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Original article

02 2021

:15;

103567

doi:

10.1016/j.arabjc.2021.103567

Irregularity molecular descriptors of Cerium oxide $C e O_{2}$ based on mathematical model and calculation

Qi Zhang^{^a}, Muhammad Mobeen Munir^{^b,⁎}, Haseeb Ahmad^{^c}, Jia-Bao Liu^{^d}

a College of Mathematics and Computer Science, Tongling University, Anhui 244000, PR China

b Department of Mathematics, University of the Punjab, Lahore 54500, Pakistan

c Division of Science and Technology, University of Education, Lahore 54000, Pakistan

d School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, PR China

⁎Corresponding author. mmunir.math@pu.edu.pk (Muhammad Mobeen Munir),

Received: 2021-8-17, Accepted: 2021-11-10,

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

The irregularities of graphs and their dependencies on the size parameters are recently attracting attention of not only mathematician but also of theoretical chemists. It is found that these irregularities are related with the properties of the substance involved. Cerium oxide is a rare-earth metal having formula $C e O_{2}$ and it is light yellow-white powder. In the present article we are concerned with computing the closed forms of irregularity measures of general form of crystal structure of Cerium Oxide $C e O_{2}$ based on mathematical model and calculation.

Keywords

Irregularity indices

Cerium oxide

Indices

Imbalance-based irregularity measure

Molecular computing

Mathematical model

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1 Introduction

Chemical Graph Theory is recently flourishing because of increasing number of structural dependent functions which exhibit the topological aspects of the chemical substance under consideration. Physio-chemical aspects of the materials are related to these functions in a complex way. Topological degree-based descriptors are numerical estimators which relate these properties. One class of such descriptors is irregularity measure which computes the complexity of the chemical structures. This degree of complexity determines the degree of chemical branching and bonding pattern, enthalpy, stability and strain energy, (Abdo and Dimitrov, 2014; Gutman, 2018; Reti et al., 2018; Hu et al., 2005 ). The present article computes irregularity measures of Cerium oxide in closed analytic form.

Basic structure of cerium oxide is such that each cerium ion $C e^{4 +}$ is surrounded by eight oxide ions $O_{2}$ that forms a cubic shape molecule and each oxide ion is enclosed by four $C e^{4 +}$ ions which forms a tetrahedron molecule (Sayle et al., 1994). This structure is rich in exceptional physical and chemical properties. Its optical and electrical properties are generally because of the little size and huge thickness of corners. Cerium exists both in $C e O_{2}$ and ${Ce}_{2} O_{3}$ in bulk state as demonstrated in Fig. 2. This double oxide nature is because of the electropositive qualities of ceria, ${Ce}_{2} O_{3},$ also called the “sesquioxide”, Fig. 3. The magnetic property is profoundly temperature dependent. This property is not present in the large structure as it has only + 4 state (Ribeiro and Ferreira, 2017). Every strange and special property of cerium oxide is because of the oxygen vacancy. In Cerium oxide a stable structure is due to the high oxygen storage capacity. When the size of cerium material is small, these oxygen vacancies tend to increase (Tseng et al., 2019; Wu et al., 2018).

Part (a) shows the unit cell of bulk C e O 2 , part (b) shows schematic illustration of building the solid atomic assembly of C e (cerium) and O (oxygen) atoms by randomly expanding the unit cell of bulk C e O 2 (enclosed by black solid lines) ×, y, and z times in three orthonormal directions, respectively part (c) the super cell of cubic cerium oxide nanoparticles of 1 nm for relaxation calculation (d) zoom-in picture of the solid atomic assembly in part c.

