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Computation of certain topological coindices of graphene sheet and C4C8(S) nanotubes and nanotorus

Melaku Berhe
Melaku Berhe
 y 
Chunxiang Wang
Chunxiang Wang
   | 18 dic 2019
Applied Mathematics and Nonlinear Sciences's Cover Image
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Publicado en línea: 18 dic 2019
Páginas: 455 - 468
Recibido: 05 mar 2019
Aceptado: 09 may 2019
DOI: https://doi.org/10.2478/AMNS.2019.2.00043
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© 2019 R. A. Mundewadi and Kumbinarasaiah S, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 Public License.

Fig. 1

Graphene sheet G(n; m) where, n and m denotes the number of hexagons in rows and columns.
Graphene sheet G(n; m) where, n and m denotes the number of hexagons in rows and columns.

Fig. 2

Graphene sheet G(1, m).
Graphene sheet G(1, m).

Fig. 3

Comparative model for different coindices of the graphene sheet G(n, m) (a) for n = 1 and (b) for n ≠ 1
Comparative model for different coindices of the graphene sheet G(n, m) (a) for n = 1 and (b) for n ≠ 1

Fig. 4

Two-dimensions of TUC4C8(S)[m, n] (a) nanotube and (b) nanotorus
Two-dimensions of TUC4C8(S)[m, n] (a) nanotube and (b) nanotorus

Fig. 5

Comparative model for different coindices of TUC4C8(S)[m, n] (a) nanotubes and (b) nanotorus
Comparative model for different coindices of TUC4C8(S)[m, n] (a) nanotubes and (b) nanotorus

Vertex and edge partitions of graphene sheet for n ≠ 1

Row n2 n3 m22 m23 m33
1 m + 3 3m − 1 3 2m 3m − 2
2 2 2m 1 2 3m − 1
3 2 2m 1 2 3m − 1
⋮ ⋮ ⋮ ⋮ ⋮ ⋮
n − 1 2 2m 1 2 3m − 1
n m + 3 m − 1 3 2m m − 1
Total 2m + 2n + 2 2mn − 2 n + 4 4m + 2n − 4 3nm − 2m − n − 1

Few topological coindex values of TUC4C8(S)[m, n] nanotubes

n m F M2 M1 ∏1 ∏2
2 2 11552 5652 4048 Inf Inf
2 3 26928 13182 9432 Inf Inf
2 4 48704 23848 17056 Inf Inf
3 2 24160 11892 8336 Inf Inf
3 3 55728 27438 19224 Inf Inf
3 4 100288 49384 34592 Inf Inf
4 2 41376 20436 14160 Inf Inf
4 3 94896 46878 32472 Inf Inf
4 4 170304 84136 58272 Inf Inf

Vertex and edge partitions of graphene sheet for n = 1

n2 n3 m22 m23 m33
2m + 4 2m − 2 6 4m − 4 m − 1

Few topological coindex values of TUC4C8(S)[m, n] nanotorus

n m F M2 M1 ∏1 ∏2
2 2 8064 4032 2688 Inf Inf
2 3 19008 9504 6336 Inf Inf
2 4 34560 17280 11520 Inf Inf
3 2 19008 9504 6336 Inf Inf
3 3 44064 22032 14688 Inf Inf
3 4 79488 39744 26496 Inf Inf
4 2 34560 17280 11520 Inf Inf
4 3 79488 39744 26496 Inf Inf
4 4 142848 71424 47616 Inf Inf

The edge partitions mij of G based on the vertex degrees of G, for n ≠ 1

(dG(u), dG(v)) where uv ∈ E(G) Number of edges in G
(2, 2) 2m2 + 2n2 + 4mn + 3m + 2n − 3
(3, 3) 2(mn)2 − 8mn + 2m + n + 4
(2, 3) 4m2 n + 4mn2 + 4mn − 8m − 6n

The edge partitions mij of G based on the vertex degrees of G, for n = 1

(dG(u), dG(v)) where uv ∈ E(G) Number of edges in G
(2, 2) 2m2 + 7m
(3, 3) 2m2 − 6m + 4
(2, 3) 4m2 − 4

The edge partitions mij of G based on the vertex degrees of G.

(du, dv) where uv ∈ E(G) Number of edges in; G
(2, 2) 8m2 − 4m
(3, 3) 32(mn)2 − 16mn + 2m
(2, 3) 32nm2 − 4m

MATLAB illustration: » Topological.coindices.of.graphene.sheet(3, 3)

n m F M2 M1 ∏1 ∏2
1 1 72 36 36 262144 262144
1 2 332 160 148 4.294967e + 21 3.829436e + 22
1 3 800 384 340 9.119789e + 48 1.578155e + 52
2 1 332 160 148 1.099512e + 16 2.279947e + 16
2 2 1168 558 476 4.057816e + 67 3.066407e + 73
2 3 2532 1212 996 2.518170e + 156 1.557022e + 173
3 1 800 383 340 1.593799e + 32 3.330579e + 33
3 2 2532 1211 996 7.383353e + 122 6.934815e + 135
3 3 5256 2523 2004 3.208625e + 277 Inf

Vertex and edge partitions of G

n2 n3 m22 m23 m33
4m 8nm 2m 4m 12nm − 2m
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