out across the globe. Kulli [16] studied on multiple connectivity of nanostructures such as linear [n]-anthracene, V-anthracene nanotube and nanotori for which various multiplicative degree based indices were computed. Zaheer Ahmed et al. [17] mentioned about three entities such as energy, inertia and nullity in graph theory and computed them for phenylene and anthracene and conclusion were drawn for the same. Zhang et al. [18] computed topological indices of para-line graphs of subdivided graph (para-line graph) of linear-[s] Anthracene and multiple Anthracene. The article focussed on the applications of the indices discussed in the paper. Chemical graphs were studied with the help of para-line graphs which are very important in the field of chemistry. Cheng-Peng Li et al. [19] established closed formulae for M-polynomials of linear chains of benzene, napthalene, and anthracene graphs. Using the polynomial nine degree-based topological indices for these three chains were computed.

This study focuses on coindices for the chemical compound Anthracene based on non-adjacent pair of vertex degrees. For the notations & terminologies used in this study refer [20].

4 CoM-polynomial

In this work, eleven degree based topological coindices are computed for Anthracene using CoM-polynomial, an extension of M-polynomial [21 –25]. CoM-polynomials are computed for non-adjacent pair of atoms/vertices. There have been a lot of studies carried out on coindices of various chemical compounds using specific definition of the respective coindex [26 –30]. Here various coindices are computed using a single polynomial, CoM-polynomial.

Definition 4.1. For a simple connected graph G, the CoM-polynomial [31, 32] is defined as, $CoM (G; x, y) = \bar{M} (G; x, y) = \sum_{i \leq j} {\bar{m}}_{ij} (G) x^{i} y^{j}$ where ${\bar{m}}_{ij}, i, j \geq 1$ represents the number of edges uv ∉ E (G) such that {d (u) , d (v)} = {i, j}. Here d (u), d (v) represent the degrees of the vertices u and v in the graph G.

List of various topological coindices are tabulated in Table 1 using CoM-polynomial

Table 1
The relation between various topological coindices and CoM-polynomial

Topological Formula Derivation from

coindex φ (d (u) , d (v)) f (x, y) = CoM (G ; x, y)

$\bar{M_{1}} (G)$ ∑_uv∉E(G)d (u) + d (v) (D_x + D_y) (f (x, y)) _x=1=y

$\bar{M_{2}} (G)$ ∑_uv∉E(G)d (u) d (v) (D_xD_y) (f (x, y)) _x=1=y

$\bar{m^{} M_{2}} (G)$ $\sum_{uv \notin E (G)} \frac{1}{d (u) d (v)}$ (S_xS_y) (f (x, y)) _x=1=y

$\bar{{ReZG}_{3}} (G)$ ∑_uv∉E(G)d (u) d (v) (d (u) + d (v)) D_xD_y (D_x + D_y) (f (x, y)) _x=1=y

$\bar{F (G)}$ ∑_uv∉E(G)d² (u) + d² (v) $(D_{x}^{2} + D_{y}^{2}) (f (x, y))_{x = 1 = y}$

$\bar{R_{k}} (G)$ ∑_uv∉E(G) {d (u) d (v)} ^k $(D_{x}^{k} D_{y}^{k}) (f (x, y))_{x = 1 = y}$

$\bar{{RR}_{k}} (G)$ $\sum_{uv \notin E (G)} \frac{1}{{d (u) d (v)}^{k}}$ $(S_{x}^{k} S_{y}^{k}) (f (x, y))_{x = 1 = y}$

$\bar{SDD} (G)$ $\sum_{uv \notin E (G)} \frac{d^{2} (u) + d^{2} (v)}{d (u) d (v)}$ (D_xS_y + S_xD_y) (f (x, y)) _x=1=y

$\bar{H (G)}$ $\sum_{uv \notin E (G)} \frac{2}{d (u) + d (v)}$ 2S_xJ (f (x, y)) _x=1

$\bar{I} (G)$ $\sum_{uv \notin E (G)} \frac{d (u) d (v)}{d (u) + d (v)}$ S_xJD_xD_y (f (x, y)) _x=1

$\bar{A} (G)$ $\sum_{uv \notin E (G)} {\frac{d (u) d (v)}{d (u) + d (v) - 2}}^{3}$ $S_{x}^{3} Q_{- 2} {JD}_{x}^{3} D_{y}^{3} (f (x, y))_{x = 1}$

where $D_{x} = x \frac{\partial (f (x, y))}{\partial x}$ , $D_{y} = y \frac{\partial (f (x, y))}{\partial y}$ , $S_{x} = \int_{0}^{x} \frac{f (t, y)}{t} dt$ ,

$S_{y} = \int_{0}^{y} \frac{f (x, t)}{t} dt$ , J (f (x, y)) = f (x, x), Q_k (f (x, y)) = x^kf (x, y).