Topological index is one such rich area of mathematics where it can enter in chemistry and bring revolution in determining properties of the chemical substances (Rucker and Rucker, 1999; Klavžar and Gutman, 1996; Brückler et al., 2011; Deng et al., 2011 ). The basic branch involved is the Chemical Graph Theory. In order to move on, we need to grasp some rudiments of Graph Theory. Let us denote G to be a simple connected graph having vertex set V and edge set E. Let us fix du and dv for the degrees of vertices u and v respectively. A graph is known to be regular if every vertex of the graph has the same degree. A topological invariant is called an irregularity index if this index vanishes for a regular graph, and remains non-zero for otherwise. Regular graphs are technically nicer than irregular ones and consequently have been extensively investigated, particularly in mathematics. After the discoveries of nanotubes and fullerenes, the regular graphs became more popular especially in Chemical Graph Theory. Today it is well-established that the topology of fullerenes is rich and has potential of many applications in mathematics. However, the irregular graphs remained unnoticed for a long time.

Paul Erdos, the famous Hungarian mathematician, put a lot of emphasis on the study of irregular graphs for the first time in history in (Chartrand et al., 1988). He posed a problem in (1994), that what could be the extreme size of highly irregular graph of a given order, (Majcher and Michael, 1997). This problem in fact opened gates for mathematician to study irregular graphs and the degree of irregularity as well. A particular graph would be highly strange in which in which each vertex has a different degree. Such graphs were termed as perfect graphs. The authors of (Behzad and Chartrand, 1947) demonstrated that no graph is perfect. The graphs lying in between are called quasi-perfect graphs, in which all except two vertices have different degrees (Majcher and Michael, 1997). One way of expressing irregularities of graphs is the computation of irregularity indices. These indices can be based on many pieces of information such as degrees of vertices, eccentricities of vertices etc. Some irregularity indices were studied recently in a novel way (Horoldagva et al., 2016; Liu et al., 2014; Collatz and Sinogowitz, 1957 ). The first such irregularity index was introduced in (Bell, 1992). Most of these indices which are based on degree of vertices used the concept of the imbalance of an edge defined as ${imball}_{uv} = | d u - d v |$ ; (Behzad and Chartrand, 1947; Horoldagva et al., 2016; Liu et al., 2014; Collatz and Sinogowitz, 1957; Bell, 1992; Albertson, 1997 ).

