On multiplicative degree based topological indices for planar octahedron networks

Ghulam Dustigeer; Haidar Ali; Muhammad Imran Khan; Yu-Ming Chu

doi:10.1515/mgmc-2020-0026

Open Access Published by De Gruyter December 31, 2020

On multiplicative degree based topological indices for planar octahedron networks

Ghulam Dustigeer , Haidar Ali , Muhammad Imran Khan and Yu-Ming Chu

From the journal Main Group Metal Chemistry

https://doi.org/10.1515/mgmc-2020-0026

Abstract

Chemical graph theory is a branch of graph theory in which a chemical compound is presented with a simple graph called a molecular graph. There are atomic bonds in the chemistry of the chemical atomic graph and edges. The graph is connected when there is at least one connection between its vertices. The number that describes the topology of the graph is called the topological index. Cheminformatics is a new subject which is a combination of chemistry, mathematics and information science. It studies quantitative structure-activity (QSAR) and structure-property (QSPR) relationships that are used to predict the biological activities and properties of chemical compounds. We evaluated the second multiplicative Zagreb index, first and second universal Zagreb indices, first and second hyper Zagreb indices, sum and product connectivity indices for the planar octahedron network, triangular prism network, hex planar octahedron network, and give these indices closed analytical formulas.

Keywords: topological indices; triangular prism network; planar octahedron network; hex-planar octahedron network

1 Introduction

Topological indices, provided by graph theory, are a valuable tool. Cheminformatics is a modern academic area that combines the subjects of chemistry, mathematics, and information science. It studies the relationships between quantitative structure-activity (QSAR) and structure-property (QSPR) used to suggest biological activities and chemical compound properties. That is why scholars around the world are extremely interested in it.

The molecular structures are those in which atoms are connected by covalent bonds. In graph theory, atoms are considered as vertices and covalent bonds are as edges. Chem-informatics is a new area of research in which the subjects chemistry, mathematics, and information science are combined.

At present, in the field computational chemistry topological indices have a rising interest which is actually associated to their use in nonempirical quantitative structure-property relationship and quantitative-structure activity relationship. Topological descriptor Top(G) may also be defined with the property of isomorphism i.e for every graph H isomorphic to G, Top(G) = Top(H). The idea of topological indices was first introduced by Weiner (1947) during the lab work on boiling point of paraffin and named this result as Path number which was later named as Weiner Index.

Nowadays there is an extensive research activity on ABC and GA indices and their variants, for further study of topological indices of various graphs (Ali and Sajjad, 2019; Ali et al., 2019, 2020; Akhtar and Imran, 2016; Aslam et al., 2019; Babar et al., 2020; Baig et al., 2015, 2018; Gao et al., 2017; Huo et al., 2020; Imran et al., 2016, 2018; Simonraj and George, 2012; Shirdel et al., 2013; Smith, 2019; Vetrík, 2018; Wei et al., 2019; Zheng et al., 2019). For the basic notations and definitions, see (Diudea et al., 2001; Trinajstić, 1983).

In this article, we consider the silicate structure (Manuel and Rajasingh, 2011) derived from the POH network, TP network, and hex POH network (Simon Raj and George, 2015). The method of drawing planar octahedron networks with dimension m is as follows:

Step 1: Let take a silicate network with dimension m.

Step 2: At the middle of each triangular face, fix new vertices, and connect them to oxide vertices in the corresponding triangle face.

Step 3: Link all these new vertices of the centre that lie in the same silicate cell.

Step 4: The resulting graph is called the planar octahedron network for the m dimension as shown in Figure 1. Delete all silicon vertices. We can also create the triangular prism network as shown in Figure 2 and the hex POH network as shown in Figure 3.

Figure 1

Planar octahedron network POH (2).

Figure 2

Triangular prism network TP (2).

Figure 3

Hex planar octahedron network (2).

In the graph theory, the number of edges which is occurrence to the vertex that is the degree of vertex of the graph. Let φ be a graph. Then second multiplicative Zagreb index (Gutman et al., 1975) can be defined as:

(1)II2(φ)=∏ab∈E(φ)(da×db).

First and second indices (Kulli, 2016) of a graph φ are defined as:

(2)HII1(φ)=∏ab∈E(φ)(da+db)2,

(3)HII2(φ)=∏ab∈E(φ)(da×db)2.

