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

Fig. 1

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Fig. 4

Fig. 5

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