The Albertson index denoted by, AL(G), was initially introduced by Alberston in (Albertson, 1997) as $A L (G) = \sum_{UV \in E} |d_{u} - d_{v}| .$ The irregularity index IRL(G) and IRLU(G) were introduced by Vukičević and Gasparov, (Vukičević and Graovac, 2004) as $I R L (G) = \sum_{UV \in E} |{lnd}_{u} - {lnd}_{v}|$ , and $I R L U (G) = \sum_{UV \in E} \frac{|d_{u} - d_{v}|}{m i n (d_{u,} d_{v})}$ where min is used as the minimum of the parameters given. Recently, Abdoo et al. defined the new term “total irregularity measure of a graph G”, as $IR R_{t} (G) = \frac{1}{2} \sum_{UV \in E} |d_{u} - d_{v}|$ , (Abdo et al., 2014; Abdo and Dimitrov, 2014; Abdo and Dimitrov, 2014 ). Recently, Gutman et al. introduced the IRF(G) irregularity index of the graph G, which is described as $I R F (G) = \sum_{UV \in E} {(d_{u} - d_{v})}^{2}$ in (Gutman, 2018). Reti et al. discussed the relation of irregularity indices and properties in (Reti et al., 2018). The Randic index itself is directly related to an irregularity measure, which is described as $I R A (G) = \sum_{UV \in E} {(d_{u}^{\frac{- 1}{2}} - d_{v}^{\frac{- 1}{2}})}^{2}$ in (Yang et al., 2019). Further irregularity indices of similar nature can be traced in (Gutman, 2018; Reti et al., 2018; Hu et al., 2005 ) in detail. These indices are given as $I R D I F (G) = \sum_{UV \in E} |\frac{d_{u}}{d_{v}} - \frac{d_{v}}{d_{v}}|$ ; $I R L F (G) = \sum_{UV \in E} \frac{|d_{u} - d_{v}|}{\sqrt{(d_{u} d_{v})}}$ ; $L A (G) = 2 \sum_{UV \in E} \frac{|d_{u} - d_{v}|}{(d_{u} + d_{v})}$ , $I R D 1 = \sum_{UV \in E} l n \{1 + |d_{v} - d_{v}|\}$ , $I R G A (G) = \sum_{UV \in E} l n \frac{d_{u} + d_{v}}{2 \sqrt{(d_{u} d_{v})}}$ , and $I R B (G) = \sum_{UV \in E} {(d_{u}^{\frac{1}{2}} - d_{v}^{\frac{1}{2}})}^{2}$ . Further details can be given in (Gutman, 2018; Reti et al., 2018; Hu et al., 2005; Zahid et al., 2019; Gao et al., 2019; Gao et al., 2017; Hussain et al., 2019; Hussain et al., 2020; Yang et al., 2019; Hussain et al., 2019; Estrada, 2010 Sep ). These irregularity measures have been put to use in determining an estimate of the properties of octane isomers in (Reti et al., 2018). Reti et al. actually did comparative regression analysis on the eighteen different irregularity indices of five different properties of octane isomers and concluded that these properties such as boiling point, standard enthalapy of vaporization, acentric factor, enthalapy of vaporization and entropy can be estimated with good accuracy of these isomers, (Reti et al., 2018). This side depict the actual applications of these indices. Other side of the story is to consider different matrices based on these irregularity indices and the eigen values of these matrices give us the energies related to these molecular graphs. These energies have deep applications in chemistry and physics such as to determine pi-electron energy. Authors in (Estrada, 2010 Sep) critically analyzed the irregularity of 10 protein–protein interaction networks in different organisms ranging from 50 to 3000 nodes. These indices can play helpful Job in estimating the properties of the molecular substances involved. Recently, Zahid et al. computed the irregularity indices of a nanotube (Zahid et al., 2019). Gao et al. recently computed irregularity measures of some dendrimer structures in (Gao et al., 2019) and molecular structures in (Gao et al., 2017). These structures are used as long infinite chain macromolecules in chemistry and related areas. Hussain et al. computed irregularities of different non-homeomorphic benzenoid structures in (Hussain et al., 2019; Hussain et al., 2020). Yang et al. discussed the imbalance-based irregularities of two types of boron nanotubes in (Yang et al., 2019). Hussain et al. also computed imbalance-based irregularities of nanostar dendrimers in (Hussain et al., 2019). As structures are rich in topology, so more and more investigations of irregular structures are in progress. In this article we focus on the topological structural facts of Cerium Oxide $C e O_{2} .$ More precisely, we are interested in computing the imbalance-based degree of irregularity measures of general form of Cerium Oxide $C e O_{2}$ . Following Fig. 1 shows the topological structure of $C e O_{2}$ .

In Cerium Oxide $Ce$ atoms are shown by blue circles and oxygen O atoms are shown by red circles. In the ${CeO}_{2}$ unit cell, each $Ce$ atom located at corner of a cube is coordinated with three $Ce$ atoms, and one oxygen O atom, on other hand each $Ce$ atom located at face of a cube is coordinated with four oxygen O atoms as shown in Fig. 2. Now in the next section we move towards our computational results. We also show some graphical analysis of the irregularities involved. It is worth mentioning that Cerium Oxide doped obsidian glasses are better radiation shielding materials than ordinary obsidian glass, as mass attenuation coefficients of CeO₂ doped obsidian glass decrease with increasing photon energy, (Yalcin et al., 2019). In (Yılmaz et al., 2020), authors computed some indices relating to the radiation shielding parameters like linear attenuation coefficients mass attenuation coefficients, effective atomic numbers, effective electron densities, half value of layers and mean free paths for rare earth metal oxides CeO₂ doped borosilicate (BS) glasses.