Topological	Formula	Derivation from
$\bar{M_{1}} (G)$	∑_uv∉E(G)d (u) + d (v)	(D_x + D_y) (f (x, y)) _x=1=y
$\bar{M_{2}} (G)$	∑_uv∉E(G)d (u) d (v)	(D_xD_y) (f (x, y)) _x=1=y
$\bar{m^{} M_{2}} (G)$	$\sum_{uv \notin E (G)} \frac{1}{d (u) d (v)}$	(S_xS_y) (f (x, y)) _x=1=y
$\bar{{ReZG}_{3}} (G)$	∑_uv∉E(G)d (u) d (v) (d (u) + d (v))	D_xD_y (D_x + D_y) (f (x, y)) _x=1=y
$\bar{F (G)}$	∑_uv∉E(G)d² (u) + d² (v)	$(D_{x}^{2} + D_{y}^{2}) (f (x, y))_{x = 1 = y}$
$\bar{R_{k}} (G)$	∑_uv∉E(G) {d (u) d (v)} ^k	$(D_{x}^{k} D_{y}^{k}) (f (x, y))_{x = 1 = y}$
$\bar{{RR}_{k}} (G)$	$\sum_{uv \notin E (G)} \frac{1}{{d (u) d (v)}^{k}}$	$(S_{x}^{k} S_{y}^{k}) (f (x, y))_{x = 1 = y}$
$\bar{SDD} (G)$	$\sum_{uv \notin E (G)} \frac{d^{2} (u) + d^{2} (v)}{d (u) d (v)}$	(D_xS_y + S_xD_y) (f (x, y)) _x=1=y
$\bar{H (G)}$	$\sum_{uv \notin E (G)} \frac{2}{d (u) + d (v)}$	2S_xJ (f (x, y)) _x=1
$\bar{I} (G)$	$\sum_{uv \notin E (G)} \frac{d (u) d (v)}{d (u) + d (v)}$	S_xJD_xD_y (f (x, y)) _x=1
$\bar{A} (G)$	$\sum_{uv \notin E (G)} {\frac{d (u) d (v)}{d (u) + d (v) - 2}}^{3}$	$S_{x}^{3} Q_{- 2} {JD}_{x}^{3} D_{y}^{3} (f (x, y))_{x = 1}$
where $D_{x} = x \frac{\partial (f (x, y))}{\partial x}$ ,	$D_{y} = y \frac{\partial (f (x, y))}{\partial y}$ ,	$S_{x} = \int_{0}^{x} \frac{f (t, y)}{t} dt$ ,
$S_{y} = \int_{0}^{y} \frac{f (x, t)}{t} dt$ ,	J (f (x, y)) = f (x, x),	Q_k (f (x, y)) = x^kf (x, y).

The following are the notations used in the paper $n_{i} = | V_{i} | for V_{i} = {v \in V (G) | d (v) = i} m_{ij} = | E_{ij} | for E_{ij} = {uv \in E (G) | d (u) = i and d (v) = j} {\bar{m}}_{ij} = | {\bar{E}}_{ij} | for {\bar{E}}_{ij} = {uv \in \bar{E} (G) | d (u) = i and d (v) = j}$

Lemma 4.2. For a simple connected graph G of cardinality n [27], we have

${\bar{m}}_{ij} = | {\bar{E}}_{ij} | = {\begin{matrix} \frac{n_{i} (n_{i} - 1)}{2} - m_{ii} & for & i = j \\ n_{i} n_{j} - m_{ij} & for & i < j \end{matrix}}$ where n_i, m_ij and ${\bar{m}}_{ij}$ are defined above.

5 CoM-polynomial of linear [n] anthracene

Theorem 5.1. The CoM-polynomial of linear [n] Anthracene is given by

$CoM (G; x, y) = (18 n^{2} + 21 n) x^{2} y^{2} + (48 n^{2} - 4 n - 12) x^{2} y^{3} + (32 n^{2} - 42 n + 14) x^{3} y^{3}$

Fig. 2

3D plot for CoM-polynomial of linear [n] Anthracene.