The first and second universal Zagreb index (Kulli, 2016) defined as:

(4)MZ1α(φ)=∏ab∈E(φ)(da+db)α,

(5)MZ2α(φ)=∏ab∈E(φ)(da×db)α,

where: α∈ℝ.

The sum and product connectivity of Multiplicative indices (Kulli, 2016) defined as:

(6)SCII(φ)=∏ab∈E(φ)(1da+db),

(7)PCII(φ)=∏ab∈E(φ)(1da×db).

2 Results

We research the Zagreb indices with its types, such as the second Zagreb multiplicative index, the first Zagreb hyper index, the second Zagreb hyper index, the first and second Zagreb universal indices and the multiplicative indices sum and product connectivity for the planar octahedron network, triangular prism network, hex planar octahedron network.

2.1 Results for planar octahedron network POH(m)

In this section, we calculate edge partition of topological indices of the dimension m for the planar octahedron network. In the coming theorems, we compute some important indices for planar octahedron network.

Theorem 2.1.1

Consider the planar octahedral network POH(m), then its second multiplicative Zagreb index is equal to:

II2(φ1)=382205952m6−169869312m4.

Proof. Let φ₁ ≅ POH(m). From Eq. 1, we have:

II2(φ1)=∏ab∈E(φ1)(da×db).

Using Table 1, we have:

Table 1

Edge partition of POH(m) network.

(d_a,d_b)	Number of edges
E₁ = (4,4)	18m² + 12m
E₂ = (4,8)	36m²
E₃ = (8,8)	18m² − 12m

II2(φ1)=(4×4)|E1(POH(m))|×(4×8)|E2(POH(m))| ×(8×8)|E3(POH(m))|,=16|E1(POH(m))|×32|E2(POH(m))|×64|E3(POH(m))|,=16(18m2+12m)×32(36m2)×64(18m2−12m).

This value is what, we get after calculations:

⇒II2(φ)=382205952m6−169869312m4.

Theorem 2.1.2

Consider the POH(m) network, then its first and second hyper Zegreb indices are equal to:

HII1(φ1)=27518828544m6−12230590464m4,HII2(φ1)=1.25×1013m6−5.57×1012m4.

Proof. Let φ₁ ≅ POH(m) network and by using Eq. 2, we have:

HII1(φ1)=∏ab∈E(φ)(da+db)2.

Using Table 1, we have:

HII1φ1=(4+4)2E1(POH(m))×(4+8)2E2(POH(m))×(8+8)2E3(POH(m)),=8218m2+12m×12236m2×16218m2−12m,=235929611664m6−5184m4.

This value is what, we get after calculations:

⇒HII1(φ1)=27518828544m6−12230590464m4.

For second hyper Zegreb index and using Eq. 3, we have:

HII2(φ1)=∏ab∈E(φ)(da×db)2.

Using Table 1, we have:

HII2φ1=(4×4)2E1(POH(m))×(4×8)2∣E2(POH(m)∣×(8×8)2E3(POH(m)),=16218m2+12m×32236m2×64218m2−12m,=107374182411664m6−5184m4.

This value is what, we get after calculations:

⇒HII2(φ1)=1.25×1013m6−5.57×1012m4.

Theorem 2.1.3

Consider the POH(m) network, then its first and second universal Zagreb indices are equal to:

MZ1α(φ1)=1536α(11664m6−5184m4),MZ2α(φ1)=32768α(11664m6−5184m4).

Proof. Let φ₁ ≅ POH(m) network. We have to prove first universal Zagreb index and using Eq. 4:

MZ1α(φ1)=∏ab∈E(φ1)(da+db)α.

Using Table 1, we have:

MZ1αφ1=(4+4)αE1(POH(m))×(4+8)αE2(POH(m))×(8+8)αE3(POH(m)),=8α18m2+12m×12α36m2×16α18m2−12m,=8α×12α×16α11664m6−5184m4.

This value is what, we get after calculations:

⇒MZ1α(φ1)=1536α(11664m6−5184m4).

For second universal-Zegreb index and by using Eq. 5, we have:

MZ2α(φ1)=∏ab∈E(φ1)(da×db)α.

Using Table 1, we have:

MZ2αφ1=(4×4)αE1(POH(m))×(4×8)αE2(POH(m))×(8×8)αE3(POH(m))=16α18m2+12m×32α36m2×64α18m2−12m,=16α×32α×64α11664m6−5184m4.

