Molecular descriptors of benzenoid systems

Molecular descriptors are being widely used in QSAR/QSPR studies in chemistry and drug designing as well as modeling of compounds. Different topological descriptors have been formulated to investigate the physio chemical properties and chemical reactivity of compounds. In this article we gave exact relations for first and second Zagreb index, hyper Zagreb index, multiplicative Zagreb indices as well as first and second Zagreb polynomials for some benzenoid systems.

Keywords:
benzenoid hydrocarbons; chemical graphs; topological indices

Authorship

Nazeran Idrees ^**e-mail: nazeranjawwad@gmail.com

Department of Mathematics, Government College University Faisalabad, Faisalabad, 38000, Punjab, PakistanGovernment College University FaisalabadPakistanPunjab, PakistanDepartment of Mathematics, Government College University Faisalabad, Faisalabad, 38000, Punjab, Pakistan

Muhammad Nawaz Naeem

Fida Hussain

Afshan Sadiq

Abdus Salam School of Mathematical Sciences, Government College University, Lahore, 54000, Punjab, PakistanGovernment College UniversityPakistanPunjab, PakistanAbdus Salam School of Mathematical Sciences, Government College University, Lahore, 54000, Punjab, Pakistan

Muhammad Kamran Siddiqui

Department of Mathematics, Comsats Institute of Information Technology, Sahiwal, 57000, Punjab, PakistanComsats Institute of Information TechnologyPakistanPunjab, PakistanDepartment of Mathematics, Comsats Institute of Information Technology, Sahiwal, 57000, Punjab, Pakistan

*e-mail: nazeranjawwad@gmail.com

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables | Formulas

Figures (3)
Tables (2)
Formulas (15)

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Figure 1
An example of benzenoid system(a) and a non - benzenoid system (b)

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Figure 2
Graph of zigzag benzenoid system Zn

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Figure 3
Graph of rhombic benzenoid system with n hexagons along each boundary

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Table 1
Edge partition of benzenoid graph Z_n based on degree of end vertices of each edge

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Table 2
Edge partition of benzenoid system R_n based on degree of end vertices of each edge

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Figure 1 An example of benzenoid system(a) and a non - benzenoid system (b)

Figure 2 Graph of zigzag benzenoid system Zn

Figure 3 Graph of rhombic benzenoid system with n hexagons along each boundary

Table 1 Edge partition of benzenoid graph Z_n based on degree of end vertices of each edge

(d_u, d_v) where uv ∈ E(G)	(2,2)	(2,3)	(3,3)
Number of edge	2(n + 2)	4n	4n - 3

Table 2 Edge partition of benzenoid system R_n based on degree of end vertices of each edge

(d_u, d_v) where uv ∈ E(G)	(2,2)	(2,3)	(3,3)
Number of edge	6	8(n - 1)	n(3n - 4) + 1

(1)

M_{1} (G) = \sum_{uv ∊ E (G)} [\deg (u) + \deg (v)]

(2)

M_{2} (G) = \sum_{uv ∊ E (G)} [\deg (u) \times \deg (v)]

(3)

HM (G) = \sum_{uv ∊ E (G)} {[\deg (u) + \deg (v)]}^{2}

(4)

{PM}_{1} (G) = \prod_{uv ∊ E (G)} [\deg (u) + \deg (v)]

(5)

{PM}_{2} (G) = \prod_{uv ∊ E (G)} [\deg (u) \times \deg (v)]

(6)

M_{1} (G, x) = \sum_{uv ∊ E (G)} X^{[\deg (u) + \deg (v)]}

(7)

M_{2} (G, x) = \sum_{uv ∊ E (G)} X^{[\deg (u) + \deg (v)]}

\begin{matrix} M_{1} (Z) = 52 n - 2 \\ M_{2} (Z_{n}) = 68 n - 11 \\ HM (Z_{n}) = 308 n + 20 \\ {PM}_{1} (Z_{n}) = 4^{2 n + 4} \times 5^{4 n} \times 6^{4 n - 3} \\ {PM}_{2} (Z_{n}) = 42^{n + 4} \times 6^{4 n} \times 9^{4 n - 3} \end{matrix}