Proof. From the Fig. 1 for a molecular graph G, it is observed that |V|=14n and |E|=18n - 2. Also, the edge set of G is divided into three partitions and are as below

Fig. 1

The molecular graph of linear [n] Anthracene.

$E_{22} = {uv \in E (G) | d (u) = 2, d (v) = 2}, E_{23} = {uv \in E (G) | d (u) = 2, d (v) = 3}, E_{33} = {uv \in E (G) | d (u) = 3, d (v) = 3},$ such that $m_{22} = 6, m_{23} = 12 n - 4, and m_{33} = 6 n - 4 .$

Similarly, vertex set can be categorized into two classes and are as follows. $n_{1} = | V_{2} | = 6 n + 4, and n_{2} = | V_{3} | = 8 n - 4 .$

Using Lemma 4.2, we have ${\bar{m}}_{22} = \frac{n_{1} (n_{1} - 1)}{2} - m_{22} = \frac{(6 n + 4) (6 n + 4 - 1)}{2} - 6 = 18 n^{2} + 21 n . {\bar{m}}_{23} = n_{1} n_{2} - m_{23} = (6 n + 4) (8 n - 4) - (12 n - 4) = 48 n^{2} - 4 n - 12 . {\bar{m}}_{33} = \frac{n_{2} (n_{2} - 1)}{2} - m_{33} = \frac{(8 n - 4) (8 n - 4 - 1)}{2} - (6 n - 4) = 32 n^{2} - 42 n + 14 .$

From the CoM-polynomial definition $CoM (G; x, y) = \bar{M} (G; x, y) = \sum_{i \leq j} {\bar{m}}_{ij} (G) x^{i} y^{j} CoM (G; x, y) = \sum_{2 \leq 2} {\bar{m}}_{22} x^{2} y^{2} + \sum_{2 \leq 3} {\bar{m}}_{23} x^{2} y^{3} + \sum_{3 \leq 3} {\bar{m}}_{33} x^{3} y^{3} = (18 n^{2} + 21 n) x^{2} y^{2} + (48 n^{2} - 4 n - 12) x^{2} y^{3} + (32 n^{2} - 42 n + 14) x^{3} y^{3} .$ Using Theorem 5.1 and Table 1, coindices of linear [n] Anthracene are in the following proposition as

Proposition 5.2. ${\bar{M}}_{1} (G) = 504 n^{2} - 188 n + 24 . {\bar{M}}_{2} (G) = 648 n^{2} - 318 n + 54 . {\bar{m M}}_{2} (G) = 776.056 n^{2} - 0.0833 n - 0.444 . {\bar{ReZG}}_{3} (G) = 3456 n^{2} - 2052 n + 396 . \bar{F} (G) = 1344 n^{2} - 640 n + 96 . \bar{R_{k}} (G) = 2^{2 k} (18 n^{2} + 21 n) + 6^{k} (48 n^{2} - 4 n - 12) + 3^{2 k} (32 n^{2} - 42 n + 14) . \bar{{RR}_{k}} (G) = \frac{(18 n^{2} + 21 n)}{2^{2 k}} + \frac{(48 n^{2} - 4 n - 12)}{6^{k}} + \frac{(32 n^{2} - 42 n + 14)}{3^{2 k}} . \bar{SDD} (G) = 204.016 n^{2} - 50.668 n + 1.996 . \bar{H} (G) = 38.866 n^{2} - 5.099 n - 0.134 . \bar{I} (G) = 123.6 n^{2} + 79.2 n + 6.6 . \bar{A} (G) = 892.512 n^{2} - 342.422 n + 63.474 .$

