Further study of eccentricity based indices for benzenoid hourglass network

Abstract

Topological Indices are the mathematical estimate related to atomic graph that corresponds biological structure with several real properties and chemical activities. These indices are invariant of graph under graph isomorphism. If top($h_1$) and top($h_2$) denotes topological index $h_1$ and $h_2$ respectively then $h_1$ approximately equal $h_2$ which implies that top($h_1$) = top($h_2$). In biochemistry, chemical science, nano-medicine, biotechnology and many other science's distance based and eccentricity-connectivity(EC) based topological invariants of a network are beneficial in the study of structure-property relationships and structure-activity relationships. These indices help the chemist and pharmacist to overcome the shortage of laboratory and equipment. In this paper we calculate the formulas of eccentricity-connectivity descriptor(ECD) and their related polynomials, total eccentricity-connectivity(TEC) polynomial, augmented eccentricity-connectivity(AEC) descriptor and further the modified eccentricity-connectivity(MEC) descriptor with their related polynomials for hourglass benzenoid network.

1. Introduction

In 1825 Farady defined benzene. He discovered a natural hydrocarbon element of crude oil. Later on then a German chemist August kekule conceit the ring network of benzene in 1865. Many organic synthetic fusion have coil of 6 carbon atoms called benzene rings. In 19th century he also accused to have blocked out of this structure. Its molecules are actinic with network $C_6{H}_6$ , it has molar mass of just over 78 g/mol. It will be like a liquid at room temperature. It have no color. It is soluble in water and also having melting point of 5.33 C. It is one of the wildly used chemical in the world United State rank. It is founded in top 20 of chemicals. It is mainly used in its original form, but almost 80% used in composing for creating other chemicals, namely alkylbenzene, cyclohexane, nitrobenzene, cumene and ethylbenzene etc. Benzene also convert longer molecules into smaller easily and safer atomic parts. Many products in daily life have benzene in few degree like paints, furniture wax, glues, detergents, cigarette smoke, gas stations and many others. A little bit benzene is also found in food such as vegetables, fruits, dairy products, nuts, fish and eggs.

In this paper, we discussed about the different structures having benzene also we give a view about its formation. Such as benzenoid ring is a class of chemical compound with only benzene ring, naphthalene, aniline and phenol etc are examples of benzenoid. Naphthalene is classified as benzenoid polycyclic aromatic - hydrocarbon (PAH). Its molar mass 128.174 with strong odor of coal tar which is catchable at attentiveness at its low point 0.08 ppm by mass. It is appeared in white solid quartz. Aniline is organic compound; it is really soluble in water with molar mass 9.313 g/mol. Its density, boiling point and melting point are 1.02, 184.1C and - 6.3C degree respectively. Brunvol et al. (1987) did classification of benzenoid hydrocarbon [1], Dias in 1996, worked on hydrocarbon to fullerene carbon. A topological index is a numerical parameter, which is invariant under graph operation - isomorphism. In 2000, Zigert and Gutman found Winner index of benzenoid [2, 3]. In 2010, Ghorbani and Ghazi computed some indices on triangular benzenoid [4]. Farahani in 2012, found cricuncoronene series of benzenoid [5]. In 2019, Imran et al. also computed topological properties of benzenoid [6]. Liu et al. defined many indices in [7]. In 2019, Tanveer et al. worked on M. polynomials [8], and in 2021 Yu et al. also discussed topological properties in [9], some other indices are computed in [10].

The study of mathematical theory with properties and applications of graph is known as graph theory. It is a representation of curve in plane with nodes and these nodes are attached together with edges. There are many types of graphs such as, regular graph, complete graph, cyclic graph, simple graph, Hamiltonian graph, finite graph, loop graph, infinite graph, connected graph, disconnected graph, and multi graph etc. A graph where each pair of nodes is connected by an edge is known to be a complete graph. If a graph having at least one non adjacent pair of nodes is called incomplete graph. A simple graph is one with no loops and having not parallel edges. In regular graph all vertices have same degree. In Hamiltonian graph each node is observed at once except beginning node, with closed walk. Here we consider simple and connected graph meaning there is no loop or multiple edges and there exist a path between every two pairs of vertices.