\begin{matrix} M_{1} (G) = \sum_{uv \in E (G)} [d_{u} + d_{v}] \\ M_{1} (Z_{n}) = \sum_{uv \in E_{1}} [d_{u} + d_{v}] + \sum_{uv \in E_{2}} [d_{u} + d_{v}] + \sum_{uv \in E_{3}} [d_{u} + d_{v}] \\ \begin{matrix} = (2 n + 4)(2 + 2) + (4 n)(2 + 3) + (4 n - 3)(3 + 3) \\ = 52 n - 2 \end{matrix} \\ M_{2} (G) = \sum_{uv \in E (G)} [d_{u} \times d_{v}] \\ M_{2} (Z_{n}) = \sum_{uv \in E_{1}} [d_{u} \times d_{v}] + \sum_{uv \in E_{2}} [d_{u} \times d_{v}] + \sum_{uv \in E_{3}} [d_{u} \times d_{v}] \\ \begin{matrix} = (2 n + 4)(2 \times 2) + (4 n)(2 \times 3) + (4 n - 3)(3 \times 3) \\ = 68 n - 11 \end{matrix} \\ HM (G) = \sum_{uv \in E (G)} [d_{u} + d_{v}]^{2} \\ HM (Z_{n}) = \sum_{uv \in E_{1}} [d_{u} + d_{v}]^{2} + \sum_{uv \in E_{2}} [d_{u} + d_{v}]^{2} + \sum_{uv \in E_{3}} [d_{u} + d_{v}]^{2} \\ = (4 n + 8)(2 + 2)^{2} + (4 n)(2 + 3)^{2} + (4 n - 3)(3 + 3)^{2} \\ = 308 n + 20 \end{matrix}

\begin{matrix} {PM}_{1} (G) = \prod_{uv \in E (G)} [d_{u} + d_{v}] \\ {PM}_{1} (Z_{n}) = \prod_{uv \in E_{1}} [d_{u} + d_{v}] \times \prod_{uv \in E_{2}} [d_{u} + d_{v}] \times \prod_{uv \in E_{3}} [d_{u} + d_{v}] \\ \begin{matrix} = (2 + 2)^{2 n + 4} \times (2 + 3)^{4 n} \times (3 + 3)^{4 n - 3} \\ = (4)^{2 n + 4} \times (5)^{4 n} \times (6)^{4 n - 3} \end{matrix} \\ {PM}_{2} (G) = \prod_{uv \in E (G)} [d_{u} \times d_{v}] \\ {PM}_{2} (Z_{n}) = \prod_{uv \in E_{1}} [d_{u} \times d_{v}] \times \prod_{uv \in E_{2}} [d_{u} \times d_{v}] \times \prod_{uv \in E_{3}} [d_{u} \times d_{v}] \\ \begin{matrix} = (2 \times 2)^{2 n + 4} \times (2 \times 3)^{4 n} \times (3 \times 3)^{4 n - 3} \\ = 4^{2 n + 4} \times 6^{4 n} \times 9^{4 n - 3} \end{matrix} \end{matrix}

\begin{matrix} M_{1} (Z_{n}, x) = (2 n + 4) x^{4} + (4 n) x^{5} + (4 n - 3) x^{6} \\ M_{2} (Z_{n}, x) = (2 n + 4) x^{4} + (4 n) x^{6} + (4 n - 3) x^{9} \end{matrix}

\begin{matrix} M_{1} (G, x) = \sum_{uv \in E (G)} x^{[d_{u} + d_{v}]} \\ M_{1} (Z_{n}, x) = \sum_{uv \in E_{1}} x^{[d_{u} + d_{v}]} + \sum_{uv \in E_{2}} x^{[d_{u} + d_{v}]} + \sum_{uv \in E_{3}} x^{[d_{u} + d_{v}]} \\ \begin{matrix} = (2 n + 4) x^{2 + 2} + (4 n) x^{2 + 3} + (4 n - 3) x^{3 + 3} \\ = (2 n + 4) x^{4} + (4 n) x^{5} + (4 n - 3) x^{6} \end{matrix} \\ M_{2} (G, x) = \sum_{uv \in E (G)} x^{[d_{u} \times d_{v}]} M_{2} (Z_{n}, x) = \sum_{uv \in E_{1}} x^{[d_{u} \times d_{v}]} + \sum_{uv \in E_{2}} x^{[d_{u} \times d_{v}]} + \sum_{uv \in E_{3}} x^{[d_{u} \times d_{v}]} \\ \begin{matrix} = (2 n + 4) x^{2 \times 2} + (4 n) x^{2 \times 3} + (4 n - 3) x^{3 \times 3} \\ = (2 n + 4) x^{4} + (4 n) x^{6} + (4 n - 3) x^{9} \end{matrix} \end{matrix}

