Analysis of irregularity measures of zigzag, rhombic, and honeycomb benzenoid systems

Zafar Hussain; Yujun Yang; Mobeen Munir; Zahid Hussain; Muhammad Athar; Ali Ahmed; Haseeb Ahmad

doi:10.1515/phys-2020-0194

Open Access Published by De Gruyter Open Access December 31, 2020

Analysis of irregularity measures of zigzag, rhombic, and honeycomb benzenoid systems

Zafar Hussain , Yujun Yang , Mobeen Munir , Zahid Hussain , Muhammad Athar , Ali Ahmed and Haseeb Ahmad

From the journal Open Physics

https://doi.org/10.1515/phys-2020-0194

Abstract

Molecular topology is a key to determine the physiochemical aspects of the molecular structure. To determine the degree of irregularity of a certain molecular structure or a network has been a key source of interest for the molecular topologists as it provides an insight of the key features that are used to guess the properties of the structures. In this article, we are interested in formulating closed forms of irregularity measures of some popular benzenoid systems, namely, zigzag, rhombic, and honeycomb benzenoid systems. We also compared our results graphically and concluded that benzenoid system among above is more asymmetric than the others.

Keywords: benzenoid system; irregularity measure; complexity of structure; zigzag benzenoid system; rhombic benzenoid system; honeycomb benzenoid system

1 Introduction

Benzenoid hydrocarbons have consistently attracted the attention of both chemists and pure mathematicians because of the structural complexities. People want to study these structures combinatorially and topologically. Research in benzenoid hydrocarbons is currently expanding because of the innovative developments. Benzenoid systems are molecular structures having nice geometrical properties. These are constructed with a definite rule from a benzene molecule which is its fundamental building block. A benzenoid system is defined to be a connected planar simple graph obtained by regular hexagons with either two such hexagons share a common edge or disjoint.

All benzenoid systems partition the plane into one non-compact external region and many internal compact regular hexagonal regions. Let h be the number of hexagons in a benzenoid system then for h = 1 we have a single non-isomorphic benzenoid system that is the single benzene molecule. For h = 2, 3, and 4, we obtain the following non-isomorphic benzenoid systems in Figure 1 ([1] p. 11–5).

Figure 1

Non-isomorphic benzenoid structures for h = 2, 3, 4.

In the chemical graph theory, we use tools of algebra and topology to approximate the properties of hydrocarbons [2,3,4,5,6,7,8,9]. Topological indices are the structure preserving maps which capture some key combinatorial and topological aspects of the structure and determine properties of chemical compounds without the help of quantum mechanics as the final product [2,3]. One of the most important types is degree-based index [3,4,5,6,7]. Weiner and hyper wiener indices are topological invariants based on the distances and are used to forecast the boiling points of alkanes [5]. Randic estimated the degree of molecular branching in ref. [5]. Estrada studied the facts that are used for determining energies and connectivity indices of branched alkanes in ref. [6] and determined enthalpy of formation of alkanes using atom–bond connectivity index in ref. [7].

A broad class of indices is the irregularity indices which critically depend on the topological arrangements of molecular graphs which are hydrogen-suppressed graphs. These indices are somewhat the measure of asymmetry in the molecular structure. Let G be a simple connected graph with vertex set V and edge set E. A graph is said to be regular if all its vertices are of same degree otherwise it is irregular. However, one can be deeply interested to know the degree of irregularity in the graph [19]. Recently, it has been depicted that graph irregularity indices show close accurate results about some properties such as entropy, standard enthalpy, vaporization, and acentric factor of octane isomers [10]. Gao et al. discussed some irregularity indices of different families of molecular graphs in ref. [14] and some dendrimers in ref. [13]. These structures are used as long infinite chain macromolecules in chemistry and related areas. Total irregularity index has been introduced recently and some well-stabilized graphs have been studied. Due to ever-increasing interests in complexity of molecular structures, one can be interested to know the degree of complexity in the underlined structure. One can compute the irregularity indices of the structure. Bell discussed the most irregular graphs according to some irregularity measures [11]. Liu et al. computed recently many irregularity indices for three types of benzenoid systems, namely, hour-glass, jagged-rectangle, and triangular benzenoid systems and proved that hour-glass is the most irregular structure for most of the indices [15]. Iqbal et al. computed some irregularity indices for some nanotubes using sigma index [16]. Gutman gave some fundamental results for irregularity indices of some graphs in ref. [17]. Dimitirov and Reti computed some irregularity indices of some general graphs in ref. [18].