Proof. Let, $f (x, y) = CoM (G; x, y) = (18 n^{2} + 21 n) x^{2} y^{2} + (48 n^{2} - 4 n - 12) x^{2} y^{3} + (32 n^{2} - 42 n + 14) x^{3} y^{3} then, D_{x} f (x, y) = 2 (18 n^{2} + 21 n) x^{2} y^{2} + 2 (48 n^{2} - 4 n - 12) x^{2} y^{3} + 3 (32 n^{2} - 42 n + 14) x^{3} y^{3}, D_{y} f (x, y) = 2 (18 n^{2} + 21 n) x^{2} y^{2} + 3 (48 n^{2} - 4 n - 12) x^{2} y^{3} + 3 (32 n^{2} - 42 n + 14) x^{3} y^{3}, (D_{x} + D_{y}) f (x, y) = 4 (18 n^{2} + 21 n) x^{2} y^{2} + 5 (48 n^{2} - 4 n - 12) x^{2} y^{3} + 6 (32 n^{2} - 42 n + 14) x^{3} y^{3}, D_{y} D_{x} f (x, y) = 4 (18 n^{2} + 21 n) x^{2} y^{2} + 6 (48 n^{2} - 4 n - 12) x^{2} y^{3} + 9 (32 n^{2} - 42 n + 14) x^{3} y^{3}, (D_{x}^{2} + D_{y}^{2}) f (x, y) = 8 (18 n^{2} + 21 n) x^{2} y^{2} + 13 (48 n^{2} - 4 n - 12) x^{2} y^{3} + 18 (32 n^{2} - 42 n + 14) x^{3} y^{3}, D_{x}^{k} D_{y}^{k} f (x, y) = 2^{2 k} (18 n^{2} + 21 n) x^{2} y^{2} + 6^{k} (48 n^{2} - 4 n - 12) x^{2} y^{3} + 3^{2 k} (32 n^{2} - 42 n + 14) x^{3} y^{3}, D_{x} D_{y} (D_{x} + D_{y}) f (x, y) = 16 (18 n^{2} + 21 n) x^{2} y^{2} + 30 (48 n^{2} - 4 n - 12) x^{2} y^{3} + 54 (32 n^{2} - 42 n + 14) x^{3} y^{3}, S_{x} S_{y} f (x, y) = \frac{(18 n^{2} + 21 n)}{4} x^{2} y^{2} + \frac{(48 n^{2} - 4 n - 12)}{6} x^{2} y^{3} + \frac{(32 n^{2} - 42 n + 14)}{9} x^{3} y^{3}, S_{x}^{k} S_{y}^{k} f (x, y) = \frac{(18 n^{2} + 21 n)}{2^{2 k}} x^{2} y^{2} + \frac{(48 n^{2} - 4 n - 12)}{6^{k}} x^{2} y^{3} + \frac{(32 n^{2} - 42 n + 14)}{3^{2 k}} x^{3} y^{3}, (S_{y} D_{x} + S_{x} D_{y}) f (x, y) = \frac{8 (18 n^{2} + 21 n)}{4} x^{2} y^{2} + \frac{13 (48 n^{2} - 4 n - 12)}{6} x^{2} y^{3} + \frac{18 (32 n^{2} - 42 n + 14)}{9} x^{3} y^{3},$ $S_{x} Jf (x, y) = \frac{(18 n^{2} + 21 n)}{4} x^{4} + \frac{(48 n^{2} - 4 n - 12)}{5} x^{5} + \frac{(32 n^{2} - 42 n + 14)}{6} x^{6}, S_{x} {JD}_{x} D_{y} f (x, y) = \frac{4 (18 n^{2} + 21 n)}{4} x^{4} + \frac{6 (48 n^{2} - 4 n - 12)}{5} x^{5} + \frac{9 (32 n^{2} - 42 n + 14)}{6} x^{6}, S_{x}^{3} Q_{- 2} {JD}_{x}^{3} D_{y}^{3} f (x, y) = \frac{(4)^{3} (18 n^{2} + 21 n)}{(2)^{3}} x^{2} + \frac{(6)^{3} (48 n^{2} - 4 n - 12)}{(3)^{3}} x^{3} + \frac{(9)^{3}}{(4)^{3}} (32 n^{2} - 42 n + 14) x^{4},$

