The measure of irregularities of nanosheets

Zahid Iqbal; Muhammad Ishaq; Adnan Aslam; Muhammad Aamir; Wei Gao

doi:10.1515/phys-2020-0164

Article Open Access

The measure of irregularities of nanosheets

Zahid Iqbal , Muhammad Ishaq , Adnan Aslam , Muhammad Aamir and Wei Gao

Published/Copyright: August 3, 2020

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Open Physics Volume 18 Issue 1

Abstract

Nanosheets are two-dimensional polymeric materials, which are among the most active areas of investigation of chemistry and physics. Many diverse physicochemical properties of compounds are closely related to their underlying molecular topological descriptors. Thus, topological indices are fascinating beginning points to any statistical approach for attaining quantitative structure–activity (QSAR) and quantitative structure–property (QSPR) relationship studies. Irregularity measures are generally used for quantitative characterization of the topological structure of non-regular graphs. In various applications and problems in material engineering and chemistry, it is valuable to be well-informed of the irregularity of a molecular structure. Furthermore, the estimation of the irregularity of graphs is helpful for not only QSAR/QSPR studies but also different physical and chemical properties, including boiling and melting points, enthalpy of vaporization, entropy, toxicity, and resistance. In this article, we compute the irregularity measures of graphene nanosheet, H-naphtalenic nanosheet, SiO 2 nanosheet, and the nanosheet covered by C 3 and C 6 .

Keywords: molecular graph; irregularity indices; nanosheets

1 Introduction

Nanotechnology is a fast-flourishing field that needs the production, design, and exploitation of structures at the nanoscale. A nano-object with all three external dimensions in the nanoscale is described as a nanoparticle. One of the most outstanding applications of nanotechnology is in the field of medicine [15]. Nanomedicine plays an appreciable role in cancer treatment, drug delivery, and so on. Nanostructures, which have the scale less than 100 nm, include nanosheets, nanoparticles, and nanotubes. Nanosheets have a sharp edge and a large surface; due to these factors, they perform a prominent role in many applications such as optoelectronics [54], bioelectronics [35], energy storage [26,50,55], and catalysis [11]. Generally, nanosheets are inorganic materials, which can be formed from bulk crystalline layered materials that have excellent electrochemical performance, fascinating properties, high potential for separation applications, and functionalities because of their unusual molecular properties [57].

A modern trend in computational and mathematical chemistry is the use of topological approaches to characterize a molecular structure. These descriptors also have useful applications in quantitative structure–activity/quantitative structure–property (QSAR/QSPR) studies convenient for unknown drug discovery, molecular design, and hazard assessment of chemicals. Mathematical characterization of molecular graphs can be successfully obtained by the graph invariants [25,51]. For isomorphic graphs, the graph invariants of both graphs are either identical or have the same value. A topological index is a numeric measure that linked with a graph and characterizes its topology [28]. These graph invariants are sensitive to such structural aspects of molecules as symmetry, shape, size, the degree of complexity of atomic neighbourhoods, the content of heteroatoms, and bonding pattern. Topological descriptors have gained respectable significance in the last few years due to the ease of generation and the speed with which these calculations can be accomplished. These molecular descriptors are an appreciable part of the chemical graph theory. There are a lot of graph-related numerical descriptors, which are vitally important in nanotechnology and theoretical chemistry. Therefore, computation of these numerical descriptors is one of the attractive areas of research. Some prominent categories of numerical descriptors of graphs are distance-based, degree-based, and counting-related. Among degree-based descriptors, irregularity measures may play a prominent role in chemistry, particularly in the QSPR/QSAR studies [22,47] as well as in the network theory [9,12,48].

For a comprehensive study of different types of topological indices, see [17,18,19,20,24,31,33,36,58,59] and references therein. Throughout this article, all graphs are finite, undirected, and simple. In the chemical graph theory, a graph in which every vertex has the degree less than 5 is called as a molecular graph. Molecular graphs of fullerenes, cycloalkanes, and annulenes are examples of regular molecular graphs. A large majority of molecular graphs is non-regular; some are more non-regular than the others. Let M = ( ν ( M ) , ℰ ( M ) ) be such a graph with vertex set ν ( M ) and edge set ℰ ( M ) . The order M is the number of elements in its vertex set and size is the number of edges in its edge set. The vertices of M correspond to atoms, and an edge is associated with the chemical bond between two vertices. Let us denote the degree of any vertex x of a graph M by ω M ( x ) and is defined as the number of edges incident with x . For the detailed discussions about these graph invariants, we refer to [2,3,4,5,21,39,40,45,46,49,60]. Although plenty of studies have been executed on the distance-based and degree-based indices of molecular graphs, the analyses of irregularity measures for chemical structures still need attention. In [4,16,22,32], the irregularity measures of various chemical structures were investigated.

$Figure 1 2D structure of graphene M 1 ( a , c ) {M}_{1}(a,c) with c rows and a benzene rings in each row.$

Figure 1

2D structure of graphene M 1 ( a , c ) with c rows and a benzene rings in each row.

2 Irregularity indices of graphene

In this section, we find the irregularity indices of graphene. First, we describe the significance of graphene. It is constructed by carbon atoms. It is one of the most eye-catching nanomaterials due to its remarkable properties. It is used in a wide spectrum of implementations ranging from electronics to optics, biodevices, and sensors. Furthermore, it is the most efficient material for electromagnetic interference shielding [41,42,56]. Nowadays, graphene is attracting the interest of a lot of scientists, researchers, and industries worldwide due to its wide range of potential applications in different areas. Some topological aspects and applications of this structure are discussed in [6,34,43,44]. We denote graphene nanosheet by M 1 ( a , c ) with c rows of benzene rings and a benzene rings in each row (Figure 1).

Theorem 2.1

For c = 1 and a ≥ 1 , the irregularity indices of graphene M 1 ( a , c ) are as follows:

IRDIF ( M 1 ( a , c ) ) = 10 a − 10 3 ,
AL ( M 1 ( a , c ) ) = 4 a − 4 ,
IRL ( M 1 ( a , c ) ) = ( 4 a − 4 ) ln 2 ln 3 ,
IRLU ( M 1 ( a , c ) ) = 2 a − 2 ,
IRLF ( M 1 ( a , c ) ) = 4 a − 4 6 ,
IRF ( M 1 ( a , c ) ) = 4 a − 4 ,
IRLA ( M 1 ( a , c ) ) = 8 a − 8 5 ,
IRDI ( M 1 ( a , c ) ) = ( 4 a − 4 ) ln 2 ,
IRA ( M 1 ( a , c ) ) = 10 − 4 2 3 a − 10 + 4 2 3 3 ,
IRGA ( M 1 ( a , c ) ) = ( 4 a − 4 ) ln 5 6 12 ,
IRB ( M 1 ( a , c ) ) = − 8 2 3 + 20 a + 8 2 3 − 20 , a n d
IRR t ( M 1 ( a , c ) ) = 2 a − 2 .

Proof

For c = 1 and a ≥ 1 , let us consider that the subset ℰ k l ⊂ ℰ ( M 1 ( a , c ) ) contains the edges having end vertices of degrees k and l . We divide ℰ ( M 1 ( a , c ) ) into subsets on the basis of the degrees of end vertices of edges. The cardinalities of these subsets are | ℰ 22 | = 6 , | ℰ 23 | = 4 a − 4 , and | ℰ 33 | = a − 1 . With the help of these data and the representations of irregularity indices shown in Table 1, the precise values of these indices can be computed in the following way:

IRDIF ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ω M 1 ( a , c ) ( d ) ω M 1 ( a , c ) ( e ) − ω M 1 ( a , c ) ( e ) ω M 1 ( a , c ) ( d ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 23 + ∑ d e ∈ ℰ 33 ω M 1 ( a , c ) ( d ) ω M 1 ( a , c ) ( e ) − ω M 1 ( a , c ) ( e ) ω M 1 ( a , c ) ( d ) = 6 2 2 − 2 2 + ( 4 a − 4 ) 2 3 − 3 2 + ( a − 1 ) 3 3 − 3 3 = 10 a − 10 3 .

AL ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) | ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) | = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 23 + ∑ d e ∈ ℰ 33 ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) = 6 | 2 − 2 | + ( a − 1 ) | 3 − 3 | + ( 4 a − 4 ) | 2 − 3 | = 4 a − 4 .