Authors gave general formulas for M-polynomials of dendrimers in ref. [23] and polyhex nanotubes in ref. [24]. Let G be a connected simple graph and d _u and d _v the degree of vertices u and v, then basic identities relating irregularity indices can be given in Table 1, and for detailed description of these indices please see refs. [10,11,12,13,14,15]. Table 1 provides us with the list of six different irregularity indices with their key defining relations.

Table 1

Irregularity indices selected for QSPR studies

Irregularity indices
IRDIF( G ) = ∑ U V ∈ E d u d v − d v d v	IRR( G ) = ∑ U V ∈ E i m b ( e ) ∴ i m b ( e ) = \| d u − d v \|
IRL( G ) = ∑ U V ∈ E \| ln d u − ln d v \|	IRLU ( G ) = ∑ U V ∈ E \| d u − d v \| min ( d u , d v )
IRLF( G ) = ∑ U V ∈ E \| d u − d v \| ( d u d v )	σ G = ∑ U V ∈ E d u − d v 2

The present article focuses on the irregularity of three different benzenoid systems, namely, zigzag, rhombic, and honeycomb networks. Presently, this system has been the subject matter of many recent results. In ref. [25], authors focused on symmetry and asymmetry of benzenoid systems. Authors discussed some topological indices and polynomials of honeycomb network [22]. Some computational results of some famous nanotubes have been done in [20,21].

2 Main results

In this section, we give our main computational results.

Figure 2

Zigzag benzenoid system.

Table 2

Edge partition of zigzag benzenoid system

Number of edges ( d u , d v )	Number of indices
(2,2)	2 ( n + 2 )
(2,3)	4 n
(3,3)	( 4 n − 3 )

Theorem 1

IRDIF ( Z p ) = 10 3 p
IRR ( Z p ) = 4 p
IRL( Z p ) = 1.621860432 p
IRLU( Z p ) = 2 p
IRLF ( Z p ) = 4 6 p
σ ( Z p ) = 4 p .

Proof

In order to prove the aforementioned theorem, we have to consider Figure 2.

Table 2 gives edge partition of zigzag benzenoid systems with |( Z _p)| = 2(n + 2) + 4n + (4n − 3).

Now using Table 2 and definitions from Table 1 we have,

IRDIF( G ) = ∑ U V ∈ E d u d v − d v d v
IRDIF( Z p ) = 2 ( p + 2 ) 2 2 − 2 2 + 4 p 3 2 − 2 3 + ( 4 p − 3 ) 3 3 − 3 3 = 2 ( p + 2 ) | 0 | + 4 p 3 2 − 2 3 + ( 4 p − 3 ) | 0 | = 4 p 3 2 − 2 3
AL( G ) = ∑ U V ∈ E | d u − d v |
IRR ( Z p ) = 2 ( p + 2 ) | 2 − 2 | + 4 p | 3 − 2 | + ( 4 p − 3 ) | 3 − 3 | = 2 ( p + 2 ) | 0 | + 4 p | 1 | + ( 4 p − 3 ) | 0 | = 4 p
IRL ( G ) = ∑ U V ∈ E | ln d u − ln d v |
IRL ( Z p ) = 2 ( p + 2 ) | ln 2 − ln 2 | + 4 p | ln 3 − ln 2 | + ( 4 p − 3 ) | ln 3 − ln 3 | = 4 p ln 3 2
IRLU( G ) = ∑ U V ∈ E | d u − d v | min ( d u d v )
IRLU ( Z p ) = 2 ( p + 2 ) | 2 − 2 | 2 + 4 p | 3 − 2 | 2 + ( 4 p − 3 ) | 3 − 3 | 2 = 4 p 1 2 = 2 p
IRLF( G ) = ∑ U V ∈ E | d u − d v | ( d u d v )
IRLU( Z p ) = 2 ( p + 2 ) | 2 − 2 | ( 4 ) + 4 p | 3 − 2 | ( 6 ) + ( 4 p − 3 ) | 3 − 3 | 9 = 4 p 1 ( 6 )
σ ( G ) = ∑ U V ∈ E ( d u − d v ) 2

IRF ( Z p ) = 2 ( p + 2 ) ( 2 − 2 ) 2 + 4 p ( 3 − 2 ) 2 + ( 4 p − 3 ) ( 3 − 3 ) 2 = 4 p ( 1 ) 2 = 4 p .