2 Main results

In this section, we present our main theoretical results. Following main result gives closed formulas for as many as twelve irregularity measures of ${CeO}_{2} [p, q, t]$ .

Theorem:

Let ${CeO}_{2} [p, q, t]$ be the crystal structure of Cerium Oxide. Tthen the irregularity indices of ${CeO}_{2}$ are:

$I R D I F ({CeO}_{2} [p, q, t] .) = 16.3 p q t + 1.83 p q + 1.83 p t + 1.83 q t + 1.6 p + 1.6 q + 1.6 t + 9.83$
$A L ({CeO}_{2} [p, q, t] .) = 40 p q t + 4 p q + 4 p t + 4 q t + 4 p + 4 q + 4 t + 24$ .
$I R L ({CeO}_{2} [p, q, t] .) = 7.873 p q t + 0.928 p q + 0.928 p t + 0.928 q t + 0.8109 p + 0.8109 q + 0.8109 t + 4.747797$ .
$I R L F ({CeO}_{2} [p, q, t] .) = 10 p q t + p q + p t + q t + p + q + t + 6$ .
$I R L U ({CeO}_{2} [p, q, t] .) = 7.946182 p q t + 0.9258 p q + 0.9258 p t + 0.9258 q t + 0.816 p + 0.816 q + 0.816 t + 4.78958 .$
$I R F ({CeO}_{2} [p, q, t] .) = 96 p q t + 8 p + 8 q + 8 t + 56$ .
$I R L A ({CeO}_{2} [p, q, t] .) = 7.73 p q t + 0.933 p q + 0.933 p t + 0.933 q t + 0.8 p + 0.8 q + 0.8 t + 4.6667$ .
$I R D 1 = 20.796668 p q t + 2.785 p q + 2.785 p t + 2.785 q t + 2.1972 p + 2.1972 q + 2.1972 t + 12.5955$ .
$I R A ({CeO}_{2} [p, q, t] .) = 0.17758 p q t + 0.0122 p q + 0.0122 p t + 0.0122 q t + 0.01683 p + 0.01683 q + 0.01683 t + 0.105627$ .
$I R G A ({CeO}_{2} [p, q, t] .) = 3.61953 p q t + 0.639 p q + 0.639 p t + 0.639 q t + 0.4082 p + 0.4082 q + 0.4082 t + 2.218017 .$
$I R B ({CeO}_{2} [p, q, t] .) = 4.60524 p q t + 0.12187 p q + 0.12187 p t + 0.12187 q t + 0.40408 p + 0.40408 q + 0.40408 t + 2.7067$ .
$IR R_{t} ({CeO}_{2} [p, q, t] .) = 20 p q t + 2 p q + 2 p t + 2 q t + 2 p + 2 q + 2 t + 12 .$ .

Proof

In order to prove the above theorem, we have to consider Figs. 1 and 2 from where some partitions of edge sets and vertex sets can be computed.

Now, using above Table 1 and the above definitions, we have:

$I R D I F (G) = \sum_{UV \in E} |\frac{d_{u}}{d_{v}} - \frac{d_{v}}{d_{v}}|$

I R D I F ({CeO}_{2} [p, q, t]) = 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 |\frac{4}{4} - \frac{4}{4}| + 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 |\frac{6}{4} - \frac{4}{6}| + 2 p q t - p q - p t - q t + 1 |\frac{8}{4} - \frac{4}{8}| + 2 p q t - p q - p t - q t + p + q + t - 2 |\frac{6}{6} - \frac{6}{6}|

= 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 |\frac{6}{4} - \frac{4}{6}| + 2 p q t - p q - p t - q t + 1 |\frac{8}{4} - \frac{4}{8}|

