Topological properties of reverse-degree-based indices for sodalite materials network

Ali N.A. Koam; Moin A. Ansari; Azeem Haider; Ali Ahmad; Muhammad Azeem

doi:10.1016/j.arabjc.2022.104160

View/Download PDF

Buy Reprints

PDF

Translate this page into:

Original article

10 2022

:15;

104160

doi:

10.1016/j.arabjc.2022.104160

Topological properties of reverse-degree-based indices for sodalite materials network

Ali N.A. Koam^{^a}, Moin A. Ansari^{^a,⁎}, Azeem Haider^{^a}, Ali Ahmad^{^b}, Muhammad Azeem^{^c}

a Department of Mathametics, College of Science, Jazan University, Jazan, Saudi Arabia

b College of Compiuter Science, Jazan University, Jazan, Saudi Arabia

c Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Riphah International University Lahore, Pakistan

⁎Corresponding author at: Department of Mathematics, College of Science, Jazan University, Jazan, Saudi Arabia. maansari@jazanu.edu.sa (Moin A. Ansari),

Received: 2022-3-1, Accepted: 2022-7-27,

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

Natural zeolites are frequently referred to as macromolecular sieves. Because of relatively inexpensive cost of installation, zeolites networks are highly trendy chemical networks. Amongst the most researched types of zeolite networks is the sodalite network. It contributes to the elimination of greenhouse gases. To investigate the above widely tested, we employ an authentic mathematical tool known as topological descriptors or index, which displays some physical and chemical properties numerically. To fully comprehend the structure, we compare different legitimate properties via topological indices, concluding in the form of figurative comparisons. In particularly, we measure some reverse degree-based indices for the sodalite materials network and put some light on the behavior of this network, in the form of topological descriptors and numerically designed equation of such network. This research is novel in terms of no one studied reverse-degree-based topological indices for this chosen structure.

Keywords

Sodalite materials network

Topological indices

Reverse degree

Natural zeolites

Show Related Articles from PubMed

1 Introduction

Macromolecular sieves, typically described as natural zeolites, are being examined and investigated deeper (Neeraj et al., 2002; Karagiaridi et al., 2012; Eddaoudi et al., 2002 ). Zeolite’s design atomic capture enables it noteworthy and broadly useful. Natural zeolites are helpful in high volume mineral usage because of their cheaper cost, according to Catlow (1992). Zeolites are classified into distinct classes based on their complexity and size. Stilbite, tschernichite, modernite, faujasite, sodalite, chabazite, for example, the one most beneficial in industrial purposes are molecular sieves, or natural zeolites, according to Dyer (1988). From those of the zeolite structures mentioned previously, synthesised compounds and mineral crystal structures with sodalite frameworks are the most extensively investigated substances (Jaeger, 1929; Pauling, 1930). The authors of Prabhu et al. (2020), Torrens and Castellano (2006), Torrens and Castellano (2019), discussed for certain chemical and physical characteristics of zeolite type structures particularly sodalite when it comes to the form of topological indices.

Sodalites already have best thermodynamic consistency and are regarded as one of the highest rated structures across all zeolites (Barth, 1932; Hassan and Grundy, 1984). Because of their crystallographic positions, sodalites are also significant, according to Taylor (1967). The unit cell of the sodalite structure is comprised of two cages. Each cage’s fundamental structure consists of six and four membered rings. Two parallel cages also share these rings. For capture molecules, cavities in sodalite are built by using a custom-made mixture, Arockiaraj et al. (2021). Tap molecules, such as water and ${CO}_{2}$ , already have capacity to desorb and adsorb molecules without causing damage to the crystal structures. As a result, Arockiaraj et al. (2021) for the use of zeolites in the removal of greenhouse gases and water. See Jiri et al. (2007), Keeffe (1991), for more research on zeolites, sodalite, and their practical uses.

Topological descriptor is a mechanism that, especially quantitative form, illustrates the physical, chemical and biological properties of a provided chemical graph in domain in a comprehensive way. The following definitions include few more topological indices that are both fascinating and usually praised, as well as their introducers. According to Ajmal et al. (2016), the researchers investigated polyhex and prism structures in terms of various indices. In Munir et al. (2016), titania nanotubes are analysed in terms of various topological descriptors. Authors of Munir et al. (2016), investigated the structure of dendrimers of nanostar and structure of nanotubes provided by polyhex. We refer to Nadeem et al. (2021), Nadeem et al. (2021), for a few intriguing discussion of chemical structures and their graphical features and diverse forms. There are several methods and different kinds of computing topological indices for different structures, such as in Shabbir et al. (2020), measured the edge-degree-based topological indices, Nadeem et al. (2021), Hong et al. (2020), metal organic structure is found in terms of topological indices, graphs resulted in the line graph operation and their results on the topological descriptors are available in Nadeem et al. (2016), Nadeem et al. (2019). Relative topics of applied graph theory in chemistry or using the chemical structures are found in Koam et al. (2022), Koam et al. (2022), Hameed et al. (2022), Alshehri et al. (2022), Koam et al. (2022), Azeem et al. (2022). Certain new research which are available at Liu et al. (2020), and also for further information on latest networks and their topological descriptors.

2 Preliminaries

To determine these primary outcomes, we must first specify certain fundamentals, let $χ$ be a corresponded graph of a network having the bonds E, and atoms labels as V. Where the notion $| V |$ symbolized as the total numbers of nodes or atoms and $| E |$ contributed as edge or bonds count. Let the notion $d_{θ}$ contributed as degree of a vertex $θ$ and defined as the count of edges attached to this vertex. The researcher from Kulli (2018), introduced another concept labeled as reverse-degree and symbolized as $r_{θ}$ . This newly developed notion is mathematically stated as $r_{θ} = 1 + Δ - d_{θ}$ , in which the parameter $Δ$ denotes the utmost degree of a vertex of a network. A notion $E_{r_{θ}, r_{ϑ}}$ labeled as an edge partition of a network designed on the concept of reverse-degree of end vertices of an edge $θ ϑ \in E$ . Similarly, the notion $| E_{r_{θ}, r_{ϑ}} |$ contributed as its cardinality.

