(β ,α)−Connectivity Index of Graphs

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Fig. 5

Partition of paths of length 2 of the graph Y_

(dY (u), dY(v), dY (w)) where uvw ∈ E2(Y)	(2,2,3)	(3,3,2)	(3,3,3)
Number of paths of length 2 in Y	4p	8p	(36pq — 26p)

Number of vertices and edges_

Graph	Number of vertices	Number of edges
2D-lattice of TUC4C8 [p,q]	4pq	6pq − p − q
TUC4C8[p,q] nanotube	4pq	6 pq — p
TUC4C8[p,q] nanotorus	4pq	6pq

Partition of paths of length 2 of the graph X_

(dx(u), dX (v), dX (w)) where uvw ∈ E2(X)	(2,2,2)	(2,2,3)	(3,3,2)	(3,3,3)
Number of paths of length 2 in X	8	4(p + q — 2)	8(p + q — 2)	(36 pq — 26 p — 26q + 16)

Partition of paths of length 2 of the graph A = L(S(Tn,k)) for k = 1

(dA(u), dA(v), dA (w)) where uvw ∈ E2(A)	(1,3,3)	(2,3,3)	(2,2,3)	(3,3,3)	(2,2,2)
Number of paths of length 2 in A	2	4	2	3	2n − 4

Partition of paths of length 2 of the graph A = L(S(Tn,k)) for k > 1

(dA(u), dA(v), dA(w)) where uvw ∈ E2(A)	(1,2,2)	(2,3,3)	(2,2,3)	(3,3,3)	(2,2,2)
Number of paths of length 2 in A	1	6	3	3	2n + 2k – 8

Partition of paths of length 2 of the graph C_

(dC(u),dC(v),dC(w))where uvwe E2(C)	(2,2,2)	(2,2,3)	(2,3,3)	(3,3,3)
Number of paths of length 2 in C	4	4	8	18n — 44

Partition of paths of length 2 of the graph B_

(dB(u),dB(v),dB(w)) where uvw e E2(B)	(3,3,3)	(3,3,n)	(3,n,n)	(n,n,n)
Number of paths of length 2 in B	7n	2n	n(n —1)	n(n−1)(n−2)2$\frac{{n\left( {n - 1} \right)\left( {n - 2} \right)}}{2}$

Experimental values of the acentric factor and corresponding values of 2χ1 of octane isomers_

Alkane	AcentFac	2χ1
n-octane	0.397898	40
2-methyl-heptane	0.377916	47
3-methyl-heptane	0.371002	54
4-methyl-heptane	0.371504	56
3-ethyl-hexane	0.362472	64
2,2-dimethyl-hexane	0.339426	64
2,3-dimethyl-hexane	0.348247	70
2,4-dimethyl-hexane	0.344223	63
2,5-dimethyl-hexane	0.35683	54
3,3-dimethyl-hexane	0.322596	80
3,4-dimethyl-hexane	0.340345	78
2-methyl-3-ethyl-pentane	0.332433	81
3-methyl-3-ethyl-pentane	0.306899	96
2,2,3-trimethyl-pentane	0.300816	96
2,2,4-trimethyl-pentane	0.30537	75
2,3,3-trimethyl-pentane	0.293177	103
2,3,4-trimethyl-pentane	0.317422	87
2,2,3,3-tetramethylbutane	0.255294	120