Using the above results in Table 1, we obtain $\bar{M_{1}} (G) = (D_{x} + D_{y}) f (x, y) |_{x = 1 = y} = 504 n^{2} - 188 n + 24 . \bar{M_{2}} (G) = (D_{x} D_{y}) f (x, y) |_{x = 1 = y} = 648 n^{2} - 318 n + 54 . \bar{m M_{2}} (G) = (S_{x} S_{y}) f (x, y) |_{x = 1 = y} = 776.056 n^{2} - 0.0833 n - 0.444 . \bar{{ReZG}_{3}} (G) = D_{x} D_{y} (D_{x} + D_{y}) f (x, y) |_{x = 1 = y} = 3456 n^{2} - 2052 n + 396 . \bar{F} (G) = (D_{x}^{2} + D_{y}^{2}) f (x, y) |_{x = 1 = y} = 1344 n^{2} - 640 n + 96 . \bar{R_{k}} (G) = (D_{x}^{k} D_{y}^{β}) f (x, y) |_{x = 1 = y} = 2^{2 k} (18 n^{2} + 21 n) + 6^{k} (48 n^{2} - 4 n - 12) + 3^{2 k} (32 n^{2} - 42 n + 14) . \bar{{RR}_{k}} (G) = (S_{x}^{k} S_{y}^{k}) f (x, y) |_{x = 1 = y} = \frac{(18 n^{2} + 21 n)}{2^{2 k}} + \frac{(48 n^{2} - 4 n - 12)}{6^{k}} + \frac{(32 n^{2} - 42 n + 14)}{3^{2 k}} . \bar{SDD} (G) = (D_{x} S_{y} + S_{x} D_{y}) f (x, y) |_{x = 1 = y} = 204.016 n^{2} - 50.668 n + 1.996 . \bar{H} (G) = 2 S_{x} Jf (x, y) |_{x = y = 1} = 38.866 n^{2} - 5.099 n - 0.134 . \bar{I} (G) = S_{x} {JD}_{x} D_{y} f (x, y) |_{x = y = 1} = 123.6 n^{2} + 79.2 n + 6.6 . \bar{A} (G) = S_{x}^{3} Q_{- 2} {JD}_{x}^{3} D_{y}^{3} f (x, y) |_{x = y = 1} = 892.512 n^{2} - 342.422 N + 63.474 .$ □

6 CoM-polynomial of multiple anthracene

Theorem 6.1. The CoM-polynomial of multiple Anthracene is given by

$CoM (G; x, y) = (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{2} y^{2} + (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{2} y^{3} + (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{3} y^{3} .$

Fig. 4

3D plot for CoM-polynomial of multiple Anthracene.

Proof. From the Fig. 3 for a molecular graph G, it is observed that |V|=14mn and |E|=21mn - 4m - n. Also, the edge set of G may be divided into three partitions are as below

Fig. 3

The molecular graph of multiple Anthracene.

$E_{22} = {uv \in E (G) | d (u) = 2, d (v) = 2}, E_{23} = {uv \in E (G) | d (u) = 2, d (v) = 3}, E_{33} = {uv \in E (G) | d (u) = 3, d (v) = 3},$ such that $m_{22} = 2 n + 4, m_{23} = 4 m + 12 n - 8, and m_{33} = 21 mn - 8 m - 15 n + 4 .$ Similarly, vertex set can be categorized into two classes are as follows. Using Lemma 4.2, we have ${\bar{m}}_{22} = \frac{n_{1} (n_{1} - 1)}{2} - m_{22} = \frac{(4 m + 6 n) (4 m + 6 n - 1)}{2} - (2 n + 4) = 8 m^{2} + 18 n^{2} + 28 mn - 2 m - 5 n - 4 . {\bar{m}}_{23} = n_{1} n_{2} - m_{23} = (4 m + 6 n) (14 mn - 4 m - 6 n) - (4 m + 12 n - 8) = 56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8 . {\bar{m}}_{33} = \frac{n_{2} (n_{2} - 1)}{2} - m_{33} = \frac{(14 mn - 4 m - 6 n) (14 mn - 4 m - 6 n - 1)}{2} - (21 mn - 8 m - 15 n + 4) = 98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} - 4 mn + 8 m^{2} + 18 n^{2} - 6 m + 18 n - 4 .$ From the CoM-polynomial definition, $CoM (G; x, y) = \bar{M} (G; x, y) = \sum_{i \leq j} {\bar{m}}_{ij} (G) x^{i} y^{j} CoM (G; x, y) = \sum_{2 \leq 2} {\bar{m}}_{22} x^{2} y^{2} + \sum_{2 \leq 3} {\bar{m}}_{23} x^{2} y^{3} + \sum_{3 \leq 3} {\bar{m}}_{33} x^{3} y^{3} = (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{2} y^{2} + (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{2} y^{3} + (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{3} y^{3} .$ □

Using Theorem 6.1 and Table 1, coindices of the multiple Anthracene are in the following proposition as