IRL ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ln ( ω M 1 ( a , c ) ( d ) ) − ln ( ω M 1 ( a , c ) ( e ) ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 ln ( ω M 1 ( a , c ) ( d ) ) − ln ( ω M 1 ( a , c ) ( e ) ) = 6 | ln 2 − ln 2 | + ( a − 1 ) | ln 3 − ln 3 | + ( 4 a − 4 ) | ln 2 − ln 3 | = ( 4 a − 4 ) ln 2 ln 3 .

IRLF ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) | ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) | ( ω M 1 ( a , c ) ( d ) ω M 1 ( a , c ) ( e ) ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 | ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) | ( ω M 1 ( a , c ) ( d ) ω M 1 ( a , c ) ( e ) ) = 6 | 2 − 2 | 2 × 2 + ( a − 1 ) | 3 − 3 | 3 × 3 + ( 4 a − 4 ) | 2 − 3 | 2 × 3 = 4 a − 4 6 .

IRF ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ( ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) ) 2 = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 ( ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) ) 2 = 6 ( 2 − 2 ) 2 + ( a − 1 ) ( 3 − 3 ) 2 + ( 4 a − 4 ) ( 2 − 3 ) 2 = 4 a − 4 .

IRLA ( M 1 ( a , c ) ) = 2 ∑ d e ∈ ℰ ( M 1 ( a , c ) ) | ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) | ( ω M 1 ( a , c ) ( d ) + ω M 1 ( a , c ) ( e ) ) = 2 ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 | ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) | ( ω M 1 ( a , c ) ( d ) + ω M 1 ( a , c ) ( e ) ) = 2 6 | 2 − 2 | 2 + 2 + ( a − 1 ) | 3 − 3 | 3 + 3 + ( 4 a − 4 ) | 2 − 3 | 2 + 3 = 8 a − 8 5 .

IRDI ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ln { 1 + | ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) | } = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 ln { 1 + | ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) | } = 6 ln { 1 + | 2 − 2 | } + ( 4 a − 4 ) ln { 1 + | 2 − 3 | } + ( a − 1 ) ln { 1 + | 3 − 3 | } = ( 4 a − 4 ) ln 2 .

IRA ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) 1 ω M 1 ( a , c ) ( d ) − 1 ω M 1 ( a , c ) ( e ) 2 = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 1 ω M 1 ( a , c ) ( d ) − 1 ω M 1 ( a , c ) ( e ) 2 = 6 1 2 − 1 2 2 + ( a − 1 ) 1 3 − 1 3 2 + ( 4 a − 4 ) 1 2 − 1 3 2 = 10 − 4 2 3 a − 10 + 4 2 3 3 .

IRGA ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ln ω M 1 ( a , c ) ( d ) + ω M 1 ( a , c ) ( e ) 2 ω M 1 ( a , c ) ( d ) ω M 1 ( a , c ) ( e ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 × ln ω M 1 ( a , c ) ( d ) + ω M 1 ( a , c ) ( e ) 2 ω M 1 ( a , c ) ( d ) ω M 1 ( a , c ) ( e ) = 6 ln 2 + 2 2 2 × 2 + ( a − 1 ) ln 3 + 3 2 3 × 3 + ( 4 a − 4 ) ln 2 + 3 2 2 × 3 = ( 4 a − 4 ) ln 5 6 12 .

IRB ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ( ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) ) 2 = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 23 + ∑ d e ∈ ℰ 33 ( ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) ) 2 = 6 2 − 2 2 + ( a − 1 ) 3 − 3 2 + ( 4 a − 4 ) 2 − 3 2 = − 8 2 3 + 20 a + 8 2 3 − 20 .

IRR t ( M 1 ( a , c ) ) = 1 2 ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) = 1 2 ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) = 1 2 ( 6 | 2 − 2 | + ( a − 1 ) | 3 − 3 | + ( 4 a − 4 ) | 2 − 3 | ) = 2 a − 2 . □

Theorem 2.2

For c ≥ 2 and a ≥ 1 , the irregularity indices of graphene M 1 ( a , c ) are as follows:

IRDIF ( M 1 ( a , c ) ) = 10 a + 5 c − 10 3 ,
AL ( M 1 ( a , c ) ) = 4 a + 2 c − 4 ,
IRL ( M 1 ( a , c ) ) = ( 4 a + 2 c − 4 ) ln 2 ln 3 ,
IRLU ( M 1 ( a , c ) ) = 2 a + c − 2 ,
IRLF ( M 1 ( a , c ) ) = 6 ( 4 a + 2 c − 4 ) 6 ,
IRF ( M 1 ( a , c ) ) = 4 a + 2 c − 4 ,
IRLA ( M 1 ( a , c ) ) = 2 ( 4 a + 2 c − 4 ) 5 ,
IRDI ( M 1 ( a , c ) ) = ( 4 a + 2 c − 4 ) ln 2 ,
IRA ( M 1 ( a , c ) ) = 5 − 2 2 3 c + 10 − 4 2 3 a − 10 + 4 2 3 3 ,
IRGA ( M 1 ( a , c ) ) = ( 4 a + 2 c − 4 ) ln 5 6 12 ,
IRB ( M 1 ( a , c ) ) = ( − 4 2 3 + 10 ) c + ( − 8 2 3 + 20 ) a + 8 2 3 − 20 , a n d
IRR t ( M 1 ( a , c ) ) = 2 a + c − 2 .

Proof

For c ≥ 2 and a ≥ 1 , let us consider that the subset ℰ k l ⊂ ℰ ( M 1 ( a , c ) ) contains the edges having end vertices of degrees k and l . We divide ℰ ( M 1 ( a , c ) ) into subsets on the basis of the degrees of end vertices of edges. The cardinalities of these subsets are | ℰ 22 | = c + 4 , | ℰ 23 | = 4 a + 2 c − 4 , and | ℰ 33 | = 3 a c − 2 a − c − 1 . With the help of these data and the representations of irregularity indices shown in Table 1, the precise values of these indices can be computed in the following way:

IRDIF ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ω M 1 ( a , c ) ( d ) ω M 1 ( a , c ) ( e ) − ω M 1 ( a , c ) ( e ) ω M 1 ( a , c ) ( d ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 23 + ∑ d e ∈ ℰ 33 ω M 1 ( a , c ) ( d ) ω M 1 ( a , c ) ( e ) − ω M 1 ( a , c ) ( e ) ω M 1 ( a , c ) ( d ) = ( c + 4 ) 2 2 − 2 2 + ( 3 a c − 2 a − c − 1 ) 3 3 − 3 3 + ( 4 a + 2 c − 4 ) 2 3 − 3 2 = 10 a + 5 c − 10 3 .

AL ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 23 + ∑ d e ∈ ℰ 33 ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) = ( c + 4 ) | 2 − 2 | + ( 3 a c − 2 a − c − 1 ) | 3 − 3 | + ( 4 a + 2 c − 4 ) | 2 − 3 | = 4 a + 2 c − 4 .

IRL ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ln ( ω M 1 ( a , c ) ( d ) ) − ln ( ω M 1 ( a , c ) ( e ) ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 ln ( ω M 1 ( a , c ) ( d ) ) − ln ( ω M 1 ( a , c ) ( e ) ) = ( c + 4 ) | ln 2 − ln 2 | + ( 4 a + 2 c − 4 ) | ln 2 − ln 3 | + ( 3 a c − 2 a − c − 1 ) | ln 3 − ln 3 | = ( 4 a + 2 c − 4 ) ln 2 ln 3 .

IRLF ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) | ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) | ( ω M 1 ( a , c ) ( d ) ω M 1 ( a , c ) ( e ) ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 | ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) | ( ω M 1 ( a , c ) ( d ) ω M 1 ( a , c ) ( e ) ) = ( c + 4 ) | 2 − 2 | 2 × 2 + ( 3 a c − 2 a − c − 1 ) | 3 − 3 | 3 × 3 + ( 4 a + 2 c − 4 ) | 2 − 3 | 2 × 3 = 6 ( 4 a + 2 c − 4 ) 6 .

IRF ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ( ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) ) 2 = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 ( ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) ) 2 = ( c + 4 ) ( 2 − 2 ) 2 + ( 3 a c − 2 a − c − 1 ) ( 3 − 3 ) 2 + ( 4 a + 2 c − 4 ) ( 2 − 3 ) 2 = 4 a + 2 c − 4 .