\begin{matrix} M_{1} (R_{n}) = 18 n^{2} + 16 n - 10 \\ M_{2} (R_{n}) = 27 n^{2} + 12 n - 15 \\ HM (R_{n}) = 108 n^{2} + 56 n - 68 \\ {PM}_{1} (R_{n}) = 4^{6} \times 5^{8 (n - 1)} \times 6^{n (3 n - 4) + 1} \\ {PM}_{2} (R_{n}) = 4^{6} \times 6^{8 (n - 1)} \times 9^{n (3 n - 4) + 1} \end{matrix}

\begin{matrix} M_{1} (H) = \sum_{uv \in E (G)} [d_{u} + d_{v}] \\ \begin{matrix} M_{1} (R_{n}) = 6 (2 + 2) + 8 (n - 1)(2 + 3) + (3 n^{2} - 4 n + 1)(3 + 3) \\ = 18 n^{2} + 16 n - 10 \end{matrix} \\ M_{2} (H) = \sum_{uv \in E (G)} [d_{u} \times d_{v}] \\ \begin{matrix} M_{2} (R_{n}) = 6 (2 \times 2) + 8 (n - 1)(2 \times 3) + (3 n^{2} - 4 n + 1)(3 \times 3) \\ = 27 n^{2} + 12 n - 15 \end{matrix} \\ HM (H) = \sum_{uv \in E (G)} [d_{u} + d_{v}]^{2} \\ \begin{matrix} HM (R_{n}) = 6 (2 + 2)^{2} + 8 (n - 1)(2 + 3)^{2} + (3 n^{2} - 4 n + 1)(3 + 3)^{2} \\ = 108 n^{2} + 56 n - 68 \end{matrix} \\ {PM}_{1} (H) = \prod_{uv \in E (G)} [d_{u} + d_{v}] \\ {PM}_{1} (R_{n}) = \prod_{uv \in E_{1}} [d_{u} + d_{v}] \times \prod_{uv \in E_{2}} [d_{u} + d_{v}] \times \prod_{uv \in E_{3}} [d_{u} + d_{v}] \\ \begin{matrix} = (2 + 2)^{6} \times (2 + 3)^{8 (n - 1)} \times (3 + 3)^{n (3 n - 4) + 1} \\ = 4^{6} \times 5^{8 (n - 1)} \times 6^{n (3 n - 4) + 1} \end{matrix} \\ {PM}_{2} (H) = \prod_{uv \in E (G)} [d_{u} \times d_{v}] \\ {PM}_{2} (R_{n}) = \prod_{uv \in E_{1}} [d_{u} \times d_{v}] \times \prod_{uv \in E_{2}} [d_{u} \times d_{v}] \times \prod_{uv \in E_{3}} [d_{u} \times d_{v}] \\ \begin{matrix} = (2 \times 2)^{6} \times (2 \times 3)^{8 (n - 1)} \times (3 \times 3)^{n (3 n - 4) + 1} \\ = 4^{6} \times 6^{8 (n - 1)} \times 9^{n (3 n - 4) + 1} \end{matrix} \end{matrix}

\begin{matrix} M_{1} (H, x) = \sum_{uv \in E (G)} x^{[d_{u} + d_{v}]} \\ \begin{matrix} M_{1} (R_{n}, x) = \sum_{uv \in E_{1}} x^{[d_{u} + d_{v}]} + \sum_{uv \in E_{2}} x^{[d_{u} + d_{v}]} + \sum_{uv \in E_{3}} x^{[d_{i} + d_{v}]} \\ \begin{matrix} = 6 x^{2 + 2} + 8 (n - 1) x^{2 + 3} + (3 n^{2} - 4 n + 1) x^{3 + 3} \\ = 6 x^{4} + 8 (n - 1) x^{5} + (3 n^{2} - 4 n + 1) x^{6} \end{matrix} \end{matrix} \\ M_{2} (H, x) = \sum_{uv \in E (G)} x^{[d_{u} \times d_{v}]} \\ M_{2} (R_{n}, x) = \sum_{uv \in E_{1}} x^{[d_{u} \times d_{v}]} + \sum_{uv \in E_{2}} x^{[d_{u} \times d_{v}]} + \sum_{uv \in E_{3}} x^{[d_{u} \times d_{v}]} \\ \begin{matrix} = 6 x^{2 \times 2} + 8 (n - 1) x^{2 \times 3} + (3 n^{2} - 4 n + 1) x^{3 \times 3} \\ = 6 x^{4} + 8 (n - 1) x^{6} + (3 n^{2} - 4 n + 1) x^{9} \end{matrix} \end{matrix}

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[1] *e-mail: nazeranjawwad@gmail.com