Proposition 6.2. ${\bar{M}}_{1} (G) = 588 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} - 168 mn + 8 m + 28 n . {\bar{M}}_{2} (G) = 882 m^{2} n^{2} - 168 m^{2} n - 252 {mn}^{2} - 228 mn + 8 m^{2} + 18 n^{2} + 22 m + 70 n - 4 . {\bar{m M}}_{2} (G) = 10.888 m^{2} n^{2} + 3.108 m^{2} n + 4.667 {mn}^{2} + 0.223 m^{2} + 0.5022 n^{2} - 2.441 mn - 0.5 m - 1.249 n - 0.112 . {\bar{ReZG}}_{3} (G) = 5292 m^{2} n^{2} - 1344 m^{2} n - 2016 {mn}^{2} - 1272 mn + 80 m^{2} + 180 n^{2} + 172 m + 532 n - 40 . \bar{F} (G) = 1764 m^{2} n^{2} - 280 m^{2} n - 420 {mn}^{2} - 504 mn + 40 m + 128 n . \bar{R_{k}} (G) = 2^{2 k} (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) + 6^{k} (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) + 3^{2 k} (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) . \bar{{RR}_{k}} (G) = \frac{1}{2^{2 k}} (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) + \frac{1}{6^{k}} (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) + \frac{1}{3^{2 k}} (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) . \bar{SDD} (G) = 196 m^{2} n^{2} + 9.335 m^{2} n + 14.003 {mn}^{2} - 64.002 mn - 2.667 m^{2} - 6.001 n^{2} - 0.667 m - 0.0004 n + 1.334 . \bar{H} (G) = 32.663 m^{2} n^{2} + 3.735 m^{2} n + 5.603 {mn}^{2} - 8.533 mn + 0.266 m^{2} + 0.599 n^{2} - 0.6 m - 1.301 n - 0.133 . \bar{I} (G) = 147 m^{2} n^{2} - 16.8 m^{2} n - 25.2 {mn}^{2} - 39.6 mn + 0.8 m^{2} + 1.8 n^{2} + 2.2 m + 7.6 n - 0.4 . \bar{A} (G) = 1117.2 m^{2} n^{2} - 190.4 m^{2} n - 285.6 {mn}^{2} - 237.6 mn + 27.2 m^{2} + 61.2 n^{2} + 20.4 m + 69.2 n - 13.6 .$

Proof. Let, $f (x, y) = CoM (G; x, y) = (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{2} y^{2} + (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{2} y^{3} + (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{3} y^{3} then, D_{x} f (x, y) = 2 (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{2} y^{2} + 2 (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{2} y^{3} + 3 (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{3} y^{3}, D_{y} f (x, y) = 2 (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{2} y^{2} + 3 (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{2} y^{3} + 3 (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{3} y^{3},$ $(D_{x} + D_{y}) f (x, y) = 4 (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{2} y^{2} + 5 (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{2} y^{3} + 6 (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{3} y^{3}, D_{y} D_{x} f (x, y) = 4 (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{2} y^{2} + 6 (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{2} y^{3} + 9 (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{3} y^{3}, (D_{x}^{2} + D_{y}^{2}) f (x, y) = 8 (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{2} y^{2} + 13 (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{2} y^{3} + 18 (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{3} y^{3}, D_{x}^{k} D_{y}^{k} f (x, y) = 2^{2 k} (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{2} y^{2} + 6^{k} (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{2} y^{3} + 3^{2 k} (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{3} y^{3}, D_{x} D_{y} (D_{x} + D_{y}) f (x, y) = 16 (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{2} y^{2} + 30 (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{2} y^{3} + 54 (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{3} y^{3}, S_{x} S_{y} f (x, y) = \frac{1}{4} (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{2} y^{2} + \frac{1}{6} (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{2} y^{3} + \frac{1}{9} (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{3} y^{3}, S_{x}^{k} S_{y}^{k} f (x, y) = \frac{1}{2^{2 k}} (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{2} y^{2} + \frac{1}{6^{k}} (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{2} y^{3} + \frac{1}{3^{2 k}} (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{3} y^{3}, (S_{y} D_{x} + S_{x} D_{y}) f (x, y) = \frac{8}{4} (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{2} y^{2} + \frac{13}{6} (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{2} y^{3} + \frac{18}{9} (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{3} y^{3}, S_{x} Jf (x, y) = \frac{1}{4} (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{4} + \frac{1}{5} (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{5} + \frac{1}{6} (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{6},$ $S_{x} {JD}_{x} D_{y} f (x, y) = \frac{4}{4} (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{4} + \frac{6}{5} (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{5} + \frac{9}{6} (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{6}, S_{x}^{3} Q_{- 2} {JD}_{x}^{3} D_{y}^{3} f (x, y) = \frac{(4)^{3}}{(2)^{3}} (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) x^{2} + \frac{(6)^{3}}{(3)^{3}} (56 m^{2} n + 84 {mn}^{2} - 16 m^{2} - 36 n^{2} - 48 mn - 4 m - 12 n + 8) x^{3} + \frac{(9)^{3}}{(4)^{3}} (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) x^{4},$