IRLA ( M 1 ( a , c ) ) = 2 ∑ d e ∈ ℰ ( M 1 ( a , c ) ) | ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) | ( ω M 1 ( a , c ) ( d ) + ω M 1 ( a , c ) ( e ) ) = 2 ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 | ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) | ( ω M 1 ( a , c ) ( d ) + ω M 1 ( a , c ) ( e ) ) = 2 ( c + 4 ) | 2 − 2 | 2 + 2 + ( 3 a c − 2 a − c − 1 ) | 3 − 3 | 3 + 3 + (4 a + 2 c − 4) |2 − 3| 2 + 3 = 2(4 a + 2 c − 4) 5 .

IRDI ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ln { 1 + | ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) | } = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 ln { 1 + | ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) | } = ( c + 4 ) ln { 1 + | 2 − 2 | } + ( 4 a + 2 c − 4 ) ln { 1 + | 2 − 3 | } + ( 3 a c − 2 a − c − 1 ) ln { 1 + | 3 − 3 | } = ( 4 a + 2 c − 4 ) ln 2 .

IRA ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) 1 ω M 1 ( a , c ) ( d ) − 1 ω M 1 ( a , c ) ( e ) 2 = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 × 1 ω M 1 ( a , c ) ( d ) − 1 ω M 1 ( a , c ) ( e ) 2 = ( c + 4 ) 1 2 − 1 2 2 + ( 3 a c − 2 a − c − 1 ) × 1 3 − 1 3 2 + ( 4 a + 2 c − 4 ) 1 2 − 1 3 2 = 5 − 2 2 3 c + 10 − 4 2 3 a − 10 + 4 2 3 3 .

IRGA ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ln ω M 1 ( a , c ) ( d ) + ω M 1 ( a , c ) ( e ) 2 ω M 1 ( a , c ) ( d ) ω M 1 ( a , c ) ( e ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 × ln ω M 1 ( a , c ) ( d ) + ω M 1 ( a , c ) ( e ) 2 ω M 1 ( a , c ) ( d ) ω M 1 ( a , c ) ( e ) = ( c + 4 ) ln 2 + 2 2 2 × 2 + ( 3 a c − 2 a − c − 1 ) ln 3 + 3 2 3 × 3 + ( 4 a + 2 c − 4 ) ln 2 + 3 2 2 × 3 = ( 4 a + 2 c − 4 ) ln 5 6 12 .

IRB ( M 1 ( a , c ) ) = ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ( ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) 2 = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 23 + ∑ d e ∈ ℰ 33 × ( ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) 2 = ( c + 4 ) ( 2 − 2 2 + ( 3 a c − 2 a − c − 1 ) ( 3 − 3 2 + ( 4 a + 2 c − 4 ) 2 − 3 2 = − 4 2 3 + 10 c + − 8 2 3 + 20 a + 8 2 3 − 20 .

IRR t ( M 1 ( a , c ) ) = 1 2 ∑ d e ∈ ℰ ( M 1 ( a , c ) ) ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) = 1 2 ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 ω M 1 ( a , c ) ( d ) − ω M 1 ( a , c ) ( e ) = 1 2 ( ( c + 4 ) | 2 − 2 | + ( 3 a c − 2 a − c − 1 ) | 3 − 3 | + ( 4 a + 2 c − 4 ) | 2 − 3 | ) = 2 a + c − 2 . □

Table 1

Irregularity indices

Irregularity indices	Mathematical form
IRDIF index [47]	IRDIF ( M ) = ∑ d e ∈ ℰ ( M ) ω M ( d ) ω M ( e ) − ω M ( e ) ω M ( d )
AL index [7]	AL ( M ) = ∑ d e ∈ ℰ ( M ) \| ω M ( d ) − ω M ( e ) \|
IRL index [53]	IRL ( M ) = ∑ d e ∈ ℰ ( M ) \| ln ( ω M ( d ) ) − ln ( ω M ( e ) ) \|
IRLU index [53]	IRLU ( M ) = ∑ d e ∈ ℰ ( M ) \| ω M ( d ) − ω M ( e ) \| min ( ω M ( d ) , ω M ( e ) )
IRLF index [47]	IRLF ( M ) = ∑ d e ∈ ℰ ( M ) \| ω M ( d ) − ω M ( e ) \| ( ω M ( d ) ω M ( e ) )
IRF index [14,23]	IRF ( M ) = ∑ d e ∈ ℰ ( M ) ( ω M ( d ) − ω M ( e ) ) 2
IRLA index [47]	IRLA ( M ) = 2 ∑ d e ∈ ℰ ( M ) \| ω M ( d ) − ω M ( e ) \| ( ω M ( d ) + ω M ( e ) )
IRDI index [47]	IRDI ( M ) = ∑ d e ∈ ℰ ( M ) ln { 1 + \| ω M ( d ) − ω M ( e ) \| }
IRA index [47]	IRA ( M ) = ∑ d e ∈ ℰ ( M ) 1 ω M ( d ) − 1 ω M ( e ) 2
IRGA index [47]	IRGA ( M ) = ∑ d e ∈ ℰ ( M ) ln ω M ( d ) + ω M ( e ) 2 ω M ( d ) ω M ( e )
IRB index [47]	IRB ( M ) = ∑ d e ∈ ℰ ( M ) ω M ( d ) − ω M ( e ) 2
IR R t index [1]	IRR t ( M ) = 1 2 ∑ d e ∈ ℰ ( M ) \| ω M ( d ) − ω M ( e ) \|

3 Irregularity indices of H-naphtalenic nanosheet

A carbon nanotube is made by a layer of carbon atoms, which are bonded together in a hexagonal mesh. These were discovered in [37]. A H-naphtalenic nanosheet H ( r , t ) is formed by alternating squares C 4 , octagons C 8 , and hexagons C 6 , as shown in Figure 2. The vertex set V ( H ( r , t ) ) has the cardinality 10 r t , where r stands for the number of paired hexagons in each alternant row with C 4 cycle and t represents the number of rows consisting of C 4 . Some topological aspects of this structure are discussed in [30,29].

Theorem 3.1

The irregularity indices of H-Naphtalenic nanosheet H ( r , t ) are as follows:

IRDIF ( H ( r , t ) ) = 20 r + 10 t − 20 3 ,
AL ( H ( r , t ) ) = 8 r + 4 t − 8 ,
IRL ( H ( r , t ) ) = ( 8 r + 4 t − 8 ) ln 2 ln 3 ,
IRLU ( H ( r , t ) ) = 4 r + 2 t − 4 ,
IRLF ( H ( r , t ) ) = 3 2 + 4 3 + ( 6 r + 8 t − 9 ) 6 6 ,
IRF ( H ( r , t ) ) = 8 r + 4 t − 8 ,
IRLA ( H ( r , t ) ) = 16 r + 8 t − 16 5 ,
IRDI ( H ( r , t ) ) = ln 2 ( 8 r + 4 t − 8 ) ,
IRA ( H ( r , t ) ) = 10 3 − 4 2 3 3 t + 20 3 − 8 2 3 3 r − 20 3 + 8 2 3 3 ,
IRGA ( H ( r , t ) ) = 4 ln 5 6 12 t + 8 ln 5 6 12 r − 8 ln 5 6 12 ,
IRB ( H ( r , t ) ) = − 8 2 3 + 20 t + − 16 2 3 + 40 r + 16 2 3 − 40 , a n d
IRR t ( H ( r , t ) ) = 3 r + 4 t − 3 .

Proof

Let us consider that the subset ℰ k l ⊂ ℰ ( H ( r , t ) ) contains the edges having end vertices of degrees k and l . We divide ℰ ( H ( r , t ) ) into subsets on the basis of the degrees of end vertices of edges. The cardinalities of these subsets are | ℰ 22 | = 2 t + 4 , | ℰ 23 | = 4 ( 2 r + t − 2 ) , and | ℰ 33 | = r ( 15 t − 14 ) − 4 ( t − 1 ) . With the help of these data and the representations of irregularity indices shown in Table 1, the precise values of these indices can be computed in the following way:

IRDIF ( H ( r , t ) ) = ∑ d e ∈ ℰ ( H ( r , t ) ) ω H ( r , t ) ( d ) ω H ( r , t ) ( e ) − ω H ( r , t ) ( e ) ω H ( r , t ) ( d ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 23 + ∑ d e ∈ ℰ 33 ω H ( r , t ) ( d ) ω H ( r , t ) ( e ) − ω H ( r , t ) ( e ) ω H ( r , t ) ( d ) = ( 2 t + 4 ) 2 2 − 2 2 + ( 4 ( 2 r + t − 2 ) ) 2 3 − 3 2 + ( r ( 15 t − 14 ) − 4 ( t − 1 ) ) 3 3 − 3 3 = 20 r + 10 t − 20 3 .