Let $H$ be a hourglass benzenoid simple chemical structure with a vertex set $V\left(H\right)$ and an edge set $E\left(H\right)$. A Benzenoid Hourglass network has three sections, beginning from broad rang, then move to slim and end with expand edges. In this structure one mid benzene particle is fix and the molecules are increased at the same time at both sides, above and below. It is a simple chemical structure with vertices collection denoted by V (Bhn) and edges collection denoted by E(Bhn). The 1st and 2nd Zagreb indices were expressed to be used as an important part of the study of medicine, medication subatomic structures which are vastly beneficial for narcotic and curing advisers to manage the biological effect and fabrication of the new medicines. These capabilities are basically tested in developing countries where vast amount of cash is entailed to manage the price of the applicable chemical indicators and apparatus. The researchers lately discovered that topological indices are used for two-dimensional RNA networks, drug protein. Further, to inscribe protein surface knowledge and for protein interaction structure (PINs).

We calculate here the formulas of ECD with its linked polynomials in variable z, the TEC polynomial, the AEC descriptor and the MEC descriptor with their associated polynomials in variable z, for hourglass benzenoid network.

The graph $H$ has the eccentric-connectivity index $\xi \left(H\right)$, which is shown in Ref. [11].

$$\xi \left(H\right)=\sum _{\ddot{v}\in V\left(H\right)}\epsilon \left(\ddot{v}\right)d_{\ddot{v}}$$ (1)

The EC polynomial is described as [12].

$$ECP\left(H,z\right)=\sum _{\ddot{v}\in V\left(H\right)}{d}_{\ddot{v}}{z}^{\epsilon \left(\ddot{v}\right)}$$ (2)

The TEC polynomial of graph $H$ is [12].

$$TECP\left(H,z\right)=\sum _{\ddot{v}\in V\left(H\right)}{z}^{\epsilon \left(\ddot{v}\right)}$$ (3)

Gupta [13] defined the AEC descriptor of $H$ as.

$$A_{\epsilon }\left(H\right)=\sum _{\ddot{v}\in V\left(H\right)}\frac{M\left(\ddot{v}\right)}{\epsilon \left(\ddot{v}\right)}$$ (4)

Where $M\left(\ddot{v}\right)$ stands for the product of degrees of all neighborhood vertices of vertex $\ddot{v}$. Many other topological indices have been discussed in [14,15]. The modified versions of the eccentric-connectivity index is shown as.

$$\mathrm{\Lambda }\left(H\right)=\sum _{\ddot{v}\in V\left(H\right)}{S}_{\ddot{v}}\epsilon \left(\ddot{v}\right)$$ (5)

The MEC polynomial described in [16]. Further, some other topological indices and their polynomial are studied in [17].

$$MECP\left(H,z\right)=\sum _{\ddot{v}\in V\left(H\right)}{S}_{\ddot{v}}{z}^{\epsilon \left(\ddot{v}\right)}$$ (6)

2. Results and discussion

In this section we discuss about the mains results for hourglass benzenoid structure which has $V_n\left(H\right)=2\left(n^2+4n-2\right)$ and $E_n\left(H\right)=3n^2+9n-6$, complete number of vertices and edges respectively. In this network there are $3n-2$ types of vertices on neighborhood degree based and $4n-1$ types of vertices based on $M\left(\ddot{v}\right)$ and $S\left(\ddot{v}\right)$ for $n>2$ where $n\in N$. We can construct tables of this network for computing different topological indices [18,19] and also their polynomials, $M\left(\ddot{v}\right)$ in the product of degrees of all neighborhood vertices of vertex $\ddot{v}$, $S\left(\ddot{v}\right)$ is the sum of degrees of all neighborhood vertices of vertex $\ddot{v}$, and their respective polynomials. Table [1] shows vertex degree based calculations with respect to eccentricity, and their count respectively. In Table [2] $M\left(\ddot{v}\right)$ and $S\left(\ddot{v}\right)$ based calculation are included with respect to eccentricity, and their count.