A general topological descriptor $I (χ)$ , based on the reverse-degree of a vertex stated as Mufti et al. (2022):

(1)

I (χ) = \sum_{θ ϑ \in E (χ)} Π (r_{θ}, r_{ϑ}) .

Given by Zhao et al. (2021), Wang et al. (2020)

(2)

Let Π (r_{θ}, r_{ϑ}) = {(r_{θ} \times r_{ϑ})}^{α} . Then I (χ) symbolises the general reverse Randic'

\begin{matrix} descriptor (r R_{α} (χ)) for α = \frac{1}{2}, - \frac{1}{2}, 1, - 1 . If α = 1, then it is known \\ as the second reverse Zagreb descriptor (r M_{2} (χ)) . \end{matrix}

Found in Koam et al. (2021)

(3)

Let Π (r_{θ}, r_{ϑ}) = \sqrt{\frac{r_{θ} + r_{ϑ} - 2}{r_{θ} \times r_{ϑ}}} . Then I (χ) symbolises the reverse atom-bond

connectivity descriptor (r ABC (χ)) .

Described in Jayanna and Swamy (2021), Gao et al. (2018)

(4)

Let Π (r_{θ}, r_{ϑ}) = \frac{2 \sqrt{r_{θ} \times r_{ϑ}}}{r_{θ} + r_{ϑ}} . Then I (χ) symbolises the reverse geometric -

(5)

\begin{matrix} arithmetic descriptor (r GA (χ)) . \\ Let Π (r_{θ}, r_{ϑ}) = & (r_{θ} + r_{ϑ}) . Then I (χ) symbolises the first reverse Zagreb \end{matrix}

descriptor (r M_{1} (χ)) .

Found in Wang et al. (2020), Poojary et al. (2022)

(6)

Let Π (r_{θ}, r_{ϑ}) = {(r_{θ} + r_{ϑ})}^{2} . Then I (χ) symbolises the reverse hyper

(7)

\begin{matrix} Zagreb descriptor (r HM (χ)) . \\ Let Π (r_{θ}, r_{ϑ}) = & ({(r_{θ})}^{2} + {(r_{ϑ})}^{2}) . Then I (χ) symbolises the reverse forgotten \end{matrix}

descriptor (r F (χ)) .

Formulated in Wang et al. (2020)

(8)

Let Π (r_{θ}, r_{ϑ}) = {(r_{θ} + r_{ϑ})}^{α} {(r_{θ} \times r_{ϑ})}^{β} . Then I (χ) symbolises

\begin{matrix} the first reverse redefined descriptor (r {ReZ}_{1} (χ)) for α = 1, β = - 1; \\ the second reverse redefined descriptor (r {ReZ}_{2} (χ)) for α = - 1, β = 1 . \\ the third reverse redefined descriptor (r {ReZ}_{3} (χ)) for α = 1, β = 1 . \end{matrix}

Furthermore, the flowchart of the methodology defined above is shown in the Fig. 1.

Flowchart to measure a reverse-degree-based topological descriptor.

3 Structure of sodalite materials

The three-dimensional sodalite network $χ_{l, w, h}$ is shown in Fig. 2. It has a total of $4 (lw + lh + wh) + 12 lwh$ vertices as well as the total number of edges. Where $l, w, h$ are parameters to show the copies of structure into vertically (length), inside the page (width) and horizontally (height), respectively. It has two types of atoms and three types of bonds, as illustrated in the Fig. 2, and these are beneficial to the key outcomes. The Fig. 2 is investigated in detail given by Arockiaraj et al. (2021), as well as to investigate several beneficial structures linked to zeolites and sodalite. Very first outcome is the sodalite network’s primary and general reverse-degree-based topological indices $χ_{l, w, h}$ .

4 Main results

Throughout this study, we looked at the sodalite three-dimensional structure, that is a component of the zeolite network, in terms of the various reverse-degree-based topological indices mentioned previously. Furthermore, we describe the structure further explicitly by using various numerical instances and conducting a comparison examination of computed M-polynomials, and we finish in the diagrammatical visualizations of generated mathematical formulation. For latest results on topological indices see Ahmad (2020), Zhang et al. (2020), Zahra et al. (2020), Randic (1975), Siddiqui et al. (2016), Aslam et al. (2017), Javaid et al. (2021), Liu et al. (2020), Refaee and Ahmad (2021).

Lemma 4.1

Let $χ_{l, w, h}$ be a sodalite network. Then

$I_{1} (χ_{l, w, h}) = 24 lwh (Π (1, 1)) + 4 (lw + wh + lh) (- 3 Π (1, 1) + 2 Π (1, 2) + 2 Π (2, 2)) + 4 (l + w + h) (Π (1, 1) - 2 Π (1, 2) + Π (2, 2))$ .

Proof

The graph $χ_{l, w, h}$ contains $24 lwh + 4 (lw + wh + lh)$ edges and maximum degree in $χ_{l, w, h}$ graph is 4. The graph of $χ_{l, w, h}$ contained two forms of reverse-degree vertices which are 1 and 2. Let us partitioned the atoms of $χ_{l, w, h}$ based on its reverse-degrees as:

(9)

E_{1, 1} = {θ ϑ \in E (χ_{l, w, h}) : r_{θ} = 1, r_{ϑ} = 1}

(10)

E_{1, 2} = {θ ϑ \in E (χ_{l, w, h}) : r_{θ} = 1, r_{ϑ} = 2}

(11)

E_{2, 2} = {θ ϑ \in E (χ_{l, w, h}) : r_{θ} = 2, r_{ϑ} = 2}

Note that

E (χ_{l, w, h}) = E_{1, 1} \cup E_{1, 2} \cup E_{2, 2}

and

| E_{1, 1} | = 24 lwh - 12 (lw + wh + lh) + 4 (l + w + h), | E_{1, 2} | = 8 (lw + wh + lh - l - w - h), | E_{2, 2} | = 8 (lw + wh + lh) + 4 (l + w + h)