AL ( H ( r , t ) ) = ∑ d e ∈ ℰ ( H ( r , t ) ) ω H ( r , t ) ( d ) − ω H ( r , t ) ( e ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 23 + ∑ d e ∈ ℰ 33 ω H ( r , t ) ( d ) − ω H ( r , t ) ( e ) = ( 2 t + 4 ) | 2 − 2 | + ( r ( 15 t − 14 ) − 4 ( t − 1 ) ) | 3 − 3 | + ( 4 ( 2 r + t − 2 ) ) | 2 − 3 | = 8 r + 4 t − 8 .

IRL ( H ( r , t ) ) = ∑ d e ∈ ℰ ( H ( r , t ) ) ln ( ω H ( r , t ) ( d ) ) − ln ( ω H ( r , t ) ( e ) ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 ln ( ω H ( r , t ) ( d ) ) − ln ( ω H ( r , t ) ( e ) ) = ( r ( 15 t − 14 ) − 4 ( t − 1 ) ) | ln 3 − ln 3 | + ( 4 ( 2 r + t − 2 ) ) | ln 2 − ln 3 | + ( 2 t + 4 ) | ln 2 − ln 2 | = ( 8 r + 4 t − 8 ) ln 2 ln 3 .

IRLF ( H ( r , t ) ) = ∑ d e ∈ ℰ ( H ( r , t ) ) | ω H ( r , t ) ( d ) − ω H ( r , t ) ( e ) | ( ω H ( r , t ) ( d ) ω H ( r , t ) ( e ) ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 | ω H ( r , t ) ( d ) − ω H ( r , t ) ( e ) | ( ω H ( r , t ) ( d ) ω H ( r , t ) ( e ) ) = ( r ( 15 t − 14 ) − 4 ( t − 1 ) ) | 3 − 3 | 3 × 3 + ( 4 ( 2 r + t − 2 ) ) | 2 − 3 | 2 × 3 + ( 2 t + 4 ) | 2 − 2 | 2 × 2 = 3 2 + 4 3 + ( 6 r + 8 t − 9 ) 6 6 .

IRF ( H ( r , t ) ) = ∑ d e ∈ ℰ ( H ( r , t ) ) ( ω H ( r , t ) ( d ) − ω H ( r , t ) ( e ) ) 2 = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 ( ω H ( r , t ) ( d ) − ω H ( r , t ) ( e ) ) 2 = ( r ( 15 t − 14 ) − 4 ( t − 1 ) ) ( 3 − 3 ) 2 + ( 4 ( 2 r + t − 2 ) ) ( 2 − 3 ) 2 + ( 2 t + 4 ) ( 2 − 2 ) 2 = 8 r + 4 t − 8 .

IRLA ( H ( r , t ) ) = 2 ∑ d e ∈ ℰ ( H ( r , t ) ) | ω H ( r , t ) ( d ) − ω H ( r , t ) ( e ) | ( ω H ( r , t ) ( d ) + ω H ( r , t ) ( e ) ) = 2 ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 | ω H ( r , t ) ( d ) − ω H ( r , t ) ( e ) | ( ω H ( r , t ) ( d ) + ω H ( r , t ) ( e ) ) = 2 ( 2 t + 4 ) | 2 − 2 | 2 + 2 + ( r ( 15 t − 14 ) − 4 ( t − 1 ) ) | 3 − 3 | 3 + 3 + ( 4 ( 2 r + t − 2 ) ) | 2 − 3 | 2 + 3 = 16 r + 8 t − 16 5 .

IRDI ( H ( r , t ) ) = ∑ d e ∈ ℰ ( H ( r , t ) ) ln { 1 + | ω H ( r , t ) ( d ) − ω H ( r , t ) ( e ) | } = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 ln 1 + ω H ( r , t ) ( d ) − ω H ( r , t ) ( e ) = ( 2 t + 4 ) ln { 1 + | 2 − 2 | } + ( 4 ( 2 r + t − 2 ) ) ln { 1 + | 2 − 3 | } + ( r ( 15 t − 14 ) − 4 ( t − 1 ) ) ln { 1 + | 3 − 3 | } = ln 2 ( 8 r + 4 t − 8 ) .

IRA ( H ( r , t ) ) = ∑ d e ∈ ℰ ( H ( r , t ) ) 1 ω H ( r , t ) ( d ) − 1 ω H ( r , t ) ( e ) 2 = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 1 ω H ( r , t ) ( d ) − 1 ω H ( r , t ) ( e ) 2 = ( 2 t + 4 ) 1 2 − 1 2 2 + ( r ( 15 t − 14 ) − 4 ( t − 1 ) ) 1 3 − 1 3 2 + ( 4 ( 2 r + t − 2 ) ) 1 2 − 1 3 2 = 10 3 − 4 2 3 3 t + 20 3 − 8 2 3 3 r − 20 3 + 8 2 3 3 .

IRGA ( H ( r , t ) ) = ∑ d e ∈ ℰ ( H ( r , t ) ) ln ω H ( r , t ) ( d ) + ω H ( r , t ) ( e ) 2 ω H ( r , t ) ( d ) ω H ( r , t ) ( e ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 ln ω H ( r , t ) ( d ) + ω H ( r , t ) ( e ) 2 ω H ( r , t ) ( d ) ω H ( r , t ) ( e ) = ( 2 t + 4 ) ln 2 + 2 2 2 × 2 + ( r ( 15 t − 14 ) − 4 ( t − 1 ) ) ln 3 + 3 2 3 × 3 + ( 4 ( 2 r + t − 2 ) ) ln 2 + 3 2 2 × 3 = 4 ln 5 6 12 t + 8 ln 5 6 12 r − 8 ln 5 6 12 .

IRB ( H ( r , t ) ) = ∑ d e ∈ ℰ ( H ( r , t ) ) ( ω H ( r , t ) ( d ) − ω H ( r , t ) ( e ) 2 = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 23 ( ω H ( r , t ) ( d ) − ω H ( r , t ) ( e ) 2 = ( 2 t + 4 ) 2 − 2 2 + ( r ( 15 t − 14 ) − 4 ( t − 1 ) ) 3 − 3 2 + ( 4 ( 2 r + t − 2 ) ) 2 − 3 2 = − 8 2 3 + 20 t + − 16 2 3 + 40 r + 16 2 3 − 40 .

IRR t ( H ( r , t ) ) = 1 2 ∑ d e ∈ ℰ ( H ( r , t ) ) ω H ( r , t ) ( d ) − ω H ( r , t ) ( e ) = ∑ d e ∈ ℰ 22 + ∑ d e ∈ ℰ 33 + ∑ d e ∈ ℰ 23 1 2 ω H ( r , t ) ( d ) − ω H ( r , t ) ( e ) = 1 2 ( ( 2 t + 4 ) | 2 − 2 | + ( r ( 15 t − 14 ) − 4 ( t − 1 ) ) | 3 − 3 | + ( 4 ( 2 r + t − 2 ) ) | 2 − 3 | ) = 4 r + 2 t − 4 . □

Figure 2

H-Naphtalenic nanosheet H [ r , t ] .

4 Irregularity indices of SiO 2 nanosheet

Silicon dioxide (SiO 2 ) is a prominent material. It has various applications in biology and semiconductor industry. It is used in tablet-making, as an anti-caking agent, as a disintegrant, as an absorbent, and in drugs as an inactive filler. In recent years, silica-based nanomaterials have gained significant interest because of their specific surface area, tunable particle size, high drug loading capability, abundant Si–OH bonds on the particle surface, chemical and thermal stability, and sustained drug release thereby enhancing the bioavailability of drugs [10,27,38,52,61]. It has a giant covalent structure in which every oxygen atom is covalently bonded to two silicon atoms, and every silicon atom is covalently bonded to four oxygen atoms. The molecular structure of silicon dioxide is an octagon structure, and these octagons construct a SiO 2 layer structure. Let M 3 ( q , n ) be the molecular graph of the SiO 2 layer structure, where the number of rows is denoted by q and the number of columns by n, and the structure in Figure 4 is of dimension ( 5 , 6 ) . Some topological aspects of this structure are discussed in [8,13]. We now move on to attain the precise values of irregularity indices of SiO 2 nanosheet (Figure 4).