Table 1.

Vertex degree and Eccentricity with their count.

$\epsilon \left(\ddot{v}\right)$	$D\left(\ddot{v}\right)$	count
4 $n$-1	2	2($n$ +2)
4 $n$-2	2	4
4 $n$-2	3	2($n$-1)
4 $n$-3	3	2($n$ +1)
4 $n$-4	2	4
4 $n$-4	3	2($n$-2)
4 $n$-5	3	2 $n$
. . .	. . .	. . .
2 $n$ +2	2	4
2 $n$ +2	3	2
2 $n$ +1	3	6

Table 2.

Vertex M($\ddot{v}$),S $\left(\ddot{v}\right)$ and Eccentricity with their count.

$\epsilon \left(\ddot{v}\right)$	$M\left(\ddot{v}\right)$	$S\left(\ddot{v}\right)$	count
4 $n$-1	9	6	2($n$-2)
4 $n$-1	6	5	4
4 $n$-1	4	4	4
4 $n$-2	12	7	2($n$-1)
4 $n$-2	6	5	4
4 $n$-3	27	9	2($n$-1)
4 $n$-3	12	7	4
4 $n$-4	27	9	2($n$-2)
4 $n$-4	9	6	4
4 $n$-5	27	9	2($n$-2)
4 $n$-5	12	7	4
. . .	. . .	. . .	. . .
2 $n$ +2	9	6	4
2 $n$ +1	27	9	2
2 $n$ +1	18	8	4

Theorem $\mathbf{1}$: Let $H$ be the graph of Hourglass Benzenoid Network, then the eccentric-connectivity index $\xi \left(H\right)$ of $H$ is

$$\xi \left(H\right)=\sum \left(4n+8\right)\left(4n-1\right)+8\sum _p\left(4n-p\right)$$

$$+6\sum _{p,s}\left(4n-p\right)\left(n-s\right)+6\sum _{q,r}\left(4n-q\right)\left(n-r\right)$$

where

$$p=\mathrm{2,4,6},...2\left(n-1\right),q=\mathrm{3,5,7},...\left(2\right)n-1,s=1,2,3,...n-1,r=-1,0,1,...n-3$$

Proof. Let $H$ be the graph of Hourglass Benzenoid Network. Now we can calculate the eccentric-connectivity index by using information from Table [1] in Equation [1].

$$\xi \left(H\right)=\sum _{\ddot{v}\in V\left(H\right)}\epsilon \left(\ddot{v}\right)d_{\ddot{v}}$$

$$=\sum \left(4n-1\right)\left(2\right)+\sum \left(4n-2\right)\left(2\right)+\sum \left(4n-2\right)\left(3\right)$$

$$+\sum \left(4n-3\right)\left(3\right)+\sum \left(4n-4\right)\left(2\right)+\sum \left(4n-4\right)\left(3\right)$$

$$+\sum \left(4n-5\right)\left(3\right)+\sum \left(4n-6\right)\left(2\right)+...+\sum \left(2n+1\right)\left(3\right)$$

$$=4\left(n+2\right)\left(4n-1\right)+8\left(4n-2\right)+6\left(n-1\right)\left(4n-2\right)$$

$$+6\left(n+1\right)\left(4n-3\right)+8\left(4n-4\right)+6\left(n-2\right)\left(4n-4\right)$$

$$+6n\left(4n-5\right)+8\left(4n-6\right)+...+18\left(2n+1\right)$$

$$=\sum \left(4n+8\right)\left(4n-1\right)+8\sum _p\left(4n-p\right)$$

$$+6\sum _{p,s}\left(4n-p\right)\left(n-s\right)+6\sum _{q,r}\left(4n-q\right)\left(n-r\right)$$

where.