. Hence,

\begin{matrix} I_{1} (χ_{l, w, h}) = \sum_{θ ϑ \in E (χ_{l, w, h})} Π (r_{θ}, r_{ϑ}) \\ = \sum_{θ ϑ \in E_{1, 1}} Π (1, 1) + \sum_{θ ϑ \in E_{1, 2}} Π (1, 2) + \sum_{θ ϑ \in E_{2, 2}} Π (2, 2) \\ = (24 lwh - 12 (lw + wh + lh) + 4 (l + w + h)) Π (1, 1) \\ + 8 (lw + wh + lh - l - w - h) Π (1, 2) \\ + (8 (lw + wh + lh) + 4 (l + w + h)) Π (2, 2) . \end{matrix}

After simplification, we get

$I_{1} (χ_{l, w, h}) = 24 lwh (Π (1, 1)) + 4 (lw + wh + lh) (- 3 Π (1, 1) + 2 Π (1, 2) + 2 Π (2, 2)) + 4 (l + w + h) (Π (1, 1) - 2 Π (1, 2) + Π (2, 2))$ . □

Theorem 4.2

The general reverse Randić index of $χ_{l, w, h}$ is equal to $r R_{α} (χ_{l, w, h}) = \{\begin{matrix} 24 lwh + 36 (lw + wh + lh) + 4 (l + w + h), & for α = 1 \\ 24 lwh + 4 (lw + wh + lh) (1 + 2 \sqrt{2}) + 4 (l + w + h) (1 - 2 \sqrt{2}), & for α = \frac{1}{2} \\ 24 lwh + 4 (lw + wh + lh) (- 2 + \sqrt{2}) + 4 (l + w + h) (\frac{3}{2} - \sqrt{2}), & for α = \frac{- 1}{2} \\ 24 lwh - 6 (lw + wh + lh) + 2 (l + w + h), & for α = - 1 . \end{matrix}$

Proof

For $r R_{α} (χ_{l, w, h})$ that is a general reverse Randić index of $χ_{l, w, h}$ , from Eq. (2), we will get $Π (r_{θ}, r_{ϑ}) = {(r_{θ} \times r_{ϑ})}^{α}$ , therefore $Π (1, 1) = 1, Π (1, 2) = 2^{α}$ and $Π (2, 2) = 2^{2 α}$ . Thus by Lemma 4.1, $r R_{α} (χ_{l, w, h})) = 24 lwh + 4 (lw + wh + lh) (- 3 + 2^{α + 1} + 2^{2 α + 1}) + 4 (l + w + h) (1 - 2^{α + 1} + 2^{2 α}) .$ Put $α = 1$ , we will get $r R_{1} (χ_{l, w, h})) = 24 lwh + 36 (lw + wh + lh) + 4 (l + w + h) .$ Put $α = \frac{1}{2}$ , we will get $r R_{\frac{1}{2}} (χ_{l, w, h})) = 24 lwh + 4 (lw + wh + lh) (1 + 2 \sqrt{2}) + 4 (l + w + h) (1 - 2 \sqrt{2}) .$ Put $α = \frac{- 1}{2}$ , we will get $r R_{\frac{- 1}{2}} (χ_{l, w, h})) = 24 lwh + 4 (lw + wh + lh) (- 2 + \sqrt{2}) + 4 (l + w + h) (\frac{3}{2} - \sqrt{2}) .$ Put $α = - 1$ , we will get $r R_{- 1} (χ_{l, w, h})) = 24 lwh - 6 (lw + wh + lh) + 2 (l + w + h) .$ □

Theorem 4.3

Let $χ_{l, w, h}$ be a sodalite materials networks. Then the reverse atom-bond connectivity index: $r ABC (χ_{l, w, h}) = 8 \sqrt{2} (lw + wh + lh) - 2 \sqrt{2} (l + w + h)$ the reverse geometric-arithmetic index: $r GA (χ_{l, w, h}) = 24 lwh + 4 (- 1 + \frac{4 \sqrt{2}}{3}) (lw + wh + lh) + 4 (l + w + h) (2 - \frac{4 \sqrt{2}}{3})$ the first reverse Zagreb index: $r M_{1} (χ_{l, w, h})) = 48 lwh + 32 (lw + wh + lh)$ the reverse hyper Zagreb index: $r HM (χ_{l, w, h}) = 94 lwh + 152 (lw + wh + lh) + 8 (l + w + h)$ the reverse forgotten index: $r F (χ_{l, w, h}) = 48 lwh + 80 (lw + wh + lh) .$

Proof

For $r ABC (χ_{l, w, h})$ that is a reverse atom-bond connectivity index of $χ_{l, w, h}$ , from Eq. (3), we will get $Π (r_{θ}, r_{ϑ}) = \sqrt{\frac{r_{θ} + r_{ϑ} - 2}{r_{θ} \times r_{ϑ}}}$ , therefore $Π (1, 1) = 0$ and $Π (1, 2) = Π (2, 2) = \frac{1}{\sqrt{2}}$ . Thus by Lemma 4.1 and after simplification, $r ABC (χ_{l, w, h}) = 8 \sqrt{2} (lw + wh + lh) - 2 \sqrt{2} (l + w + h) .$ For $r GA (χ_{l, w, h})$ that is a reverse geometric-arithmetic index of $χ_{l, w, h}$ , from Eq. (4), we will get $Π (r_{θ}, r_{ϑ}) = \frac{2 \sqrt{r_{θ} \times r_{ϑ}}}{r_{θ} + r_{ϑ}}$ , therefore $Π (1, 1) = 1, Π (1, 2) = \frac{2 \sqrt{2}}{3}$ and $Π (2, 2) = 1$ . Thus by Lemma 4.1 and after simplification, $r GA (χ_{l, w, h}) = 24 lwh + 4 (- 1 + \frac{4 \sqrt{2}}{3}) (lw + wh + lh) + 4 (l + w + h) (2 - \frac{4 \sqrt{2}}{3}) .$ For $r M_{1} (χ_{l, w, h})$ that is a first reverse Zagreb index of $χ_{l, w, h}$ , from Eq. (5), we will get $Π (r_{θ}, r_{ϑ}) = (r_{θ} + r_{ϑ})$ , therefore $Π (1, 1) = 2, Π (1, 2) = 3$ and $Π (2, 2) = 4$ . Thus by Lemma 4.1 and after simplification, $r M_{1} (χ_{l, w, h})) = 48 lwh + 32 (lw + wh + lh) .$ For $r HM (χ_{l, w, h})$ that is a first reverse hyper Zagreb index of $χ_{l, w, h}$ , from Eq. (6), we will get $Π (r_{θ}, r_{ϑ}) = {(r_{θ} + r_{ϑ})}^{2}$ , therefore $Π (1, 1) = 4, Π (1, 2) = 9$ and $Π (2, 2) = 16$ . Thus by Lemma 4.1 and after simplification, $r HM (χ_{l, w, h}) = 94 lwh + 152 (lw + wh + lh) + 8 (l + w + h) .$ For $r F (χ_{l, w, h})$ that is a reverse forgotten index of $χ_{l, w, h}$ , from Eq. (7), we will get $Π (r_{θ}, r_{ϑ}) = ({(r_{θ})}^{2} + {(r_{ϑ})}^{2})$ , therefore $Π (1, 1) = 2, Π (1, 2) = 5$ and $Π (2, 2) = 8$ . Thus by Lemma 4.1 and after simplification, $r F (χ_{l, w, h}) = 48 lwh + 80 (lw + wh + lh) .$ □