Theorem 4.1

The irregularity indices of M 3 ( q , n ) are as follows:

IRDIF ( M 3 ( q , n ) ) = 21 q + 21 n + 12 q n + 30 2 ,
AL ( M 3 ( q , n ) ) = 10 q + 10 n + 12 + 8 q n ,
IRL ( M 3 ( q , n ) ) = − 2 ln 2 ( 2 q n + 3 q + 3 n + 4 ) ,
IRLU ( M 3 ( q , n ) ) = 8 q + 8 n + 12 + 4 q n ,
IRLF ( M 3 ( q , n ) ) = 3 q + 3 n + 6 + ( 2 q + 2 n + 4 q n ) 2 2 ,
IRF ( M 3 ( q , n ) ) = 26 q + 26 n + 52 ,
IRLA ( M 3 ( q , n ) ) = 2 ( 28 q + 28 n + 36 + 20 q n ) 15 ,
IRDI ( M 3 ( q , n ) ) = ( 4 ln 2 + 2 ln 3 ) n + ( 4 ln 2 + 2 ln 3 ) q + 8 ln 2 + 4 q n ln 3 ,
IRA ( M 3 ( q , n ) ) = − 2 2 + 3 q n − q 2 − n 2 + 2 q + 2 n + 1 ,
IRGA ( M 3 ( q , n ) ) = 2 ln 5 4 + 2 ln 3 4 2 ) n + 2 ln 5 4 + 2 ln 3 4 2 q + 4 ln 5 4 + 4 ln 3 4 2 q n ,
IRB ( M 3 ( q , n ) ) = − 16 2 + 24 q n + 14 − 8 2 n + 14 − 8 2 q + 4 , a n d
IRR t ( M 3 ( q , n ) ) = 5 q + 5 n + 6 + 4 q n .

Proof

Let us consider that the subset ℰ k l ⊂ ℰ ( M 3 ( q , n ) ) contains the edges having end vertices of degrees k and l . We divide ℰ ( M 3 ( q , n ) ) into subsets on the basis of the degrees of end vertices of edges. The cardinalities of these subsets are | ℰ 14 | = 2 q + 2 n + 4 and | ℰ 24 | = 2 q + 2 n + 4 q n . With the help of these data and the representations of irregularity indices shown in Table 1, the precise values of these indices can be computed in the following way:

IRDIF ( M 3 ( q , n ) ) = ∑ d e ∈ ℰ ( M 3 ( q , n ) ) ω M 3 ( q , n ) ( d ) ω M 3 ( q , n ) ( e ) − ω M 3 ( q , n ) ( e ) ω M 3 ( q , n ) ( d ) = ∑ d e ∈ ℰ 14 + ∑ d e ∈ ℰ 24 ω M 3 ( q , n ) ( d ) ω M 3 ( q , n ) ( e ) − ω M 3 ( q , n ) ( e ) ω M 3 ( q , n ) ( d ) = ( 2 q + 2 n + 4 ) 1 4 − 4 1 + ( 2 q + 2 n + 4 q n ) 2 4 − 4 2 = 21 q + 21 n + 12 q n + 30 2 .

AL ( M 3 ( q , n ) ) = ∑ d e ∈ ℰ ( M 3 ( q , n ) ) ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) = ∑ d e ∈ ℰ 14 + ∑ d e ∈ ℰ 24 ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) = ( 2 q + 2 n + 4 ) | 1 − 4 | + ( 2 q + 2 n + 4 q n ) | 2 − 4 | = 10 q + 10 n + 12 + 8 q n .

IRL ( M 3 ( q , n ) ) = ∑ d e ∈ ℰ ( M 3 ( q , n ) ) ln ( ω M 3 ( q , n ) ( d ) ) − ln ( ω M 3 ( q , n ) ( e ) ) = ∑ d e ∈ ℰ 14 + ∑ d e ∈ ℰ 24 ln ( ω M 3 ( q , n ) ( d ) ) − ln ( ω M 3 ( q , n ) ( e ) ) = ( 2 q + 2 n + 4 ) | ln 1 − ln 4 | + ( 2 q + 2 n + 4 q n ) | ln 2 − ln 4 | = − 2 ln 2 ( 2 q n + 3 q + 3 n + 4 ) .

IRLU ( M 3 ( q , n ) ) = ∑ d e ∈ ℰ ( M 3 ( q , n ) ) | ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) | min ( ω M 3 ( q , n ) ( d ) , ω M 3 ( q , n ) ( e ) ) = ∑ d e ∈ ℰ 14 + ∑ d e ∈ ℰ 24 | ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) | min ( ω M 3 ( q , n ) ( d ) , ω M 3 ( q , n ) ( e ) ) = ( 2 q + 2 n + 4 ) | 1 − 4 | min ( 1 , 4 ) + ( 2 q + 2 n + 4 q n ) | 2 − 4 | min ( 2 , 4 ) = 8 q + 8 n + 12 + 4 q n .

IRLF ( M 3 ( q , n ) ) = ∑ d e ∈ ℰ ( M 3 ( q , n ) ) | ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) | ( ω M 3 ( q , n ) ( d ) ω M 3 ( q , n ) ( e ) ) = ∑ d e ∈ ℰ 14 + ∑ d e ∈ ℰ 24 | ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) | ( ω M 3 ( q , n ) ( d ) ω M 3 ( q , n ) ( e ) ) = ( 2 q + 2 n + 4 ) | 1 − 4 | 1 × 4 + ( 2 q + 2 n + 4 q n ) | 2 − 4 | 2 × 4 = 3 q + 3 n + 6 + ( 2 q + 2 n + 4 q n ) 2 2 .

IRF ( M 3 ( q , n ) ) = ∑ d e ∈ ℰ ( M 3 ( q , n ) ) ( ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) ) 2 = ∑ d e ∈ ℰ 14 + ∑ d e ∈ ℰ 24 ( ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) ) 2 = ( 2 q + 2 n + 4 ) ( 1 − 4 ) 2 + ( 2 q + 2 n + 4 q n ) ( 2 − 4 ) 2 = 26 q + 26 n + 36 + 16 q n .

IRLA ( M 3 ( q , n ) ) = 2 ∑ d e ∈ ℰ ( M 3 ( q , n ) ) | ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) | ( ω M 3 ( q , n ) ( d ) + ω M 3 ( q , n ) ( e ) ) = 2 ∑ d e ∈ ℰ 14 + ∑ d e ∈ ℰ 24 | ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) | ( ω M 3 ( q , n ) ( d ) + ω M 3 ( q , n ) ( e ) ) = 2 ( 2 q + 2 n + 4 ) | 1 − 4 | 1 + 4 + ( 2 q + 2 n + 4 q n ) | 2 − 4 | 2 + 4 = 2 ( 28 q + 28 n + 36 + 20 q n ) 15 .

IRDI ( M 3 ( q , n ) ) = ∑ d e ∈ ℰ ( M 3 ( q , n ) ) ln 1 + ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) = ∑ d e ∈ ℰ 14 + ∑ d e ∈ ℰ 24 ln 1 + ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) = ( 2 q + 2 n + 4 ) ln { 1 + | 1 − 4 | } + ( 2 q + 2 n + 4 q n ) ln { 1 + | 2 − 4 | } = ( 4 ln 2 + 2 ln 3 ) n + ( 4 ln 2 + 2 ln 3 ) q + 8 ln 2 + 4 q n ln 3 .

IRA ( M 3 ( q , n ) ) = ∑ d e ∈ ℰ ( M 3 ( q , n ) ) 1 ω M 3 ( q , n ) ( d ) − 1 ω M 3 ( q , n ) ( e ) 2 = ∑ d e ∈ ℰ 14 + ∑ d e ∈ ℰ 24 1 ω M 3 ( q , n ) ( d ) − 1 ω M 3 ( q , n ) ( e ) 2 = ( 2 q + 2 n + 4 ) 1 2 − 1 4 2 + ( 2 q + 2 n + 4 q n ) 1 2 − 1 4 2 = ( − 2 2 + 3 ) q n − q 2 − n 2 + 2 q + 2 n + 1 .

IRGA ( M 3 ( q , n ) ) = ∑ d e ∈ ℰ ( M 3 ( q , n ) ) ln ω M 3 ( q , n ) ( d ) + ω M 3 ( q , n ) ( e ) 2 ω M 3 ( q , n ) ( d ) ω M 3 ( q , n ) ( e ) = ∑ d e ∈ ℰ 14 + ∑ d e ∈ ℰ 24 ln ω M 3 ( q , n ) ( d ) + ω M 3 ( q , n ) ( e ) 2 ω M 3 ( q , n ) ( d ) ω M 3 ( q , n ) ( e ) = ( 2 q + 2 n + 4 ) ln 1 + 4 2 1 × 4 + ( 2 q + 2 n + 4 q n ) ln 2 + 4 2 2 × 4 = ln 5 4 + ln 3 4 2 ) 2 n + ln 5 4 + ln 3 4 2 2 q + 4 ln 5 4 + 4 ln 3 4 2 q n .