Using the above results in Table 1, we obtain $\bar{M_{1}} (G) = (D_{x} + D_{y}) f (x, y) |_{x = 1 = y} = 588 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} - 168 mn + 8 m + 28 n . \bar{M_{2}} (G) = (D_{x} D_{y}) f (x, y) |_{x = 1 = y} = 882 m^{2} n^{2} - 168 m^{2} n - 252 {mn}^{2} - 228 mn + 8 m^{2} + 18 n^{2} + 22 m + 70 n - 4 . \bar{m M_{2}} (G) = (S_{x} S_{y}) f (x, y) |_{x = 1 = y} = 10.888 m^{2} n^{2} + 3.108 m^{2} n + 4.667 {mn}^{2} + 0.223 m^{2} + 0.502 n^{2} - 2.441 mn - 0.5 m - 1.249 n - 0.112 . \bar{{ReZG}_{3}} (G) = D_{x} D_{y} (D_{x} + D_{y}) f (x, y) |_{x = 1 = y} = 5292 m^{2} n^{2} - 1344 m^{2} n - 2016 {mn}^{2} - 1272 mn + 80 m^{2} + 180 n^{2} + 172 m + 532 n - 40 . \bar{F} (G) = (D_{x}^{2} + D_{y}^{2}) f (x, y) |_{x = 1 = y} = 1764 m^{2} n^{2} - 280 m^{2} n - 420 {mn}^{2} - 504 mn + 40 m + 128 n . \bar{R_{k}} (G) = (D_{x}^{k} D_{y}^{β}) f (x, y) |_{x = 1 = y} = 2^{2 k} (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) + 6^{k} (48 n^{2} - 4 n - 12) + 3^{2 k} (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) . \bar{{RR}_{k}} (G) = (S_{x}^{k} S_{y}^{k}) f (x, y) |_{x = 1 = y} = \frac{1}{2^{2 k}} (8 m^{2} + 18 n^{2} + 24 mn - 2 m - 5 n - 4) + \frac{1}{6^{k}} (48 n^{2} - 4 n - 12) + \frac{1}{3^{2 k}} (98 m^{2} n^{2} - 56 m^{2} n - 84 {mn}^{2} + 8 m^{2} + 18 n^{2} - 4 mn + 6 m + 18 n - 4) . \bar{SDD} (G) = (D_{x} S_{y} + S_{x} D_{y}) f (x, y) |_{x = 1 = y} = 196 m^{2} n^{2} + 9.335 m^{2} n + 14.003 {mn}^{2} - 64.002 mn - 2.667 m^{2} - 6.001 n^{2} - 0.667 m - 0.0004 n + 1.334 . \bar{H} (G) = 2 S_{x} Jf (x, y) |_{x = y = 1} = 32.663 m^{2} n^{2} + 3.735 m^{2} n + 5.603 {mn}^{2} - 8.533 mn + 0.266 m^{2} + 0.599 n^{2} - 0.6 m - 1.301 n - 0.133 . \bar{I} (G) = S_{x} {JD}_{x} D_{y} f (x, y) |_{x = y = 1} = 147 m^{2} n^{2} - 16.8 m^{2} n - 25.2 {mn}^{2} - 39.6 mn + 0.8 m^{2} + 1.8 n^{2} + 2.2 m + 7.6 n - 0.4 . \bar{A} (G) = S_{x}^{3} Q_{- 2} {JD}_{x}^{3} D_{y}^{3} f (x, y) |_{x = y = 1} = 1117.2 m^{2} n^{2} - 190.4 m^{2} n - 285.6 {mn}^{2} - 237.6 mn + 27.2 m^{2} + 61.2 n^{2} + 20.4 m + 69.2 n - 13.6 .$ □

7 Numerical comparison of coindices

In this section, we compare the values of eleven coindices of anthracene. The comparison of values is shown to understand the behaviour of the coincides for the mentioned compound. It gives the researchers a clear picture about the trend of the indices considered for further studies in the same area.