Theorem 4.4

Let $χ_{l, w, h}$ be a sodalite materials networks. Then the first reverse redefined index: $r {ReZ}_{1} (χ_{l, w, h}) = 48 lwh - 4 (lw + wh + lh)$ the second reverse redefined index: $r {ReZ}_{2} (χ_{l, w, h}) = 12 lwh + \frac{22}{3} (lw + wh + lh) + \frac{2}{3} (l + w + h)$ the third reverse redefined index: $r {ReZ}_{3} (χ_{l, w, h}) = 48 lwh + 152 (lw + wh + lh) + 24 (l + w + h) .$

Proof

For $r {ReZ}_{1} (χ_{l, w, h})$ that is a first reverse redefined index of $χ_{l, w, h}$ , from Eq. (8), we will get $Π (r_{θ}, r_{ϑ}) = {(r_{θ} + r_{ϑ})}^{1} {(r_{θ} \times r_{ϑ})}^{- 1}$ , therefore $Π (1, 1) = 2, Π (1, 2) = \frac{3}{2}$ and $Π (2, 2) = 1$ . Thus by Lemma 4.1 and after simplification, $r {ReZ}_{1} (χ_{l, w, h}) = 48 lwh - 4 (lw + wh + lh)$ For $r {ReZ}_{2} (χ_{l, w, h})$ that is a second reverse redefined index of $χ_{l, w, h}$ , from Eq. (8), we will get $Π (r_{θ}, r_{ϑ}) = {(r_{θ} + r_{ϑ})}^{- 1} {(r_{θ} \times r_{ϑ})}^{1}$ , therefore $Π (1, 1) = \frac{1}{2}, Π (1, 2) = \frac{2}{3}$ and $Π (2, 2) = 1$ . Thus by Lemma 4.1 and after simplification, $r {ReZ}_{2} (χ_{l, w, h}) = 12 lwh + \frac{22}{3} (lw + wh + lh) + \frac{2}{3} (l + w + h)$ For $r {ReZ}_{3} (χ_{l, w, h})$ that is a third reverse redefined index of $χ_{l, w, h}$ , from Eq. (8), we will get $Π (r_{θ}, r_{ϑ}) = (r_{θ} + r_{ϑ}) (r_{θ} \times r_{ϑ})$ , therefore $Π (1, 1) = 2, Π (1, 2) = 6$ and $Π (2, 2) = 16$ . Thus by Lemma 4.1 and after simplification, $r {ReZ}_{3} (χ_{l, w, h}) = 48 lwh + 152 (lw + wh + lh) + 24 (l + w + h) .$ □

5 Numerical and graphical representation and discussion

In the Figs. 3 and 4, we did some comparative study and by using particular values of $l, w, h$ . For all the theorems of the main results we set some parameteric values shown in the Tables 1 and 2, like $l, w, h \in {1, 2, \dots, 10}$ . The figures generated from the chosen values and equations of theorem results are shown the lowest and peak point of which topological index gain. The Fig. 3 shows that the highest value gain by $r GA$ and the lowest value $r ABC$ , while in the next Fig. 4 $r {ReZ}_{2}$ and $r {ReZ}_{3}$ , respectively.

Graphical analysis of r ABC , r GA , r M 1 , r HM and r F for sodalite materials networks, x-axis shows the numeral values of parameters l , w , h and y-axis shows the resulted values of descriptor.

Graphical analysis of reverse redefined indices for sodalite materials networks, x-axis shows the numeral values of parameters l , w , h and y-axis shows the resulted values of descriptor.

Table 1 Statistical analysis of

r ABC, r GA, r M_{1}, r HM

and

r F

for sodalite materials networks.

[l,w,h]	$r ABC$	$r GA$	$r M_{1}$	$r HM$	$r F$
$[1, 1, 1]$	25.456	81.254	144	574	288
$[2, 2, 2]$	118.79	327.76	768	2624	1344
$[3, 3, 3]$	280.01	883.53	2160	6714	3456
$[4, 4, 4]$	509.11	1892.5	4608	13408	6912
$[5, 5, 5]$	806.09	3498.8	8400	23270	12000
$[6, 6, 6]$	1171.0	5846.3	13824	36864	19008
$[7, 7, 7]$	1603.7	9079.1	21168	54754	28224
$[8, 8, 8]$	2104.3	13341.0	30720	77504	39936
$[9, 9, 9]$	2672.8	18776.0	42768	105678	54432
$[10, 10, 10]$	3309.2	25529.0	57600	139840	72000

Table 2 Statistical analysis of reverse redefined indices for sodalite materials networks.