IRB ( M 3 ( q , n ) ) = ∑ d e ∈ ℰ ( M 3 ( q , n ) ) ( ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) 2 = ∑ d e ∈ ℰ 14 + ∑ d e ∈ ℰ 24 ( ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) 2 = ( 2 q + 2 n + 4 ) 1 − 4 2 + ( 2 q + 2 n + 4 q n ) 2 − 4 2 = − 16 2 + 24 q n + ( 14 − 8 2 ) n + ( 14 − 8 2 ) q + 4 .

IRR t ( M 3 ( q , n ) ) = 1 2 ∑ d e ∈ ℰ ( M 3 ( q , n ) ) ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) = 1 2 ∑ d e ∈ ℰ 14 + ∑ d e ∈ ℰ 24 ω M 3 ( q , n ) ( d ) − ω M 3 ( q , n ) ( e ) = 1 2 ( ( 2 q + 2 n + 4 ) | 1 − 4 | + ( 2 q + 2 n + 4 q n ) | 2 − 4 | ) = 5 q + 5 n + 6 + 4 q n . □

$Figure 3 M 3 ( 4 , 6 ) {M}_{3}(4,6) .$

Figure 3

M 3 ( 4 , 6 ) .

$Figure 4 M 4 ( 5 , 6 ) {M}_{4}(5,6) .$

Figure 4

M 4 ( 5 , 6 ) .

5 Irregularity indices of nanosheet covered by C 3 and C 6

In this section, we compute irregularity indices of nanosheet covered by C 3 and C 6 . We represent the molecular graph of it by M 4 ( p , r ) , where p stands for the number of hexagons in each row and r stands for the number of hexagons in each column. Chemical graph of this nanosheet has consecutive columns of hexagons C 6 and triangles C 3 .

Theorem 5.1

The irregularity indices of nanosheet M 4 ( p , r ) are as follows:

IRDIF ( M 4 ( p , r ) ) = 14 p + 27 r − 21 6 ,
AL ( M 4 ( p , r ) ) = 4 p + 6 r − 6 ,
IRL ( M 4 ( p , r ) ) = 4 r ln 2 3 + ( 4 p + 2 r − 6 ) ln 3 4 ,
IRLU ( M 4 ( p , r ) ) = 4 p + 8 r − 6 3 ,
IRLF ( M 4 ( p , r ) ) = 3 r + 2 6 r + 2 p 3 − 3 3 3 ,
IRF ( M 4 ( p , r ) ) = 4 p + 6 r − 6 ,
IRLA ( M 4 ( p , r ) ) = 2 ( 20 p + 38 r − 30 ) 35 ,
IRDI ( M 4 ( p , r ) ) = ln 2 ( 4 p + 6 r − 6 ) ,
IRA ( M 4 ( p , r ) ) = − 21 + 12 3 + ( 14 − 8 3 ) p + ( 27 − 4 3 − 8 2 3 ) r 6 ,
IRGA ( M 4 ( p , r ) ) = 2 ln 7 3 12 + 4 ln 5 6 12 r + 4 ln 5 3 12 p − 6 ln 7 3 12 ,
IRB ( M 4 ( p , r ) ) = ( − 8 2 3 − 8 3 + 34 ) r + ( − 16 3 + 28 ) p + 24 3 − 42 , a n d
IRR t ( M 4 ( p , r ) ) = 2 p + 3 r − 3 .

Proof

Let us consider that the subset ℰ j k ⊂ ℰ ( M 4 ( p , r ) ) contains the edges having end vertices of degrees k and l . We divide ℰ ( M 4 ( p , r ) ) into subsets on the basis of the degrees of end vertices of edges. The cardinalities of these subsets are | ℰ 22 | = 4 , | ℰ 33 | = 4 p − 6 , | ℰ 34 | = 4 p + 2 r − 6 , | ℰ 23 | = 4 r , and | ℰ 44 | = 6 p r − 6 p − 7 r + 7 . With the help of these data and the representations of irregularity indices shown in Table 1, the precise values of these indices can be computed in the following way:

IRDIF ( M 4 ( p , r ) ) = ∑ d e ∈ ℰ ( M 4 ( p , r ) ) ω M 4 ( p , r ) ( d ) ω M 4 ( p , r ) ( e ) − ω M 4 ( p , r ) ( e ) ω M 4 ( p , r ) ( d ) = 4 2 2 − 2 2 + 4 r 2 3 − 3 2 + ( 4 p − 6 ) 3 3 − 3 3 + ( 4 p + 2 r − 6 ) 3 4 − 4 3 + ( 6 p r − 6 p − 7 r + 7 ) 4 4 − 4 4 = 14 p + 27 r − 21 6 .

AL ( M 4 ( p , r ) ) = ∑ d e ∈ ℰ ( M 4 ( p , r ) ) ω M 4 ( p , r ) ( d ) − ω M 4 ( p , r ) ( e ) = 4 | 2 − 2 | + ( 4 p − 6 ) | 3 − 3 | + 4 r | 2 − 3 | + ( 4 p + 2 r − 6 ) | 3 − 4 | + ( 6 p r − 6 p − 7 r + 7 ) | 4 − 4 | = 4 p + 6 r − 6 .

IRLF ( M 4 ( p , r ) ) = ∑ d e ∈ ℰ ( M 4 ( p , r ) ) ω M 4 ( p , r ) ( d ) − ω M 4 ( p , r ) ( e ) ( ω M 4 ( p , r ) ( d ) ω M 4 ( p , r ) ( e ) ) = 4 | 2 − 2 | 2 × 2 + ( 4 p − 6 ) | 3 − 3 | 3 × 3 + 4 r | 2 − 3 | 2 × 3 + ( 4 p + 2 r − 6 ) | 3 − 4 | 3 × 4 + ( 6 p r − 6 p − 7 r + 7 ) | 4 − 4 | 4 × 4 = 3 r + 2 6 r + 2 p 3 − 3 3 3 .

IRF ( M 4 ( p , r ) ) = ∑ d e ∈ ℰ ( M 4 ( p , r ) ) ( ω M 4 ( p , r ) ( d ) − ω M 4 ( p , r ) ( e ) ) 2 = 4 ( 2 − 2 ) 2 + ( 4 p − 6 ) ( 3 − 3 ) 2 + 4 r ( 2 − 3 ) 2 + ( 4 p + 2 r − 6 ) ( 3 − 4 ) 2 + ( 6 p r − 6 p − 7 r + 7 ) ( 4 − 4 ) 2 = 4 p + 6 r − 6 .

IRLA ( M 4 ( p , r ) ) = 2 ∑ d e ∈ ℰ ( M 4 ( p , r ) ) ω M 4 ( p , r ) ( d ) − ω M 4 ( p , r ) ( e ) ( ω M 4 ( p , r ) ( d ) + ω M 4 ( p , r ) ( e ) ) = 2 4 | 2 − 2 | 2 + 2 + ( 4 p − 6 ) | 3 − 3 | 3 + 3 + 4 r | 2 − 3 | 2 + 3 + 2 ( 4 p + 2 r − 6 ) | 3 − 4 | 3 + 4 + ( 6 p r − 6 p − 7 r + 7 ) | 4 − 4 | 4 + 4 = 2 ( 20 p + 38 r − 30 ) 35 .

IRDI ( M 4 ( p , r ) ) = ∑ d e ∈ ℰ ( M 4 ( p , r ) ) ln 1 + ω M 4 ( p , r ) ( d ) − ω M 4 ( p , r ) ( e ) = 4 ln { 1 + | 2 − 2 | } + ( 4 p − 6 ) ln { 1 + | 3 − 3 | } + ( 4 p + 2 r − 6 ) ln { 1 + | 3 − 4 | } + 4 r ln { 1 + | 2 − 3 | } + ( 6 p r − 6 p − 7 r + 7 ) ln { 1 + | 4 − 4 | } = ln 2 ( 4 p + 6 r − 6 ) .

IRA ( M 4 ( p , r ) ) = ∑ d e ∈ ℰ ( M 4 ( p , r ) ) 1 ω M 4 ( p , r ) ( d ) − 1 ω M 4 ( p , r ) ( e ) 2 = 4 1 2 − 1 2 2 + ( 4 p − 6 ) 1 3 − 1 3 2 + 4 r 1 2 − 1 3 2 + ( 4 p + 2 r − 6 ) 1 3 − 1 4 2 + ( 6 p r − 6 p − 7 r + 7 ) 1 4 − 1 4 2 = − 21 + 12 3 + 14 − 8 3 p + 27 − 4 3 − 8 2 3 r 6 .