$p$ = $\mathrm{2,4,6},...2\left(n-1\right)\text{,}$ $q$ = $\mathrm{3,5,7},...\left(2\right)n-1\text{,}$ $s$ = $1,2,3,...n-1\text{,}$ $r$ = $-1,0,1,...n-3$.

Theorem $\mathbf{2}$: Let $H$ be the graph of Hourglass Benzenoid Network, then the eccentric-connectivity polynomial $ECP\left(H,z\right)$ of $H$ is,

$$ECP\left(H,z\right)=\sum \left(4n+8\right)z^{4n-1}+8\sum _p{z}^{4n-p}$$

$$+6\sum _{p,s}{z}^{-\mathrm{p}+4n}\left(-\mathrm{s}+n\right)+6\sum _{r,q}{z}^{-\mathrm{q}+4n}\left(-\mathrm{r}+n\right)$$

where.

Proof. Let $H$ be the graph of Hourglass Benzenoid Network. Now we can calculate the eccentric-connectivity polynomial in variable z by using information from Table [1] in Equation [2].

$$ECP\left(H,z\right)=\sum _{\ddot{v}\in V\left(H\right)}{d}_{\ddot{v}}{z}^{\epsilon \left(\ddot{v}\right)}$$

$$=\sum \left(2\right)z^{4n-1}+\sum 2z^{4n-2}+\sum 3z^{4n-2}$$

$$+\sum 3z^{4n-3}+\sum 2z^{4n-4}+\sum 3z^{4n-4}$$

$$+\sum 3z^{4n-5}+\sum 2z^{4n-6}+...+\sum 3z^{2n+1}$$

$$=4\left(n+2\right)z^{4n-1}+8z^{4n-2}+6\left(n-1\right)z^{4n-2}$$

$$+6\left(n+1\right)z^{4n-3}+8z^{4n-4}+6\left(n-2\right)z^{4n-4}$$

$$+6n{z}^{4n-5}+8z^{4n-6}+...+18z^{2n+1}$$

$$=\sum \left(4n+8\right)z^{4n-1}+8\sum _p{z}^{4n-p}$$

$$+6\sum _{p,s}{z}^{-\mathrm{p}+4n}\left(-\mathrm{s}+n\right)+6\sum _{r,q}{z}^{-\mathrm{q}+4n}\left(-\mathrm{r}+n\right)$$

where

$$p=\mathrm{2,4,6},...2\left(n-1\right),q=\mathrm{3,5,7},...2n-1,s=\mathrm{1,2,3},...n-1,r=-\mathrm{1,0,1},...n-3$$

Theorem $\mathbf{3}$: Let $H$ be the graph of Hourglass Benzenoid Network, then the total eccentric-connectivity polynomial $TECP\left(H,z\right)$ of $H$ is,

$$TECP\left(H,z\right)=2\sum _{r,s}\left(n-r\right)z^{4n-s}$$

where $r$ = $-2,-1,-\mathrm{1,0,0},...n-3\text{,}$ $s$ = $\mathrm{1,2,3},...2n-1$.

Proof. Let $H$ be the graph of Hourglass Benzenoid Network. Now we can calculate the polynomial version of the total eccentricity index by using information from Table [1] in Equation [3].

$$TECP\left(H,z\right)=\sum _{\ddot{v}\in V\left(H\right)}{z}^{\epsilon \left(\ddot{v}\right)}$$

$$=\sum z^{4n-1}+\sum z^{4n-2}+\sum z^{4n-3}$$

$$+\sum z^{4n-4}+\sum z^{4n-5}+...+\sum z^{2n+1}$$

$$=2\left(2+n\right)z^{4n-1}+2\left(1+n\right)z^{-2+4n}+2\left(1+n\right)z^{-3+4n}$$

$$+2n{z}^{4-n4}+2n{z}^{4n-5}+...+\sum z^{2n+1}$$

$$=2\sum _{r,s}\left(n-r\right)z^{4n-s}$$

where $r$ = $-2,-1,-\mathrm{1,0,0},...n-3\text{,}$ $s$ = $\mathrm{1,2,3},...2n-1$.