$[l, w, h]$	$r {ReZ}_{1}$	$r {ReZ}_{2}$	$r {ReZ}_{3}$
$[1, 1, 1]$	36	36	576
$[2, 2, 2]$	336	188	2352
$[3, 3, 3]$	1188	528	5616
$[4, 4, 4]$	2880	1128	10656
$[5, 5, 5]$	5700	2060	17760
$[6, 6, 6]$	9936	3396	27216
$[7, 7, 7]$	15876	5208	39312
$[8, 8, 8]$	23808	7568	54336
$[9, 9, 9]$	34020	10548	72576
$[10, 10, 10]$	46800	14220	94320

6 Conclusion

This research looked that we examined the most significant structure from zeolites structures, known as the sodalite materials network, and we represented the fetched graph of this network by $χ_{l, w, h}$ , as shown in Fig. 2. This structure’s movement parameters are $l, w, h ⩾ 1$ . We investigate this structure in terms of various reverse-degree-based topological indices derived from simple degree-based topological descriptors. Namely we did the discussion on following topological descriptors, the general reverse Randić, second reverse Zagreb descriptor, reverse atom-bond connectivity, arithmetic descriptor, first reverse Zagreb, the reverse hyper Zagreb, the reverse forgotten, first, second and third reverse redefined Zagreb descriptors. In future, some other topological descriptors are studied and discussed for this typical enriched structure.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