Table 3 gives some values of irregularity indices of zigzag benzenoid systems for different values of p.□

Theorem 2

IRDIF( R p ) = 40 6 ( p − 1 )
IRR( R p ) = 8 ( p − 1 )
IRL( R p ) = 3.243720865
IRLU( R p ) = 4 ( p − 1 )
IRLF( R p ) = 8 ( 6 ) ( p − 1 )
σ ( R p ) = 8 ( p − 1 ) .

Proof

In order to prove the aforementioned theorem, we have to consider Figure 3.

Now using Table 4 and definitions from Table 1 we have,

IRDIF( G ) = ∑ U V ∈ E d u d v − d v d v
IRDIF ( R p ) = 6 2 2 − 2 2 + 8 ( p − 1 ) 3 2 − 2 3 + ( p ( 3 p − 4 ) + 1 ) 3 3 − 3 3 = 6 | 0 | + 8 ( p − 1 ) 3 2 − 2 3 + ( p ( 3 p − 4 ) + 1 ) | 0 | = 8 ( p − 1 ) 3 2 − 2 3 = 8 ( p − 1 ) 5 6 = 40 6 ( p − 1 )
IRR( G ) = ∑ U V ∈ E | d u − d v |
AL ( R p ) = 6 | 2 − 2 | + 8 ( p − 1 ) | 3 − 2 | + ( p ( 3 p − 4 ) + 1 ) | 3 − 3 | = 6 | 0 | + 8 ( p − 1 ) | 1 | + ( p ( 3 p − 4 ) + 1 ) | 0 | = 8 ( p − 1 )
IRL( G ) = ∑ U V ∈ E | ln d u − ln d v |
IRL ( R p ) = 6 | ln 2 − ln 2 | + 8 ( p − 1 ) | ln 3 − ln 2 | + ( p ( 3 p − 4 ) + 1 ) | ln 3 − ln 3 | = 8 ( p − 1 ) ln 3 2 .
IRLU( G ) = ∑ U V ∈ E | d u − d v | min ( d u d v )
IRLU ( R p ) = 6 | 2 − 2 | 2 + 8 ( p − 1 ) | 3 − 2 | 2 + ( p ( 3 p − 4 ) + 1 ) | 3 − 3 | 2 = 8 ( p − 1 ) 1 2 = 4 ( p − 1 )
IRLU( G ) = ∑ U V ∈ E | d u − d v | ( d u d v )
IRLU ( R p ) = 6 | 2 − 2 | ( 4 ) + 8 ( p − 1 ) | 3 − 2 | ( 6 ) + ( p ( 3 p − 4 ) + 1 ) | 3 − 3 | 9 = 8 ( p − 1 ) 1 ( 6 )
σ ( G ) = ∑ U V ∈ E ( d u − d v ) 2

σ ( R p ) = 6 ( 2 − 2 ) 2 + 8 ( p − 1 ) ( 3 − 2 ) 2 + ( p ( 3 p − 4 ) + 1 ) ( 3 − 3 ) 2 = 8 ( p − 1 ) ( 1 ) 2 = 8 ( p − 1 ) . □

Table 5 gives some values of irregularity indices of rhombic benzenoid systems for different values of p.