Theorem $\mathbf{4}$: Let $H$ be the graph of Hourglass Benzenoid Network, then the augmented eccentric-connectivity index $A_{\epsilon }\left(H\right)$ of $H$ is,

$$A_{\epsilon }\left(H\right)=2\left[\frac{9n+2}{4n-1}\right]+\frac{24n}{4n-2}+4\sum _{i,j}\left[\frac{12}{4n-i}+\frac{9}{4n-j}\right]$$

$$+2\sum _{k,l}\left(n-k\right)\left[\frac{27}{4n-l}\right]+4\left[\frac{18}{2n+1}\right]$$

where $i$ = $\mathrm{3,5,7},...2n-3$, $j$ = $\mathrm{4,6,8},...2\left(-1+n\right)\text{,}$ $k$ = $1,2,2,3,3,...,n-1\text{,}$.

$$l=\mathrm{3,4,5},..,2n-1$$

Proof. Let $H$ be the graph of Hourglass Benzenoid Network. Now we can calculate the augmented eccentric-connectivity index of $H$ by using information from Table [2] in Equation $\left(4\right)$.

$$A_{\epsilon }\left(H\right)=\sum _{\ddot{v}\in V\left(H\right)}\frac{M\left(\ddot{v}\right)}{\epsilon \left(\ddot{v}\right)}$$

$$=\sum \frac{9}{4n-1}+\sum \frac{6}{4n-1}+\sum \frac{4}{4n-1}$$

$$+\sum \frac{12}{4n-2}+\sum \frac{6}{4n-2}+\sum \frac{27}{4n-3}$$

$$+\sum \frac{12}{4n-3}+\sum \frac{27}{4n-4}+\sum \frac{9}{4n-4}$$

$$+\sum \frac{27}{4n-5}+\sum \frac{12}{4n-5}+...+\sum \frac{18}{2n+1}$$

$$=2\left(n-2\right)\frac{9}{4n-1}+4\frac{6}{4n-1}+4\frac{4}{4n-1}$$

$$+2\left(n-1\right)\frac{12}{4n-2}+4\frac{6}{4n-2}+2\left(n-1\right)\frac{27}{4n-3}$$

$$+4\frac{12}{4n-3}+2\left(n-2\right)\frac{27}{4n-4}+4\frac{9}{4n-4}$$

$$+2\left(n-2\right)\frac{27}{4n-5}+4\frac{12}{4n-5}+...+4\frac{18}{2n+1}$$

$$=2\left[\frac{9n+2}{4n-1}\right]+\frac{24n}{4n-2}+4\sum _{i,j}\left[\frac{12}{4n-i}+\frac{9}{4n-j}\right]$$

$$+2\sum _{k,l}\left(n-k\right)\left[\frac{27}{4n-l}\right]+4\left[\frac{18}{2n+1}\right]$$

where $i$ = $\mathrm{3,5,7},...2n-3$, $j$ = $\mathrm{4,6,8},...2\left(-1+n\right)\text{,}$ $k$ = $1,2,2,3,3,...,n-1\text{,}$.

$$l=\mathrm{3,4,5},..,2n-1=$$

Theorem $\mathbf{5}$: Let $H$ be the graph of Hourglass Benzenoid Network, then the modified version of the eccentric-connectivity index $\mathrm{\Lambda }\left(H\right)$ of $H$ is,

$$\mathrm{\Lambda }\left(H\right)=8\left(13n^2+4n-3\right)+28\sum _{i}4n-i+24\sum _{j}4n-j$$

$$+18\sum _{k,l}\left(n-k\right)\left(4n-l\right)+32\left(2n+1\right)$$

where $i$ = $\mathrm{3,5,7},...2n-3$, $j$ = $\mathrm{4,6,8},...2\left(-1+n\right)\text{,}$ $k$ = $1,2,2,3,3,....,n-1\text{,}$.