$A L (G) = \sum_{UV \in E} |d_{u} - d_{v}|$

A L ({CeO}_{2} [p, q, t] .) = 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 |4 - 4| + 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 |6 - 4| + 2 p q t - p q - p t - q t + 1 |8 - 4| + 2 p q t - p q - p t - q t + p + q + t - 2 |6 - 6|

= 40 p q t + 4 p q + 4 p t + 4 q t + 4 p + 4 q + 4 t + 24

$I R L (G) = \sum_{UV \in E} |{lnd}_{u} - {lnd}_{v}|$

I R L ({CeO}_{2} [p, q, t]) = 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 |l n 4 - l n 4| + 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 |l n 6 - l n 4| + 2 p q t - p q - p t - q t + 1 |l n 8 - l n 4| + 2 p q t - p q - p t - q t + p + q + t - 2 |l n 6 - l n 6|

= (16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10) l n \frac{6}{4} + (2 p q t - p q - p t - q t + 1) l n \frac{8}{4}

$I R L U (G) = \sum_{UV \in E} \frac{|d_{u} - d_{v}|}{m i n (d_{u,} d_{v})}$

I R L U ({CeO}_{2} [p, q, t]) = 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 \frac{|4 - 4|}{4} + 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 \frac{|6 - 4|}{4} + 2 p q t - p q - p t - q t + 1 \frac{|8 - 4|}{4} + 2 p q t - p q - p t - q t + p + q + t - 2 \frac{|6 - 6|}{6}

= (16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10) \frac{1}{2} + (2 p q t - p q - p t - q t + 1)

$I R L F (G) = \sum_{UV \in E} \frac{|d_{u} - d_{v}|}{\sqrt{(d_{u} d_{v})}}$

I R L F ({CeO}_{2} [p, q, t]) = 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 \frac{|4 - 4|}{\sqrt{16}} + 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 \frac{|6 - 4|}{\sqrt{24}} + 2 p q t - p q - p t - q t + 1 \frac{|8 - 4|}{\sqrt{32}} + 2 p q t - p q - p t - q t + p + q + t - 2 \frac{|6 - 6|}{\sqrt{36}}

= (16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10) \frac{|2|}{\sqrt{24}} + (2 p q t - p q - p t - q t + 1) \frac{|4|}{\sqrt{32}}

$I R F (G) = \sum_{UV \in E} {(d_{u} - d_{v})}^{2}$

I R F ({CeO}_{2} [p, q, t]) = 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 {(4 - 4)}^{2} + 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 {(6 - 4)}^{2} + 2 p q t - p q - p t - q t + 1 {(8 - 4)}^{2} + 2 p q t - p q - p t - q t + p + q + t - 2 {(6 - 6)}^{2}

= 96 p q t + 8 p + 8 q + 8 t + 56

$I R L A (G) = 2 \sum_{UV \in E} \frac{|d_{u} - d_{v}|}{(d_{u} + d_{v})}$

I R L A ({CeO}_{2} [p, q, t]) = 2 [16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 \frac{|4 - 4|}{8} + 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 \frac{|6 - 4|}{10} + 2 p q t - p q - p t - q t + 1 \frac{|8 - 4|}{12} + 2 p q t - p q - p t - q t + p + q + t - 2 \frac{|6 - 6|}{12}]

= 2 [(16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10) \frac{2}{10} + (2 p q t - p q - p t - q t + 1) \frac{4}{12}]

$I R D 1 = \sum_{UV \in E} l n \{1 + |d_{v} - d_{v}|\}$

I R D 1 = 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 l n {1 + |4 - 4|} + 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 l n {1 + |6 - 4|} + 2 p q t - p q - p t - q t + 1 l n {1 + |8 - 4|} + 2 p q t - p q - p t - q t + p + q + t - 2 l n {1 + |6 - 6|}

= (16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10) l n (3) + (2 p q t - p q - p t - q t + 1) l n (5) .