IRGA ( M 4 ( p , r ) ) = ∑ d e ∈ ℰ ( M 4 ( p , r ) ) ln ω M 4 ( p , r ) ( d ) + ω M 4 ( p , r ) ( e ) 2 ω M 4 ( p , r ) ( d ) ω M 4 ( p , r ) ( e ) = 4 ln 2 + 2 2 2 × 2 + ( 4 p − 6 ) ln 3 + 3 2 3 × 3 + 4 r ln 2 + 3 2 2 × 3 + ( 4 p + 2 r − 6 ) ln 3 + 4 2 3 × 4 + ( 6 p r − 6 p − 7 r + 7 ) ln 4 + 4 2 4 × 4 = 2 ln 7 3 12 + 4 ln 5 6 12 r + 4 ln 5 3 12 p − 6 ln 7 3 12 .

IRB ( M 4 ( p , r ) ) = ∑ d e ∈ ℰ ( M 4 ( p , r ) ) ( ω M 4 ( p , r ) ( d ) − ω M 4 ( p , r ) ( e ) 2 = 4 2 − 2 2 + ( 4 p − 6 ) 3 − 3 2 + 4 r 2 − 3 2 + ( 4 p + 2 r − 6 ) 3 − 4 2 + ( 6 p r − 6 p − 7 r + 7 ) 4 − 4 2 = − 8 2 3 − 8 3 + 34 r + − 16 3 + 28 p + 24 3 − 42 .

IRR t ( M 4 ( p , r ) ) = 1 2 ∑ d e ∈ ℰ ( M 4 ( p , r ) ) ω M 4 ( p , r ) ( d ) − ω M 4 ( p , r ) ( e ) = 1 2 ( 4 | 2 − 2 | + ( 4 p − 6 ) | 3 − 3 | + 4 r | 2 − 3 | ) + 1 2 ( ( 4 p + 2 r − 6 ) | 3 − 4 | + ( 6 p r − 6 p − 7 r + 7 ) | 4 − 4 | ) = 2 p + 3 r − 3 . □

6 Conclusion

In the past few years, the problems of computational physics on specific molecular structures have gained attention in theoretical physics. Topological indices facilitate the researchers by giving knowledge about the structure of materials that reduce their workload. Reckoning the topological indices of a compound may assist in the assessment of its medicinal behaviour. Day by day, the theory of understanding compounds through topological indices achieved great significance in the area of medicine considering that it needs no chemical-related apparatus to study. Topological indices are being widely studied for various graphs, particularly for the chemical graphs. Hence, the irregularity indices for the molecular graphs of four types of nanosheets are manifested by using mathematical derivation approach in this article. These outcomes can also perform a valuable role in the findings on the significance of the considered structures.

References

[1] Abdo H, Brandt S, Dimitrov D. The total irregularity of a graph. Discret Math Theor Comput Sci. 2014;16:201–6.10.46298/dmtcs.1263Search in Google Scholar

[2] Abdo H, Cohen N, Dimitrov D. Graphs with maximal irregularity. Filomat. 2014;28:1315–22.10.2298/FIL1407315ASearch in Google Scholar

[3] Abdo H, Dimitrov D, Gutman I. Graphs with maximal σ irregularity. Discr Appl Math. 2018;250:57–64.10.1016/j.dam.2018.05.013Search in Google Scholar

[4] Abdo H, Dimitrov D, Gao W. On the irregularity of some molecular structures. Can J Chem. 2017;95:174–83.10.1139/cjc-2016-0539Search in Google Scholar

[5] Abdo H, Dimitrov D. The irregularity of graphs under graph operations. Discuss Math Graph Theory. 2014;34:263–78.10.7151/dmgt.1733Search in Google Scholar

[6] Akhter S, Imran M, Gao W, Farahani MR. On topological indices of honeycomb networks and graphene networks. Hacet J Math Stat. 2018;47(1):19–35.10.15672/HJMS.2017.464Search in Google Scholar

[7] Albertson MO. The irregularity of a graph. ARS Comb. 1997;46:219–25.Search in Google Scholar

[8] Arockiaraj M, Klavzar S, Mushtaq S, Balasubramanian K. Distance-based topological indices of nanosheets, nanotubes and nanotori of SiO2. J Math Chem. 2019;57:343–69.10.1007/s10910-018-0956-8Search in Google Scholar

[9] Criado R, Flores J, del Amo AG, Romance M. Centralities of a network and its line graph: an analytical comparison by means of their irregularity. Int J Comput Math. 2014;91:304–14.10.1080/00207160.2013.793316Search in Google Scholar

[10] Chen X, Klingeler R, Kath M, Gendy AAE, Cendrowski K, Kalenczuk RJ, et al. Magnetic silica nanotubes: synthesis, drug release, and feasibility for magnetic hyperthermia. ACS Appl Mater Interfaces. 2012;4(4):2303–9.10.1021/am300469rSearch in Google Scholar PubMed

[11] Chhowalla M, Shin HS, Eda G, Li L-J, Loh KP, Zhang H. +e chemistry of two-dimensional layered transition metal dichalcogenide nanosheets. Nat Chem. 2013;5(4):263–75.10.1038/nchem.1589Search in Google Scholar PubMed

[12] Estrada E. Randić index, irregularity and complex biomolecular networks. Acta Chim Slov. 2010;57:597–603.Search in Google Scholar

[13] Farrukh F, Farooq R, Farahani MR. On the atom-bond connectivity and geometric arithmetic indices of SiO2 layer structure. Mor J Chem. 2017;5(2):384–90.Search in Google Scholar

[14] Furtula B, Gutman I. A forgotten topological index. J Math Chem. 2015;53:1184–90.10.1007/s10910-015-0480-zSearch in Google Scholar

[15] Freitas R. Nanomedicine 2. 2000. Available online: http://www.foresight.org/nanomedicine (accessed on 18 January 2019).Search in Google Scholar

[16] Gao W, Aamir M, Iqbal Z, Ishaq M, Aslam A. On irregularity measures of some dendrimers structures. Mathematics. 2019;7:271.10.3390/math7030271Search in Google Scholar

[17] Gao W, Iqbal Z, Ishaq M, Aslam A, Aamir M, Binyamin MA. Bounds on topological descriptors of the corona product of F-sum of connected graphs. IEEE Access. 2019;7:26788–96.10.1109/ACCESS.2019.2900061Search in Google Scholar

[18] Gao W, Iqbal Z, Ishaq M, Aslam A, Sarfraz R. Topological aspects of dendrimers via distance-based descriptors. IEEE Access. 2019;7:35619–30.10.1109/ACCESS.2019.2904736Search in Google Scholar

[19] Gao W, Iqbal Z, Ishaq M, Sarfraz R, Aamir M, Aslam A. On eccentricity-based topological indices study of a class of porphyrin-cored dendrimers. Biomolecules. 2018;8:71–81.10.3390/biom8030071Search in Google Scholar PubMed PubMed Central

[20] Gao W, Iqbal Z, Akhter S, Ishaq M, Aslam A. On irregularity descriptors of derived graphs. AIMS Math. 2020;5(5):4085–107.10.3934/math.2020262Search in Google Scholar

[21] Gutman I. Stepwise irregular graphs. Appl Math Comput. 2018;325:234–8.10.1016/j.amc.2017.12.045Search in Google Scholar

[22] Gutman I, Hansen P, Mélot H. Variable neighborhood search for extremal graphs 10. Comparison of irregularity indices for chemical trees. J Chem Inf Model. 2005;45:222–30.10.1021/ci0342775Search in Google Scholar PubMed

[23] Gutman I, Togan M, Yurttas A, Cevik AS, Cangul IN. Inverse problem for sigma index. Match Commun Math Comput Chem. 2018;79:491–508.Search in Google Scholar

[24] Gutman I. Degree-based topological indices. Croatica Chem Acta. 2013;86(4):351–61.10.5562/cca2294Search in Google Scholar

[25] Harary F. Graph theory. Philippines: Addison Wesley Publishing Company, inc.; 1969.10.21236/AD0705364Search in Google Scholar

[26] Hiramatsu M, Hori M. Fabrication of carbon nanowalls using novel plasma processing. Japanese J Appl Phys. 2006;45(6):5522–7.10.1109/IMNC.2005.203831Search in Google Scholar