$$l=\mathrm{3,4,5},..,2n-1$$

Proof. Let $H$ be the graph of Hourglass Benzenoid Network. Now we can calculate the modified version of eccentric-connectivity index by using information from Table [2] in Equation [5].

$$\mathrm{\Lambda }\left(H\right)=\sum _{\ddot{v}\in V\left(H\right)}{S}_{\ddot{v}}\epsilon \left(\ddot{v}\right)$$

$$=\sum 6\left(4n-1\right)+\sum 5\left(4n-1\right)+\sum 4\left(4n-1\right)+\sum 7\left(4n-2\right)$$

$$+\sum 5\left(4n-2\right)+\sum 9\left(4n-3\right)+\sum 7\left(4n-3\right)+\sum 9\left(4n-4\right)$$

$$+\sum 6\left(4n-4\right)+\sum 9\left(4n-5\right)+\sum 7\left(4n-5\right)+...+\sum 8\left(2n+1\right)$$

$$=12\left(n-2\right)\left(4n-1\right)+20\left(4n-1\right)+16\left(4n-1\right)+14\left(n-1\right)\left(4n-2\right)$$

$$+20\left(4n-2\right)+18\left(n-1\right)\left(4n-3\right)+28\left(4n-3\right)+18\left(n-2\right)\left(4n-4\right)$$

$$+24\left(4n-4\right)+18\left(n-2\right)\left(4n-5\right)+28\left(4n-5\right)+...+32\left(2n+1\right)$$

$$=8\left(13n^2+4n-3\right)+28\sum _{i}4n-i+24\sum _{j}4n-j$$

$$+18\sum _{k,l}\left(n-k\right)\left(4n-l\right)+32\left(2n+1\right)$$

where $i$ = $\mathrm{3,5,7},...2n-3$, $j$ = $\mathrm{4,6,8},...2\left(n-1\right)\text{,}$ $k$ = $\mathrm{1,2,2,3,3},...n-1\text{,}$.

$$l=\mathrm{3,4,5},..,2n-1$$

Theorem $\mathbf{6}$: Let $H$ be the graph of Hourglass Benzenoid Network, then the modified version of the eccentric-connectivity polynomial MECP(H,z) of $H$ is,

$$MECP\left(H,z\right)=12z^{4n-1}\left(n+1\right)+2z^{4n-2}\left(7n+3\right)+18\sum _{k,l}\left(n-k\right)z^{4n-l}$$

$$+28\sum _i{z}^{4n-i}+24\sum _j{z}^{4n-j}+32z^{2n+1}$$

where $i$ = $\mathrm{3,5,7},...2n-3$, $j$ = $\mathrm{4,6,8},...2\left(-1+n\right)\text{,}$ $k$ = $1,2,2,3,3,...,n-1\text{,}$.

$$l=\mathrm{3,4,5},..,2n-1$$

Proof. Let $H$ be the graph of Hourglass Benzenoid Network. Now we can calculate the modified version of the eccentric-connectivity polynomial by using information from Table [2] in Equation [6].

$$MECP\left(H,z\right)=\sum _{\ddot{v}\in V\left(H\right)}{S}_{\ddot{v}}{z}^{\epsilon \left(\ddot{v}\right)}$$

$$=\sum 6z^{4n-1}+\sum 5z^{4n-1}+\sum 4z^{4n-1}+\sum 7z^{4n-2}$$

$$+\sum 5z^{4n-2}+\sum 9z^{4n-3}+\sum 7z^{4n-3}+\sum 9z^{4n-4}$$

$$+\sum 6z^{4n-4}+\sum 9z^{4n-5}+\sum 7z^{4n-5}+...+\sum 8z^{2n+1}$$

$$=12\left(n-2\right)z^{4n-1}+20z^{4n-1}+16z^{4n-1}+14\left(n-1\right)z^{4n-2}$$

$$+20z^{4n-2}+18\left(n-1\right)z^{4n-3}+28z^{4n-3}+18\left(n-2\right)z^{4n-4}$$

$$+24z^{4n-4}+18\left(n-2\right)z^{4n-5}+28z^{4n-5}+...+32z^{2n+1}$$

$$=12z^{4n-1}\left(n+1\right)+2z^{4n-2}\left(7n+3\right)+18\sum _{k,l}\left(n-k\right)z^{4n-l}$$