Ajmal Muhammad, Nazeer Waqas, Munir Mobeen, Kang Shin, Kwun Young, . Some algebraic polynomials and topological indices of generalized prism and toroidal polyhex networks. Symmetry. 2016;9(1):5.
[Google Scholar]
Ali Ahmad, . Comparative study of ve-degree and ev-degree topological descriptors for benzene ring embedded in p-type-surface in 2d network. Polycyclic Aromat. Compd. 2020:1-10.
[Google Scholar]
Alshehri Hamdan, Ahmad Ali, Alqahtani Yahya, Azeem Muhammad, . Vertex metric-based dimension of generalized perimantanes diamondoid structure. IEEE Access. 2022;10:43320-43326.
[Google Scholar]
Arockiaraj Micheal, Clement Joseph, Paul Daniel, Balasubramanian Krishnan, . Quantitative structural descriptors of sodalite materials. J. Mol. Struct.. 2021;1223:128766.
[Google Scholar]
Aslam Adnan, Ahmad Safyan, Gao Wei, . On certain topological indices of boron triangular nanotubes. Zeitschrift für Naturforschung A. 2017;72(8):711-716.
[Google Scholar]
Azeem Muhammad, Imran Muhammad, Nadeem Muhammad Faisal, . Sharp bounds on partition dimension of hexagonal möbius ladder. J. King Saud Univ.- Sci.. 2022;34(2):101779.
[Google Scholar]
Barth T.F.W., . The structures of the minerals of the sodalite family. Z. Krist. Cryst. Mater.. 1932;83:1-6.
[Google Scholar]
Catlow C.R.A., . Modeling of Structure and Reactivity in Zeolites. London: Academic Press; 1992.
Dyer A., . An Introduction to Zeolite Molecular Sieves. Chichester: Wiley-Interscience; 1988.
Eddaoudi M., Kim J., Vodak D., Sudik A., Wachter J., OKeeffe M., Yaghi O.M., . Geometric requirements and examples of important structures in the assembly of square building blocks. Proc. Nat. Acad. Sci.. 2002;99(8):4900-4904.
[Google Scholar]
Gao Wei, Younas Muhammad, Farooq Adeel, Virk Abaid, Nazeer Waqas, . Some reverse degree-based topological indices and polynomials of dendrimers. Mathematics. Oct 2018;6(10):214.
[Google Scholar]
Hameed Shehla, Jamil Muhammad Kamran, Waheed Muhammad, Azeem Muhammad, Swaray Senesie, . Two complex graph operations and their exact formulations on topological properties. Complexity. 2022;2022:1-15.
[Google Scholar]
Hassan I., Grundy H.D., . The crystal structures of sodalite-group minerals. Acta Crystall. Section B Struct. Sci.. 1984;40(1):6-13.
[Google Scholar]
Hong Gang, Gu Zhen, Javaid Muhammad, Awais Hafiz Muhammad, Siddiqui Muhammad Kamran, . Degree-based topological invariants of metal-organic networks. IEEE Access. 2020;8:68288-68300.
[Google Scholar]
Jaeger F.M., . On the constitution and the structure of ultramarine. Trans. Faraday Soc.. 1929;25:320.
[Google Scholar]
Javaid Muhammad, Zafar Hassan, Zhu Q., Alanazi Abdulaziz Mohammed, . Improved lower bound of LFMD with applications of prism-related networks. Mathe. Probl. Eng.. 2021;2022:1-9.
[Google Scholar]
Jayanna Gowtham Kalkere, Swamy Narahari Narasimha, . Multiplicative reverse geometric-arithmetic indices and arithmetic-geometric of silicate network. Biointerface Res. Appl. Chem.. Aug 2021;12(3):4192-4199.
[Google Scholar]
Jiri C., Herman V.B., Avelino C., Ferdi S., . Introduction to zeolite science andpractice. Stud. Surf. Sci. 2007;2(1):1058-1065.
[Google Scholar]
Karagiaridi Olga, Lalonde Marianne B., Bury Wojciech, Sarjeant Amy A., Farha Omar K., Hupp Joseph T., . Opening ZIF-8: A catalytically active zeolitic imidazolate framework of sodalite topology with unsubstituted linkers. J. Am. Chem. Soc.. Oct 2012;134(45):18790-18796.
[Google Scholar]
Keeffe M.O., . N-dimensional diamond, sodalite and rare sphere packings. Acta Crystallogr.. 1991;47:748-753.
[Google Scholar]
Koam Ali N.A., Ahmad Ali, Ahmad Yasir, . Computation of reverse degree-based topological indices of hex-derived networks. AIMS Mathematics. 2021;6(10):11330-11345.
[Google Scholar]
Koam Ali N.A., Ahmad Ali, Azeem Muhammad, Khalil Adnan, Nadeem Muhammad Faisal, . On adjacency metric dimension of some families of graph. J. Function Spaces. 2022;2022:1-8.
[Google Scholar]
Koam Ali N.A., Ahmad Ali, Asim M.A., Azeem Muhammad, . Computation of vertex and edge resolvability of benzenoid tripod structure. J. King Saud Univ.- Sci. 2022:102208.
[Google Scholar]
Koam Ali N.A., Ahmad Ali, Azeem Muhammad, Nadeem Muhammad Faisal, . Bounds on the partition dimension of one pentagonal carbon nanocone structure. Arabian J. Chem.. 2022;15(7):103923.
[Google Scholar]
Kulli V.R., . Reverse zagreb and reverse hyper-zagreb indices and their polynomials of rhombus silicate networks. Annals Pure Appl. Mathe.. 2018;16(1):47-51.
[Google Scholar]
Liu Jia-Bao, Raza Zahid, Javaid Muhammad, . Zagreb connection numbers for cellular neural networks. Discrete Dyn. Nature Soc.. 2020;2020:1-8.
[Google Scholar]
Mufti Zeeshan Saleem, Anjum Rukhshanda, Abbas Ayesha, Ali Shahbaz, Afzal Muhammad, Alam Ashraful, . Computation of vertex degree-based molecular descriptors of hydrocarbon structure. J. Chem.. 2022;2022:1-15.
[Google Scholar]
Munir Mobeen, Nazeer Waqas, Nizami Abdul, Rafique Shazia, Kang Shin, . M-polynomials and topological indices of titania nanotubes. Symmetry. Oct 2016;8(11):117.
[Google Scholar]
Munir Mobeen, Nazeer Waqas, Rafique Shazia, Kang Shin, . M-polynomial and related topological indices of nanostar dendrimers. Symmetry. 2016;8(9):97.
[Google Scholar]
Nadeem Muhammad Faisal, Zafar Sohail, Zahid Zohaib, . On topological properties of the line graphs of subdivision graphs of certain nanostructures. Appl. Mathe. Comput.. 2016;273:125-130.
[Google Scholar]
Nadeem Imran, Shaker Hani, Hussain Muhammad, Naseem Asim, . Topological indices of para-line graphs of v-phenylenic nanostructures. Open Mathe.. 2019;17(1):260-266.
[Google Scholar]
Nadeem Muhammad Faisal, Azeem Muhammad, Farman Iqra, . Comparative study of topological indices for capped and uncapped carbon nanotubes. Polycyclic Aromat. Compd. 2021:1-18.
[Google Scholar]
Nadeem Muhammad Faisal, Azeem Muhammad, Siddiqui Hafiz Muhammad Afzal, . Comparative study of zagreb indices for capped, semi-capped, and uncapped carbon nanotubes. Polycyclic Aromat. Compd. 2021:1-18.
[Google Scholar]
Nadeem Muhammad Faisal, Imran Muhammad, Siddiqui Hafiz Muhammad Afzal, Azeem Muhammad, Khalil Adnan, Ali Yasir, . Topological aspects of metal-organic structure with the help of underlying networks. Arabian J. Chem.. 2021;14(6):103157.
[Google Scholar]
Neeraj S., Noy M.L., Rao C.N.R., Cheetham A.K., . Sodalite networks formed by metal squarates. Solid State Sci.. 2002;4(10):1231-1236.
[Google Scholar]
Pauling L., . The structure of sodalite and helvite. Zeitschrift für Kristallographie. 1930;74:213-225.
[Google Scholar]
Poojary Prasanna, Shenoy Gautham B., Narahari N., Raghavendra A., Sooryanarayana B., Poojary Nandini, . Reverse topological indices of some molecules in drugs used in the treatment of h1n1. Biointerface Res. Appl. Chem.. 2022;13(1):71.
[Google Scholar]
Prabhu S., Murugan G., Cary M., Arulperumjothi M., Liu J.-B., . On certain distance and degree based topological indices of zeolite lta frameworks. Mater. Res. Express. 2020;7:055006.
[Google Scholar]
Randic Milan, . Characterization of molecular branching. J. Am. Chem. Soc.. 1975;97(23):6609-6615.
[Google Scholar]
Refaee Eshrag A., Ahmad Ali, . A study of hexagon star network with vertex-edge-based topological descriptors. Complexity. 2021;2021:1-7.
[Google Scholar]
Shabbir Ayesha, Nadeem Muhammad Faisal, Mukhtar Safyan, Raza Ahsan, . On edge version of some degree-based topological indices of HAC5c7 [p, q] and VC5c7[p, q] nanotubes. Polycyclic Aromat. Compd. 2020:1-17.
[Google Scholar]
Siddiqui Muhammad Kamran, Imran Muhammad, Ahmad Ali, . On zagreb indices, zagreb polynomials of some nanostar dendrimers. Appl. Mathe. Comput.. 2016;280:132-139.
[Google Scholar]
Taylor D., . The sodalite group of minerals. Contrib. Mineral. Petrol.. 1967;16:172-188.
[Google Scholar]
Torrens Francisco, Castellano Gloria, . Fractal dimension of active-site models of zeolite catalysts. J. Nanomater.. 2006;1–9:2006.
[Google Scholar]
Torrens F., Castellano G., . Polarizability characterization of zeolitic br awnsted acidic sites. Recent Prog. Comput. Sci. Eng. 2019;2:555-556.
[Google Scholar]
Wang Zhen, Chaudhry Faryal, Naseem Maria, Asghar Adnan, . Reverse zagreb and reverse hyper-zagreb indices for crystallographic structure of molecules. J. Chem.. Mar 2020;2020:1-12.
[Google Scholar]
Zahra Nida, Ibrahim Muhammad, Siddiqui Muhammad Kamran, . On topological indices for swapped networks modeled by optical transpose interconnection system. IEEE Access. 2020;8:200091-200099.
[Google Scholar]
Zhang Jing, Siddiqui Muhammad Kamran, Rauf Abdul, Ishtiaq Muhammad, . On ve-degree and ev-degree based topological properties of single walled titanium dioxide nanotube. J. Cluster Sci.. 2020;32(4):821-832.
[Google Scholar]
Zhao Dongming, Chu Yu-Ming, Siddiqui Muhammad Kamran, Ali Kashif, Nasir Muhammad, Younas Muhammad Tayyab, Cancan Murat, . On reverse degree based topological indices of polycyclic metal organic network. Polycyclic Aromat. Compd.. 2021;12:1-18.
[Google Scholar]