$I R A (G) = \sum_{UV \in E} {(d_{u}^{\frac{- 1}{2}} - d_{v}^{\frac{- 1}{2}})}^{2}$

I R A ({CeO}_{2} [p, q, t]) = 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 {(\frac{1}{\sqrt{4}} - \frac{1}{\sqrt{4}})}^{2} + 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 {(\frac{1}{\sqrt{6}} - \frac{1}{\sqrt{4}})}^{2} + 2 p q t - p q - p t - q t + 1 {(\frac{1}{\sqrt{8}} - \frac{1}{\sqrt{4}})}^{2} + 2 p q t - p q - p t - q t + p + q + t - 2 {(\frac{1}{\sqrt{6}} - \frac{1}{\sqrt{6}})}^{2}

= 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 {(\frac{1}{\sqrt{6}} - \frac{1}{\sqrt{4}})}^{2} + 2 p q t - p q - p t - q t + 1 {(\frac{1}{\sqrt{8}} - \frac{1}{\sqrt{4}})}^{2}

10.

$I R G A (G) = \sum_{UV \in E} l n \frac{d_{u} + d_{v}}{2 \sqrt{(d_{u} d_{v})}}$

I R G A ({CeO}_{2} [p, q, t]) = (16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10) l n \frac{|4 - 4|}{2 \sqrt{16}} + (16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10) l n \frac{|6 - 4|}{2 \sqrt{24}} + (2 p q t - p q - p t - q t + 1) l n \frac{|8 - 4|}{2 \sqrt{32}} + (2 p q t - p q - p t - q t + p + q + t - 2) l n \frac{|6 - 6|}{2 \sqrt{36}}

= (16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10) l n \frac{1}{\sqrt{24}} + (2 p q t - p q - p t - q t + 1) l n \frac{2}{\sqrt{32}}

11.

$I R B (G) = \sum_{UV \in E} {(d_{u}^{\frac{1}{2}} - d_{v}^{\frac{1}{2}})}^{2}$

I R B ({CeO}_{2} [p, q, t]) = (16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10) {(\sqrt{4} - \sqrt{4})}^{2} + (16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10) {(\sqrt{6} - \sqrt{4})}^{2} + (2 p q t - p q - p t - q t + 1) {(\sqrt{8} - \sqrt{4})}^{2} + (2 p q t - p q - p t - q t + p + q + t - 2) {(\sqrt{6} - \sqrt{6})}^{2}

= (16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10) {(\sqrt{6} - \sqrt{4})}^{2} + (2 p q t - p q - p t - q t + 1) {(\sqrt{8} - \sqrt{4})}^{2} .

12.

$IR R_{t} (G) = \frac{1}{2} \sum_{UV \in E} |d_{u} - d_{v}|$

IR R_{t} ({CeO}_{2} [p, q, t]) = \frac{1}{2} [16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 |4 - 4| + 16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10 |6 - 4| + 2 p q t - p q - p t - q t + 1 |8 - 4| + 2 p q t - p q - p t - q t + p + q + t - 2 |6 - 6|]

= 20 p q t + 2 p q + 2 p t + 2 q t + 2 p + 2 q + 2 t + 12 .

Table 1 Edge partition of Cerium Oxide

b e {C e O}_{2}

Number of edges ( $d_{u}, d_{v})$	Number of indices
(4,4)	$16 p q t + 4 p q + 4 p t + 4 q t + 2 p + 2 q + 2 t + 10$
(4,6)	$2 p q t + 2 p q + 2 p t + 2 q t - p - q - t - 5$
(4,8)	$2 p q t - p q - p t - q t + 1$
(6,6)	$2 p q t - p q - p t - q t + p + q + t - 2$

Table 2. Shows the values of these irregularity indices for some test values of parameter p, q t.