$$+28\sum _i{z}^{4n-i}+24\sum _j{z}^{4n-j}+32z^{2n+1}$$

where $i$ = $\mathrm{3,5,7},...2n-3$, $j$ = $\mathrm{4,6,8},...2\left(-1+n\right)\text{,}$ $k$ = $1,2,2,3,3,...,n-1\text{,}$.

$$l=\mathrm{3,4,5},..,2n-1$$

3. Comparison and discussion

We have calculated the numerical values of all indices of hourglass benzenoid structure for various values of $n$, in Table [3]. We can easily observe that the graphs are changing their behavior according to the value of $n$ as the value of $n$ increases the indices are also increasing at the same time. The graphs of indices are shown in Fig. 1, Fig. 2, Fig. 3.

Table 3.

For topological indices of H Eccentric-Connectivity Index Augmented Eccentric-Connectivity Index.

$n$	${\sum }_{\ddot{v}\in V\left(H\right)}\epsilon \left(\ddot{v}\right)d_{\ddot{v}}$	${\sum }_{\ddot{v}\in V\left(H\right)}\frac{M\left(\ddot{v}\right)}{\epsilon \left(\ddot{v}\right)}$	${\sum }_{\ddot{v}\in V\left(H\right)}{S}_{\ddot{v}}\epsilon \left(\ddot{v}\right)$
3	874	59.05	270
4	1920	77.25	5128
5	3544	94.75	8971
6	5866	111.92	15858
7	9084	128.91	25066

Fig. 1.

$\xi \left(H\right)$.

Fig. 2.

$A_{\epsilon }\left(H\right)$.

Fig. 3.

$\mathrm{\Lambda }\left(H\right)$.

Modified Eccentric-Connectivity Index.

4. Conclusion

Benzenoid has many industrial uses, which motivated us to study and calculate its topological properties. In this paper we calculated the vertex eccentricity labelled indices for the chemical network of hourglass benzenoid by using the method of induction we defined edge partition. The closed formulas of eccentric-connectivity index $\xi \left(H\right)$, the polynomial version of eccentric-connectivity $ECP\left(H,z\right)$, the TEC polynomial $TECP\left(H,z\right)\text{,}$ the AEC descriptor $A_{\epsilon }\left(H\right)$, the MEC descriptor with their polynomials are stated for our structure. These numerical parameters are useful to perceive physical features, chemical reaction and biological activity of a variety of molecular structures. The results can also display vital role in the recognition of the significance and utilization of concealed benzenoid hourglass network in the chemical and pharmaceutical industry. The reader is encouraged to study different molecular structures in future for $M\left(\ddot{v}\right)$, $S\left(\ddot{v}\right)$, $d\left(\ddot{v}\right)$, edge partition in combination with eccentricity based indices.

Funding statement

No specific funding available.

Author contribution statement

Hifza Iqbal: Conceived and designed the experiments.

Muhammad Haroon Aftab: Analyzed and interpreted the data.

Ali Akgul, Zeeshan Saleem Mufti, Mustafa Bayram: Contributed reagents, materials, analysis tools or data.

Iram Yaqoob: Wrote the paper; performed the experiments.

Muhammad Bilal Riaz: Analyzed and interpreted the data; Conceived and designed the experiments.

Data availability statement

Data will be made available on request.

Declaration of competing interest

On the behalf of all coauthors, I declare that all authors have no conflict of interest. Furthermore, our paper is original and has not been submitted to any other journals. We would like to publish in your journal.