Table 3

Irregularity indices for zigzag benzenoid system

Irregularity indices	p = 1	p = 2	p = 3	p = 4	p = 5
IRDIF( G ) = ∑ U V ∈ E d u d v − d v d v	3.3333	6.666	9.9999	13.3332	16.6665
IRR( G ) = ∑ U V ∈ E \| d u − d v \|	4	8	12	16	20
IRL( G ) = ∑ U V ∈ E \| ln d u − ln d v \|	1.62186	3.24372	4.86558	6.48744	8.10930
IRLU( G ) = ∑ U V ∈ E \| d u − d v \| min ( d u , d v )	2	4	6	8	10
IRLU( G ) = ∑ U V ∈ E \| d u − d v \| ( d u d v )	1.632993	3.265986	4.898979	6.531972	8.164965
σ ( G ) = ∑ U V ∈ E ( d u − d v ) 2	4	8	12	16	20

Table 4

Edge partition of rhombic benzenoid system

Number of edges ( d u , d v )	Number of indices
(2,2)	6
(2,3)	8 ( n − 1 )
(3,3)	( n ( 3 n − 4 ) + 1 )

Table 5

Irregularity indices for rhombic benzenoid system

Irregularity indices	p = 2	p = 3	p = 4	p = 5
IRDIF( G ) = ∑ U V ∈ E d u d v − d v d v	6.6667	13.3334	20	26.6668
IRR( G ) = ∑ U V ∈ E \| d u − d v \|	8	16	24	32
IRL( G ) = ∑ U V ∈ E \| ln d u − ln d v \|	3.24372	6.48744	9.731163	12.97488
IRLU( G ) = ∑ U V ∈ E \| d u − d v \| min ( d u , d v )	4	8	12	16
IRLU( G ) = ∑ U V ∈ E \| d u − d v \| ( d u d v )	3.265986	6.531972	9.797889	13.06394
σ ( G ) = ∑ U V ∈ E ( d u − d v ) 2	8	16	24	32

Figure 3

Rhombic benzenoid system.

Now we move toward irregularity indices of honeycomb networks (Figures 4 and 5).

Theorem 3

IRDIF ( H C p ) = 10 ( p − 1 )
IRR( H C p ) = 12 ( p − 1 )
IRL( H C p ) = 4.865581297 ( p − 1 )
IRLU( H C p ) = 6 ( p − 1 )
IRLU( H C p ) = 12 ( 6 ) ( p − 1 )
σ ( H C p ) = 12 ( p − 1 ) .

$Figure 4 Honeycomb benzenoid system HC 2 {\text{HC}}_{2} .$

Figure 4

Honeycomb benzenoid system HC 2 .

$Figure 5 Honeycomb benzenoid system HC 3 {\text{HC}}_{3} .$

Figure 5

Honeycomb benzenoid system HC 3 .

Proof

Now using Table 6 and definitions from Table 1 we have,

IRDIF( G ) = ∑ U V ∈ E d u d v − d v d v
IRDIF ( H C p ) = 6 2 2 − 2 2 + 12 ( p − 1 ) 3 2 − 2 3 + 9 p 2 − 15 p + 6 3 3 − 3 3 = 6 | 0 | + 12 ( p − 1 ) 3 2 − 2 3 + 9 p 2 − 15 p + 6 | 0 | = 12 ( p − 1 ) 3 2 − 2 3 = 12 ( p − 1 ) 5 6 = 10 ( p − 1 )
AL( G ) = ∑ U V ∈ E | d u − d v |
IRR ( H C p ) = 6 | 2 − 2 | + 12 ( p − 1 ) | 3 − 2 | + 9 p 2 − 15 p + 6 | 3 − 3 | = 6 | 0 | + 12 ( p − 1 ) | 1 | + 9 p 2 − 15 p + 6 | 0 | = 12 ( p − 1 )
IRL( G ) = ∑ U V ∈ E | ln d u − ln d v |
IRL ( HC p ) = 6 | ln 2 − ln 2 | + 12 ( p − 1 ) | ln 3 − ln 2 | + 9 p 2 − 15 p + 6 | ln 3 − ln 3 | = 12 ( p − 1 ) ln 3 2 .
IRLU ( G ) = ∑ U V ∈ E | d u − d v | min ( d u d v )
IRLU( H C p ) = 6 | 2 − 2 | 2 + 12 ( p − 1 ) | 3 − 2 | 2 + 9 p 2 − 15 p + 6 | 3 − 3 | 2 = 12 ( p − 1 ) 1 2 = 6 ( p − 1 ) .
IRLU( G ) = ∑ U V ∈ E | d u − d v | ( d u d v )
IRLU ( H C p ) = 6 | 2 − 2 | ( 4 ) + 12 ( p − 1 ) | 3 − 2 | ( 6 ) + 9 p 2 − 15 p + 6 | 3 − 3 | 9 = 6 | 0 | ( 4 ) + 12 ( p − 1 ) 1 ( 6 ) + 9 p 2 − 15 p + 6 | 0 | 9 = 12 ( p − 1 ) ( 6 )
σ ( G ) = ∑ U V ∈ E ( d u − d v ) 2