This value is what, we get after calculations:

⇒MZ2α(φ1)=32768α(11664m6−5184m4).

Theorem 2.1.4

Consider the POH(m) network, the sum and product connectivity of multiplicative indices described as:

SCII(φ1)=696(11664m6−5184m4),PCII(φ1)=132768(11664m6−5184m4).

Proof. Let φ₁ ≅ POH(m) network. We have to prove the sum and product connectivity of multiplicative indices and by using Eq. 6, we have:

SCII(φ1)=∏ab∈E(φ1)(1da+db).

Using Table 1, we have:

This value is what, we get after calculations:

⇒SCII(φ1)=696(11664m6−5184m4).

For product connectivity of multiplicative indices and by using Eq. 7, we have:

PCII(φ1)=∏ab∈E(φ1)(1da×db).

Using Table 1, we have:

This value is what, we get after calculations:

⇒PCII(φ1)=132768(11664m6−5184m4).

2.2 Results for triangular prism network TP(m)

In this section, we calculate edge partition of topological indices of the dimension m for the Triangular prism network. In the coming theorems, we compute some important indices for triangular prism network.

Theorem 2.2.1

Consider the triangular prism network TP(m), then its second multiplicative Zagreb index is equal to:

II2(φ2)=34012224m6−11337408m4−2519424m3.

Proof. Let φ₂ ≅ TP network and by using Eq. 1, we have:

II2(φ2)=∏ab∈E(φ2)(da×db).

Using Table 2, we have:

Table 2

Edge partition of TP network.

(d_a,d_b)	Number of edges
E₁ = (3,3)	18m² + 6m
E₂ = (3,6)	18m² + 6m
E₃ = (6,6)	18m² − 12m

II2(φ2)=(3×3)|E1(TP(m))|×(3×6)|E2(TP(m))| ×(6×6)|E3(TP(m))|,=9|E1(TP(m))|×18|E1(TP(m))|×36|E1(TP(m))|,=9(18m2+6m)×18(18m2+6m)×36(18m2−12m).

This value is what, we get after calculations:

⇒II2(φ2)=34012224m6−11337408m4−2519424m3.

Theorem 2.2.2

Consider the TP(m) network, then its first and second hyper Zegreb indices are equal to:

HII1(φ2)=2448880128m6−816293376m4−181398528m3,HII2(φ2)=1.99×1011m6−6.6×1010m4−1.47×1010m3.

Proof. Let φ₂ ≅ TP Network and by using Eq. 2, we have:

HII2(φ2)=∏ab∈E(φ2)(da×db)2.

Using Table 2, we have:

HII1φ2=(3+3)2E1(TP(m))×(3+6)2E2(TP(m))×(6+6)2E3(TP(m)),=6218m2+6m×9218m2+6m×12218m2−12m,=4199045832m6−1944m4−432m3.

This value is what, we get after calculations:

⇒HII1(φ2)=2448880128m6−816293376m4−181398528m3.

For second hyper Zegreb index and by using Eq. 3, we have:

HII2(φ2)=∏ab∈E(φ2)(da×db)2.

Using Table 2, we have:

HII2φ2=(3×3)2E1(TP(m))×(3×6)2E2(TP(m))×(6×6)2E3(TP(m)),=9218m2+6m×18218m2+6m×36218m2−12m,=340122245832m6−1944m4−432m3.

This value is what, we get after calculations:

⇒HII2(φ2)=1.99×1011m6−6.6×1010m4−1.47×1010m3.

Theorem 2.2.3

Consider the TP network, then its first and second universal Zagreb indices are equal to:

MZ1α(φ2)=648α(5832m6−1944m4−432m3),MZ2α(φ2)=5832α(5832m6−1944m4−432m3).

Proof. Let φ₂ ≅ TP network. Using Eq. 4, we have:

MZ1α(φ2)=∏ab∈E(φ2)(da+db)α.

Using Table 2, we have:

MZ1αφ2=(3+3)αE1(TP(m))×(3+6)αE2(TP(m))×(6+6)αE3(TP(m))=6α18m2+6m×9α18m2+6m×12α18m2−12m,=6α×9α×12α5832m6−1944m4−432m3.

This value is what, we get after calculations:

⇒MZ1α(φ2)=648α(5832m6−1944m4−432m3).