[27] Horcajada P, Rámila A, Pérez Pariente J, Vallet-Regi M. Influence of pore size of MCM-41 matrices on drug delivery rate. Microporous Mesoporous Mater. 2004;68(1–3):105–9.10.1016/j.micromeso.2003.12.012Search in Google Scholar

[28] Hosoya H. Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Buff Chem SOC Jpn. 1971;44:2332–9.10.1246/bcsj.44.2332Search in Google Scholar

[29] Idrees N, Saif MJ, Sadiq A, Rauf A, Hussain F. Topological indices of H-naphtalenic nanosheet. Cent Eur J Chem. 2018;16(1):1184–8.10.1515/chem-2018-0131Search in Google Scholar

[30] Idrees N, Saif MJ, Sadiq A, Rauf A, Hussain F. Topological indices of H-naphtalenic nanosheet. Open Chem. 2018;16:1184–8.10.1515/chem-2018-0131Search in Google Scholar

[31] Iqbal Z, Ishaq M, Aamir M. On eccentricity-based topological descriptors of dendrimers. Iran J Sci Technol Trans Sci. 2019;43:1523–33.10.1007/s40995-018-0621-xSearch in Google Scholar

[32] Iqbal Z, Aslam A, Ishaq M, Aamir M. Characteristic study of irregularity measures of some nanotubes. Can J Phys. 2019;97(10):1125–32.10.1139/cjp-2018-0619Search in Google Scholar

[33] Iqbal Z, Ishaq M, Aslam A, Gao W. On eccentricity-based topological descriptors of water-soluble dendrimers. Z Naturforsch C J Biosci. 2018;74:25–33.10.1515/znc-2018-0123Search in Google Scholar PubMed

[34] Jagadeesh R, Kanna MR, Indumathi RS. Some results on topological indices of graphene. Nanomater Nanotechnol. 2016;6:1–6.Search in Google Scholar

[35] Jing L, Meng L, Fei Z, Dyson PJ, Jing X, Xing L. MnO2, nanosheets as an artificial enzyme to mimic oxidase for rapid and sensitive detection of glutathione. Biosens Bioelectron. 2017;90:69–74.10.1016/j.bios.2016.11.046Search in Google Scholar PubMed

[36] Kang SM, Iqbal Z, Ishaq M, Sarfraz R, Aslam A, Nazeer W. On eccentricity-based topological indices and polynomials of phosphorus-containing dendrimers. Symmetry. 2018;10:237–46.10.3390/sym10070237Search in Google Scholar

[37] Iijima S. Helical microtubules of graphitic carbon. Nature. 1991;354:56–8.10.1038/354056a0Search in Google Scholar

[38] Parsons-Moss T, Schwaiger LK, Hubaud A, Hu YJ, Tuysuz H, Yang P, et al. Plutonium complexation by phosphonate-functionalized mesoporous silica, 241th ACS National Meeting, Anaheim, CA, USA, 2011.Search in Google Scholar

[39] Ma Y, Cao S, Shi Y, Dehmer M, Xia C. Nordhaus–Gaddum type results for graph irregularities. Appl Math Comput. 2019;343:268–72.10.1016/j.amc.2018.09.057Search in Google Scholar

[40] Matejic M, Mitic B, Milovanovic E, Milovanovic I. On Alberson irregularity measure of graphs. Sci Publ State Univ Novi Pazar Ser A Appl Mathematics Inform Mech. 2019;11(2):97–106.10.5937/SPSUNP1902097MSearch in Google Scholar

[41] Ray S. Applications of graphene and graphene-oxide based nanomaterials. Oxford, UK: William Andrew (Elsevier); 2015. ISBN: 978-0-323-37521-4.Search in Google Scholar

[42] Shen H, Zhang L, Liu M, Zhang Z. Biomedical applications of graphene. Theranostics. 2012;2(3):283–94.10.7150/thno.3642Search in Google Scholar

[43] Soldano C, Mahmood A, Dujardin E. Production, properties and potential of graphene. Carbon. 2010;48(8):2127–50.10.1016/j.carbon.2010.01.058Search in Google Scholar

[44] Sridhara G, Kanna MRR, Indumathi RS. Computation of topological indices of graphene. J Nanomater. 2015;2015:8. 10.1155/2015/969348.Search in Google Scholar

[45] Réti T, Dimitrov D. On irregularities of bidegreed graphs. Acta Polytech Hung. 2013;10:117–34.Search in Google Scholar

[46] Réti T. On some properties of graph irregularity indices with a particular regard to the σ-index. Appl Math Comput. 2019;344:107–15.10.1016/j.amc.2018.10.010Search in Google Scholar

[47] Réti T, Sharafdini R, Dregelyi-Kiss A, Haghbin H. Graph irregularity indices used as molecular descriptor in QSPR studies. Match Commun Math Comput Chem. 2018;79:509–24.Search in Google Scholar

[48] Snijders TAB. The degree variance: an index of graph heterogeneity. Soc Netw. 1981;3:163–74.10.1016/0378-8733(81)90014-9Search in Google Scholar

[49] Tavakoli M, Rahbarnia F, Mirzavaziri M, Ashrafi AR, Gutman I. Extremely irregular graphs. Kragujev J Math. 2013;37:135–39.Search in Google Scholar

[50] Tian Q, Zhang Z, Chen J, Yang L, Hirano S-i. Carbon nanowires@ultrathin SnO2 nanosheets@carbon composite and its lithium storage properties. J Power Sources. 2014;246:587–95.10.1016/j.jpowsour.2013.08.009Search in Google Scholar

[51] Trinajstić N. Chemical Graph Theory, 2nd edn. Boca Raton, Florida: CRC Press; 1992.Search in Google Scholar

[52] Vallet-Regi M, Rámila A, del Real RP, Pérez Pariente J, A new property of MCM-41: drug delivery system. Chem Mater. 2001;13(2):308–11.10.1021/cm0011559Search in Google Scholar

[53] Vukičević D, Graovac A. Valence connectivities versus Randić, Zagreb and modified Zagreb index: a linear algorithm to check discriminative properties of indices in acyclic molecular graphs. Croat Chem Acta. 2004;77:501–8.Search in Google Scholar

[54] Wang QH, Kalantar-Zadeh K, Kis A, Coleman JN, Strano MS. Electronics and optoelectronics of twodimensional transition metal dichalcogenides. Nat Nanotechnol. 2012;7(11):699–712.10.1038/nnano.2012.193Search in Google Scholar

[55] Wu Y, Qiao P, Chong T, Shen Z. Carbon nanowalls grown by microwave plasma enhanced chemical vapor deposition. Adv Mater. 2002;14(1):64–67.10.1002/1521-4095(20020104)14:1<64::AID-ADMA64>3.0.CO;2-GSearch in Google Scholar

[56] Xu YX, Shi GQ. Assembly of chemically modified graphene, methods and applications. J Mater Chem. 2011;21:3311–23.10.1039/C0JM02319ASearch in Google Scholar

[57] Yin C, Dong L, Wang Z, Chen M, Wang Y, Zhao Y. CO2-Responsive graphene oxide nanofiltration membranes for switchable rejection to cations and anions. J Membr Sci. 2019;592:117374.10.1016/j.memsci.2019.117374Search in Google Scholar

[58] Zhao D, Iqbal Z, Irfan R, Chaudhry MA, Ishaq M, Jameel MK, et al. Comparison of irregularity indices of several dendrimers structures. Processes. 2018;7(10):1–14.10.3390/pr7100662Search in Google Scholar

[59] Zheng J, Iqbal Z, Fahad A, Zafar A, Aslam A, Qureshi MI, et al. Some eccentricity-based topological indices and polynomials of poly(ethyleneamidoamine) (PETAA) dendrimers. Processes. 2019;7:433.10.3390/pr7070433Search in Google Scholar

[60] Zhou B, Luo W. On irregularity of graphs. ARS Comb. 2008;88:55–64.Search in Google Scholar

[61] Zhu Y, Shi J, Shen W, Dong X, Feng J, Ruan M, et al. Stimuli-responsive controlled drug release from a hollow mesoporous silica sphere/polyelectrolyte multilayer core–shell structure. Angew Chem Int Ed. 2005;44(32):5083–7.10.1002/anie.200501500Search in Google Scholar

Received: 2019-11-30

Revised: 2020-05-29

Accepted: 2020-06-02

Published Online: 2020-08-03

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2020-0164

Keywords for this article

molecular graph; irregularity indices; nanosheets

Creative Commons

BY 4.0