IRF ( H C p ) = 6 ( 2 − 2 ) 2 + 12 ( p − 1 ) ( 3 − 2 ) 2 + 9 p 2 − 15 p + 6 ( 3 − 3 ) 2 = 12 ( p − 1 ) ( 1 ) 2 = 12 ( p − 1 ) . □

Table 7 gives some values of irregularity indices of honeycomb benzenoid systems for different values of p.

Table 6

Edge partition of rhombic benzenoid system

Number of edges ( d u , d v )	Number of indices
(2,2)	6
(2,3)	12 ( p − 1 )
(3,3)	9 p 2 − 15 p + 6

Table 7

Irregularity indices for zigzag benzenoid system

Irregularity indices	p = 2	p = 3	p = 4	p = 5
IRDIF( G ) = ∑ U V ∈ E d u d v − d v d v	10	20	30	40
IRR( G ) = ∑ U V ∈ E \| d u − d v \|	12	24	36	48
IRL( G ) = ∑ U V ∈ E \| ln d u − ln d v \|	4.86558	9.731626	14.59674	19.46232
IRLU( G ) = ∑ U V ∈ E \| d u − d v \| min ( d u , d v )	6	12	18	24
IRLU( G ) = ∑ U V ∈ E \| d u − d v \| ( d u d v )	4.898979	9.797959	14.69693	19.595916
σ ( G ) = ∑ U V ∈ E ( d u − d v ) 2	12	24	36	48

3 Conclusions, graphical analysis, and discussion

In this section, we focus on graphical analysis and comparative approach to study the irregularity of the aforementioned calculated indices. All three benzenoid structures are one parameter dependent so we concentrate only on 2D graphs by maple. In each figure, we use red color to show the irregularity of zigzag benzenoid system, blue color to show the irregularity of rhombic benzenoid system, and green color indicates the irregularity of honeycomb benzenoid system with change in structural parameters. Figure 6 shows that rhombic structure is the most irregular, whereas zigzag is the most regular structure and honeycomb falls in between with respect to irregularity index IRDIF.

Figure 6

IRDIF.

For IRR(G), the honeycomb network is the most asymmetric, whereas the zigzag system is the most symmetric and rhombic structure sandwiched in other two structures with change in parameter of structure (Figure 7).

Figure 7

IRR(G).

Figure 8 suggests that the honeycomb network is the most asymmetric, whereas zigzag is the most symmetric for irregularity index IRL and same happens for IRLU.

Figure 8

IRL(G) and IRLU(G).

Figure 9 shows the trends of irregularity index IRLF.

Figure 9

IRLF(G).

Figure 10 represents that the honeycomb network is most irregular, whereas zigzag is the most regular and rhombic structures that remain in between for σ.

Figure 10

(G).

Figures 6–10 explain the trends of these indices and irregularity patterns can be explained easily. These facts contribute to the complexity of molecular structure in a non-trivial way.

Benzenoid structures with high irregularity indices are more non-symmetric, whereas with low irregularity index are highly symmetric. These facts can be used in modeling industry and optical materials.

Acknowledgments

This research was partially supported by the National Natural Science Foundation of China through grant no. 11671347 and the Natural Science Foundation of Shandong Province through grant no. ZR2019YQ002.

Author contributions: Mobeen Munir gave the idea. Haseeb Ahmad, Ali Ahmed, and Muhammad Athar wrote the article. Zafar Hussain and Yujun edited and verified the results.
Conflict of interest: The authors declare no conflict of interest.

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Received: 2019-04-16

Revised: 2020-09-25

Accepted: 2020-10-09

Published Online: 2020-12-31

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

Analysis of irregularity measures of zigzag, rhombic, and honeycomb benzenoid systems

Abstract

1 Introduction

2 Main results

Theorem 1

Proof

Theorem 2

Proof

Theorem 3

Proof

3 Conclusions, graphical analysis, and discussion

Acknowledgments

References

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