For second universal Zegreb index and by using Eq. 5:

MZ2α(φ2)=∏ab∈E(φ2)(da×db)α.

Using Table 2, we have:

MZ2αφ2=(3×3)αE1(TP(m))×(3×6)αE2(TP(m))×(6×6)αE3(TP(m))=9α18m2+6m×18α18m2+6m×36α18m2−12m,=9α×18α×36α5832m6−1944m4−432m3.

This value is what, we get after calculations:

⇒MZ2α(φ2)=5832α(5832m6−1944m4−432m3).

Theorem 2.2.4

Consider the TP(m) network, the sum and product connectivity of multiplicative indices described as:

SCII(φ2)=236(5832m6−1944m4−432m3),PCII(φ2)=15832(5832m6−1944m4−432m3).

Proof. Let φ₂ ≅ TP network. We have to prove the sum and product connectivity of multiplicative indices. First prove sum connectivity of multiplicative indices and by using Eq. 6:

SCII(φ2)=∏ab∈E(φ2)(1da+db).

Using Table 2, we have:

This value is what, we get after calculations:

⇒SCII(φ2)=236(5832m6−1944m4−432m3).

For product connectivity of multiplicative indices and by using Eq. 7, we have:

PCII(φ2)=∏ab∈E(φ2)(1da×db).

Using Table 2, we have:

This value is what, we get after calculations:

⇒PCII(φ2)=15832(5832m6−1944m4−432m3).

2.3 Results for hex POH network (m)

In this section, we calculate edge partition of topological indices of the dimension m for the hex POH network (m). In the coming theorems, we compute some important indices for hex POH network (m).

Theorem 2.3.1

Consider the hex POH network (m), then its second multiplicative Zagreb index is equal to:

II2(φ3)=32768(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480).

Proof. Let φ₃ ≅ hex POH(m) and by using Eq. 1, we have:

II2(φ3)=∏ab∈E(φ3)(da×db).

Using Table 3, we have:

Table 3

Edge partition of hex planar octahedron network.

(d_a,d_b)	Number of edges
E₁ = (4,4)	18m² + 18m − 30
E₂ = (4,8)	36m² − 48m + 12
E₃ = (8,8)	18m² − 36m + 18

This value is what, we get after calculations:

⇒II2(φ3)=32768(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480).

Theorem 2.3.2

Consider the hex POH network, then its first and second hyper Zegreb indices are equal to:

HII1(φ3)=2359296(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480),HII2(φ3)=1073741824(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480).

Proof. Let φ₃ ≅ hex POH network and by using Eq. 2, we have:

HII1(φ3)=∏ab∈E(φ3)(da+db)2.

Using Table 3, we have:

HII1φ3=(4+4)2E1(HexPOH(m))×(4+8)2∣E2( HexPOH (m))∣×(8+8)2∣E3( HexPOH (m))∣,=8218m2+18m−30×12236m2−48m+12×16218m2−36m+18.

This value is what, we get after calculations:

⇒HII1(φ3)=2359296(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480).

For second hyper Zegreb index and by using Eq. 3, we have:

HII2(φ3)=∏ab∈E(φ3)(da×db)2.

Using Table 3, we have:

HII2φ3=(4×4)2E1(HexPOH(m))×(4×8)2E2(HexPOH(m))×(8×8)2E3(HexPOH(m)),=16218m2+18m−30×32236m2−48m+12×64218m2−36m+18.

This value is what, we get after calculations:

⇒HII2(φ3)=1073741824(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480).

Theorem 2.3.3

Consider the hex POH network, then its first and second universal Zagreb indices are equal to:

MZ1α(φ3)=1536α(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480),MZ2α(φ3)=32768α(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480).

Proof. Let φ₃ ≅ hex POH network (m). We have to prove first universal Zagreb index and by using Eq. 4, we have:

MZ1α(φ3)=∏ab∈E(φ3)(da+db)α.

Using Table 3, we have:

MZ1αφ3=(4+4)αE1(HexPOH⁡(m))×(4+8)αE2(HexPOH⁡(m))×(8+8)α∣E3( HexPOH (m)∣,=8α18m2+18m−30×12α36m2−48m+12×18α18m2−36m+18,=8α×12α×16α11664m6−27216m5−11664m4+88128m3−97200m2+42768m−6480.