Table 2 Irregularity indices for Cerium Oxide

({C e O}_{2}

Irregularity Indices	p = 1, q = 1, t = 1	p = 2, q = 2, t = 2	p = 3, q = 3, t = 3	p = 4, q = 4, t = 4	p = 5, q = 5, t = 5
$I R D I F (G) = \sum_{U V \in E} \|\frac{d_{u}}{d_{v}} - \frac{d_{v}}{d_{v}}\|$	36.42	171.79	513.74	1160.07	2208.58
$A L (G) = \sum_{U V \in E} \|d_{u} - d_{v}\|$	88	416	1248	2824	5384
$I R L (G) = \sum_{U V \in E} \|{l n d}_{u} - {l n d}_{v}\|$	17.837497	83.733197	249.672897	562.894597	1070.636297
$I R L U (G) = \sum_{U V \in E} \frac{\|d_{u} - d_{v}\|}{m i n (d_{u,} d_{v})}$	22	104	312	706	1342
$I R L U (G) = \sum_{U V \in E} \frac{\|d_{u} - d_{v}\|}{\sqrt{(d_{u} d_{v})}}$	17.971162	84.364636	251.677097	567.575628	1097.737330
$I R F (G) = \sum_{U V \in E} {(d_{u} - d_{v})}^{2}$	176	872	2720	6296	12,176
$I R L A (G) = 2 \sum_{U V \in E} \frac{\|d_{u} - d_{v}\|}{(d_{u} + d_{v})}$	17.6757	82.5027	245.7677	553.7707	1052.8917
$I R D 1 = \sum_{U V \in E} l n \{1 + \|d_{v} - d_{v}\|\}$	48.3387	225.5720	669.07533	1503.6286	2854.0120
$I R A (G) = \sum_{U V \in E} {(d_{u}^{\frac{- 1}{2}} - d_{v}^{\frac{- 1}{2}})}^{2}$	0.370297	1.367747	5.381157	12.258307	23.470577
$I R G A (G) = \sum_{U V \in E} l n \frac{d_{u} + d_{v}}{2 \sqrt{(d_{u} d_{v})}}$	8.979147	41.291457	120.872127	269.438337	508.707267
$I R B (G) = \sum_{U V \in E} {(d_{u}^{\frac{1}{2}} - d_{v}^{\frac{1}{2}})}^{2}$	8.88909	43.43484	133.97469	308.14008	553.56245
$I R R_{t} (G) = \frac{1}{2} \sum_{U V \in E} \|d_{u} - d_{v}\|$	44	208	624	1412	2692

3 Graphical analysis and discussions

Present part is devoted to the irregularity analysis of above-mentioned Cerium Oxide $({CeO}_{2})$ . The graphs as indicated in Figs. 4–7 show irregularity behavior of any particular index with respect to the parameters involved in the structure. In the present case, the involved parameters are p, q and t. We obtain almost similar graphs for all irregularity measures which shows that irregularity tends to rise with the increase in the size parameters.

Graph of irregularity index IRRt cerium oxide when p, q are variables and t is fixed (t = 1) as shown in Figs. 4–7.

4 Conclusion

The behaviors and tendencies of topological irregularities can be determined by the graph. All graphs exhibit the similar nature of irregularity indices regardless of the particular irregularity index we prefer to study the irregularity of the cerium oxide. This graphical analysis can be used to model new nanostructures where irregularities can be controlled by controlling involved parameters. Using these graphs and above table of values, the estimate of properties of cerium oxide can be done. Due to the lack of resources regarding wet lab, we pose it as an open problem to give some families of hydrocarbons whose complexity depend on irregularity indices as has been done in (Gutman, 2018; Reti et al., 2018).

Author contributions

M.M. gave the idea, H.A. wrote the article and Jia-Bao Liu and Qi Zhang edited and verified the results.

Data availability statement

No such data has been used for this study.

Acknowledgements

We are thankful to potential reviewers for fruitful suggestions and comments. First author is thankful to the support of Uniersity of the Punjab and HEC of Pakistan.

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