The comparison of above said coindices is made for Anthracene and is tabulated through the Tables 3 for the values of n & m = n from 1 to 10. It is clear that all the coindices are varying upwards as n & m = n approaches greater value.

Table 2

Numerical comparison of coindices of linear [n] Anthracene for n = 1 to 10

n	$\bar{M_{1}}$	$\bar{M_{2}}$		$\bar{{ReZG}_{3}}$	$\bar{F}$	$\bar{R}$	$\bar{RR}$	$\bar{SDD}$	$\bar{H}$	$\bar{I}$	$\bar{A}$
1	340	384	775.528	1800	800	33.899	168.384	155.344	33.634	209.4	613.564
2	1664	2010	3103.6	10116	4192	146.559	823.314	716.724	145.133	659.4	2948.7
3	3996	4932	6983.8	25344	10272	337.747	1977.4	1686.1	334.364	1356.6	7068.8
4	7336	9150	12416	47484	19040	607.463	3630.6	3063.6	601.328	2301	12974
5	11684	14664	19401	76536	30496	955.708	5783	4849.1	946.023	3492.6	20664
6	17040	21474	27937	112500	44640	1382.5	8434.6	7042.6	1368.5	4931.4	30139
7	23404	29580	38026	155376	61472	1887.8	11585	9644.1	1868.6	6617.4	41400
8	30776	38982	49666	205164	80992	2471.6	15235	12654	2446.5	8550.6	54445
9	39156	49680	62859	261864	103200	3134	19384	16071	3102.1	10731	69275
10	48544	61674	77604	325476	128096	3874.8	24032	19897	3835.5	13159	85890

Table 3

Numerical comparison of coindices of multiple Anthracene for m = n = 1 to 10

m = n	$\bar{M_{1}}$	$\bar{M_{2}}$		$\bar{{ReZG}_{3}}$	$\bar{F}$	$\bar{R}$	$\bar{RR}$	$\bar{SDD}$	$\bar{H}$	$\bar{I}$	$\bar{A}$
1	316	348	15.1	1422	728	44	164	155	32.3	85	568
2	7688	10124	225.9	54464	20944	660	3840	3064	563	1916	13640
3	42444	58556	1071	329438	119952	3153	21150	15920	2823	10530	76550
4	139024	196044	3250	1123704	399392	9614	69270	50630	8829	34510	253500
5	345980	494156	7725	2861630	1003240	22915	172440	123800	21380	85930	635500
6	725976	1045628	15718	6094592	2117808	46710	361900	256730	44060	180430	1340200
7	1355788	1964364	28712	11500974	3971744	85440	676100	475430	81240	337150	2512400
8	2326304	3385436	48450	19886168	6836032	144340	1160400	810600	138060	578780	4323500
9	3742524	5465084	76949	32182574	11023992	229390	1867200	1297700	220470	931500	6971700
10	5723560	8380716	116470	49449600	16891280	347400	2856100	1976900	335180	1425100	10682000

Gutman and Trinajstic introduced the first and the second Zagreb coindices in the studies of total π-electron energy. These measures provide branching of the carbon atom skeleton in determining some properties and bounds of molecular graphs. Forgotten coindex is defined by De et. al., in determining the correlation between log P (logarithm of octanol-water partition coefficient) and the F-coindex values of octane isomers. F-coindex is used to predict the logP values with high precision. Many chemical characteristics of molecules are correlated using Randic indices. Various coindices are studied in this article which can be defined in a similar way to that of its classical degree counter parts.

Conclusion

This work focuses on the study of paranapthalene/Anthracene which has varied applications in pharmaceutical and dye industries. Anthracene plays a crucial role in anti carcinogenicity and dermatological infections. It is used as antibiotics in cancer treatment. The two structural forms of Anthracene considered are linear and multiple for which eleven degree based topological coindices are derived using CoM-polynomial. Numerical comparison of the above said coindices are computed and the observations are summarized. 3D-plots of CoM-polynomials of both the forms of Anthracene are illustrated. As green oil (Anthracene) has numerous applications, this study would be useful for researchers/pharmacists/chemists in further studies about the compound.

Conflict of interest

There are no conflicts of interest among the authors.

Acknowledgment

This work was supported and funded by Gachon University (No. 202110260001).

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