This value is what, we get after calculations:

⇒MZ1α(φ3)=1536α(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480).

For second universal Zegreb index and by using Eq. 5, we have:

MZ2α(φ3)=∏ab∈E(φ3)(da×db)α.

Using Table 3, we have:

MZ2αφ2=(4×4)αE1(HexPOH(m))×(4×8)αE2(HexPOH(m))×(8×8)αE3(HexPOH(m)),=16α18m2+18m−30×32α36m2−48m+12×64α18m2−36m+18,=16α×32α×64α11664m6−27216m5−11664m4+88128m3−97200m2+42768m−6480.

This value is what, we get after calculations:

⇒MZ2α(φ3)=32768α(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480).

Theorem 2.3.4

Consider the hex POH(m) network, the sum and product connectivity of multiplicative indices described as:

SCII(φ3)=116(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480),PCII(φ3)=132768(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480).

Proof. Let φ₃ ≅ hex POH(m). We have to prove the sum and product connectivity of multiplicative indices. First we prove sum connectivity of multiplicative indices and by using Eq. 6, we have:

SCII(φ3)=∏ab∈E(φ3)(1(da+db).

Using Table 3, we have:

This value is what, we get after calculations:

⇒SCII(φ3)=116(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480).

For product connectivity of multiplicative indices and by using Eq. 7:

PCII(φ3)=∏ab∈E(φ3)(1(da×db)).

Using Table 3, we have:

PCII(φ3)=14×4∣E1(HexPOH(m)) ×14×8|E2(HexPOH(m))| ×18×8|E3(HexPOH(m))|,=14×4(18m2+18m−30) ×14×8(36m2−48m+12) ×18×8(18m2−36m+18),=116×132×164(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480)=116×32×64(11664m6−27216m5−11664m4 +88128m3−97200mΛ2+42768m−6480).

This value is what, we get after calculations:

⇒PCII(φ3)=132768(11664m6−27216m5−11664m4 +88128m3−97200m2+42768m−6480).

3 Discussion

The comparison of second kind multiplicative Zagrab, first and second kind of hyper Zagreb, first and second universal Zagreb, sum and product connectivity of multiplicative indices for the planar octahedron structure, triangular prism structure and hex planar octahedron structures have been computed for the different numerical values as shown in Figures 4-9. The graphical representations show the correctness of the results.

Figure 4

Comparison of Indices for planar octahedron network.

Figure 5

Comparison of indices for planar octahedron network.

Figure 6

Comparison of indices for triangular prism network.

Figure 7

Comparison of indices for triangular prism network.

Figure 8

Comparison of indices for hex POH.

Figure 9

Comparison of indices for hex POH.

4 Conclusions

In this paper, second multiplicative Zagreb index, first hyper Zagreb and second hyper Zagreb, first universal Zagreb and second hyper Zagreb, sum and product connectivity of multiplicative indices computed for the planar octahedron network, triangular prism network and hex planar octahedron network. Chemical point of view these results may be helpful for people working in computer science and chemistry who encounter hex-derived networks. There exist many open problems for calculating the expressions of similar derived networks we are hoping to compute the other multiplicative degree based indices.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11971142

Award Identifier / Grant number: 61673169

Funding statement: The research was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 61673169).

Acknowledgement

Authors are very grateful to the referees for their careful reading with corrections and useful comments, which improved this work very much.

Author contribution: Ghulam Dustigeer: Writing – original draft; Haidar Ali: Writing – review and editing; Muhammad Imran Khan: Methodology; Yu-Ming Chu: Formal analysis, Visualization, Resources.
Conflict of interest: Authors state no conflict of interest.

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Received: 2020-11-06

Accepted: 2020-12-17

Published Online: 2020-12-31

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

On multiplicative degree based topological indices for planar octahedron networks

Abstract

1 Introduction

2 Results

2.1 Results for planar octahedron network POH(m)

Theorem 2.1.1

Theorem 2.1.2

Theorem 2.1.3

Theorem 2.1.4

2.2 Results for triangular prism network TP(m)

Theorem 2.2.1

Theorem 2.2.2

Theorem 2.2.3

Theorem 2.2.4

2.3 Results for hex POH network (m)

Theorem 2.3.1

Theorem 2.3.2

Theorem 2.3.3

Theorem 2.3.4

3 Discussion

4 Conclusions

Acknowledgement

References

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