Introduction

Preliminaries

Structure of sodalite materials

Main results

Numerical and graphical representation and discussion

Conclusion

Fulltext Views
48

PDF downloads
61

View/Download PDF
Download Citations

BibTeX
RIS

Show Sections

[1] Ajmal Muhammad, Nazeer Waqas, Munir Mobeen, Kang Shin, Kwun Young, . Some algebraic polynomials and topological indices of generalized prism and toroidal polyhex networks. Symmetry. 2016;9(1):5.
[Google Scholar]

[2] Ali Ahmad, . Comparative study of ve-degree and ev-degree topological descriptors for benzene ring embedded in p-type-surface in 2d network. Polycyclic Aromat. Compd. 2020:1-10.
[Google Scholar]

[3] Alshehri Hamdan, Ahmad Ali, Alqahtani Yahya, Azeem Muhammad, . Vertex metric-based dimension of generalized perimantanes diamondoid structure. IEEE Access. 2022;10:43320-43326.
[Google Scholar]

[4] Arockiaraj Micheal, Clement Joseph, Paul Daniel, Balasubramanian Krishnan, . Quantitative structural descriptors of sodalite materials. J. Mol. Struct.. 2021;1223:128766.
[Google Scholar]

[5] Aslam Adnan, Ahmad Safyan, Gao Wei, . On certain topological indices of boron triangular nanotubes. Zeitschrift für Naturforschung A. 2017;72(8):711-716.
[Google Scholar]

[6] Azeem Muhammad, Imran Muhammad, Nadeem Muhammad Faisal, . Sharp bounds on partition dimension of hexagonal möbius ladder. J. King Saud Univ.- Sci.. 2022;34(2):101779.
[Google Scholar]

[7] Barth T.F.W., . The structures of the minerals of the sodalite family. Z. Krist. Cryst. Mater.. 1932;83:1-6.
[Google Scholar]

[8] Catlow C.R.A., . Modeling of Structure and Reactivity in Zeolites. London: Academic Press; 1992.

[9] Dyer A., . An Introduction to Zeolite Molecular Sieves. Chichester: Wiley-Interscience; 1988.

[10] Eddaoudi M., Kim J., Vodak D., Sudik A., Wachter J., OKeeffe M., Yaghi O.M., . Geometric requirements and examples of important structures in the assembly of square building blocks. Proc. Nat. Acad. Sci.. 2002;99(8):4900-4904.
[Google Scholar]

[11] Gao Wei, Younas Muhammad, Farooq Adeel, Virk Abaid, Nazeer Waqas, . Some reverse degree-based topological indices and polynomials of dendrimers. Mathematics. Oct 2018;6(10):214.
[Google Scholar]

[12] Hameed Shehla, Jamil Muhammad Kamran, Waheed Muhammad, Azeem Muhammad, Swaray Senesie, . Two complex graph operations and their exact formulations on topological properties. Complexity. 2022;2022:1-15.
[Google Scholar]

[13] Hassan I., Grundy H.D., . The crystal structures of sodalite-group minerals. Acta Crystall. Section B Struct. Sci.. 1984;40(1):6-13.
[Google Scholar]

[14] Hong Gang, Gu Zhen, Javaid Muhammad, Awais Hafiz Muhammad, Siddiqui Muhammad Kamran, . Degree-based topological invariants of metal-organic networks. IEEE Access. 2020;8:68288-68300.
[Google Scholar]

[15] Jaeger F.M., . On the constitution and the structure of ultramarine. Trans. Faraday Soc.. 1929;25:320.
[Google Scholar]

[16] Javaid Muhammad, Zafar Hassan, Zhu Q., Alanazi Abdulaziz Mohammed, . Improved lower bound of LFMD with applications of prism-related networks. Mathe. Probl. Eng.. 2021;2022:1-9.
[Google Scholar]

[17] Jayanna Gowtham Kalkere, Swamy Narahari Narasimha, . Multiplicative reverse geometric-arithmetic indices and arithmetic-geometric of silicate network. Biointerface Res. Appl. Chem.. Aug 2021;12(3):4192-4199.
[Google Scholar]

[18] Jiri C., Herman V.B., Avelino C., Ferdi S., . Introduction to zeolite science andpractice. Stud. Surf. Sci. 2007;2(1):1058-1065.
[Google Scholar]

[19] Karagiaridi Olga, Lalonde Marianne B., Bury Wojciech, Sarjeant Amy A., Farha Omar K., Hupp Joseph T., . Opening ZIF-8: A catalytically active zeolitic imidazolate framework of sodalite topology with unsubstituted linkers. J. Am. Chem. Soc.. Oct 2012;134(45):18790-18796.
[Google Scholar]

[20] Keeffe M.O., . N-dimensional diamond, sodalite and rare sphere packings. Acta Crystallogr.. 1991;47:748-753.
[Google Scholar]

[21] Koam Ali N.A., Ahmad Ali, Ahmad Yasir, . Computation of reverse degree-based topological indices of hex-derived networks. AIMS Mathematics. 2021;6(10):11330-11345.
[Google Scholar]

[22] Koam Ali N.A., Ahmad Ali, Azeem Muhammad, Khalil Adnan, Nadeem Muhammad Faisal, . On adjacency metric dimension of some families of graph. J. Function Spaces. 2022;2022:1-8.
[Google Scholar]

[23] Koam Ali N.A., Ahmad Ali, Asim M.A., Azeem Muhammad, . Computation of vertex and edge resolvability of benzenoid tripod structure. J. King Saud Univ.- Sci. 2022:102208.
[Google Scholar]

[24] Koam Ali N.A., Ahmad Ali, Azeem Muhammad, Nadeem Muhammad Faisal, . Bounds on the partition dimension of one pentagonal carbon nanocone structure. Arabian J. Chem.. 2022;15(7):103923.
[Google Scholar]

[25] Kulli V.R., . Reverse zagreb and reverse hyper-zagreb indices and their polynomials of rhombus silicate networks. Annals Pure Appl. Mathe.. 2018;16(1):47-51.
[Google Scholar]

[26] Liu Jia-Bao, Raza Zahid, Javaid Muhammad, . Zagreb connection numbers for cellular neural networks. Discrete Dyn. Nature Soc.. 2020;2020:1-8.
[Google Scholar]

[27] Mufti Zeeshan Saleem, Anjum Rukhshanda, Abbas Ayesha, Ali Shahbaz, Afzal Muhammad, Alam Ashraful, . Computation of vertex degree-based molecular descriptors of hydrocarbon structure. J. Chem.. 2022;2022:1-15.
[Google Scholar]

[28] Munir Mobeen, Nazeer Waqas, Nizami Abdul, Rafique Shazia, Kang Shin, . M-polynomials and topological indices of titania nanotubes. Symmetry. Oct 2016;8(11):117.
[Google Scholar]

[29] Munir Mobeen, Nazeer Waqas, Rafique Shazia, Kang Shin, . M-polynomial and related topological indices of nanostar dendrimers. Symmetry. 2016;8(9):97.
[Google Scholar]

[30] Nadeem Muhammad Faisal, Zafar Sohail, Zahid Zohaib, . On topological properties of the line graphs of subdivision graphs of certain nanostructures. Appl. Mathe. Comput.. 2016;273:125-130.
[Google Scholar]

[31] Nadeem Imran, Shaker Hani, Hussain Muhammad, Naseem Asim, . Topological indices of para-line graphs of v-phenylenic nanostructures. Open Mathe.. 2019;17(1):260-266.
[Google Scholar]

[32] Nadeem Muhammad Faisal, Azeem Muhammad, Farman Iqra, . Comparative study of topological indices for capped and uncapped carbon nanotubes. Polycyclic Aromat. Compd. 2021:1-18.
[Google Scholar]

[33] Nadeem Muhammad Faisal, Azeem Muhammad, Siddiqui Hafiz Muhammad Afzal, . Comparative study of zagreb indices for capped, semi-capped, and uncapped carbon nanotubes. Polycyclic Aromat. Compd. 2021:1-18.
[Google Scholar]

[34] Nadeem Muhammad Faisal, Imran Muhammad, Siddiqui Hafiz Muhammad Afzal, Azeem Muhammad, Khalil Adnan, Ali Yasir, . Topological aspects of metal-organic structure with the help of underlying networks. Arabian J. Chem.. 2021;14(6):103157.
[Google Scholar]

[35] Neeraj S., Noy M.L., Rao C.N.R., Cheetham A.K., . Sodalite networks formed by metal squarates. Solid State Sci.. 2002;4(10):1231-1236.
[Google Scholar]

[36] Pauling L., . The structure of sodalite and helvite. Zeitschrift für Kristallographie. 1930;74:213-225.
[Google Scholar]

[37] Poojary Prasanna, Shenoy Gautham B., Narahari N., Raghavendra A., Sooryanarayana B., Poojary Nandini, . Reverse topological indices of some molecules in drugs used in the treatment of h1n1. Biointerface Res. Appl. Chem.. 2022;13(1):71.
[Google Scholar]

[38] Prabhu S., Murugan G., Cary M., Arulperumjothi M., Liu J.-B., . On certain distance and degree based topological indices of zeolite lta frameworks. Mater. Res. Express. 2020;7:055006.
[Google Scholar]

[39] Randic Milan, . Characterization of molecular branching. J. Am. Chem. Soc.. 1975;97(23):6609-6615.
[Google Scholar]

[40] Refaee Eshrag A., Ahmad Ali, . A study of hexagon star network with vertex-edge-based topological descriptors. Complexity. 2021;2021:1-7.
[Google Scholar]

[41] Shabbir Ayesha, Nadeem Muhammad Faisal, Mukhtar Safyan, Raza Ahsan, . On edge version of some degree-based topological indices of HAC5c7 [p, q] and VC5c7[p, q] nanotubes. Polycyclic Aromat. Compd. 2020:1-17.
[Google Scholar]

[42] Siddiqui Muhammad Kamran, Imran Muhammad, Ahmad Ali, . On zagreb indices, zagreb polynomials of some nanostar dendrimers. Appl. Mathe. Comput.. 2016;280:132-139.
[Google Scholar]

[43] Taylor D., . The sodalite group of minerals. Contrib. Mineral. Petrol.. 1967;16:172-188.
[Google Scholar]

[44] Torrens Francisco, Castellano Gloria, . Fractal dimension of active-site models of zeolite catalysts. J. Nanomater.. 2006;1–9:2006.
[Google Scholar]

[45] Torrens F., Castellano G., . Polarizability characterization of zeolitic br awnsted acidic sites. Recent Prog. Comput. Sci. Eng. 2019;2:555-556.
[Google Scholar]

[46] Wang Zhen, Chaudhry Faryal, Naseem Maria, Asghar Adnan, . Reverse zagreb and reverse hyper-zagreb indices for crystallographic structure of molecules. J. Chem.. Mar 2020;2020:1-12.
[Google Scholar]

[47] Zahra Nida, Ibrahim Muhammad, Siddiqui Muhammad Kamran, . On topological indices for swapped networks modeled by optical transpose interconnection system. IEEE Access. 2020;8:200091-200099.
[Google Scholar]

[48] Zhang Jing, Siddiqui Muhammad Kamran, Rauf Abdul, Ishtiaq Muhammad, . On ve-degree and ev-degree based topological properties of single walled titanium dioxide nanotube. J. Cluster Sci.. 2020;32(4):821-832.
[Google Scholar]

[49] Zhao Dongming, Chu Yu-Ming, Siddiqui Muhammad Kamran, Ali Kashif, Nasir Muhammad, Younas Muhammad Tayyab, Cancan Murat, . On reverse degree based topological indices of polycyclic metal organic network. Polycyclic Aromat. Compd.. 2021;12:1-18.
[Google Scholar]

Topological properties of reverse-degree-based indices for sodalite materials network

Abstract

Keywords

Sodalite materials network

Topological indices

Reverse degree

Natural zeolites

1 Introduction

2 Preliminaries

3 Structure of sodalite materials

4 Main results

5 Numerical and graphical representation and discussion

6 Conclusion

Declaration of Competing Interest

References

Suggested read for related articles: