*4.1. Methodology of Carbon Graphite HAC*5*C*6*C*7[*p*, *q*] *Formulas*

For the Nanotube *HAC*5*C*6*C*7[*p*, *q*],(*p*, *q* ≥ 1) (see Figure 2), we have *<sup>V</sup>*(*HAC*5*C*6*C*7[*p*, *q*]) = 8*pq* + *p* and *<sup>E</sup>*(*HAC*5*C*6*C*7[*p*, *q*]) = 12*pq* − *p*, and its edge set can be partitioned as follows. *<sup>E</sup>*1*HAC*5*C*6*C*7[*p*, *q*] = *sr* ∈ *<sup>E</sup>HAC*5*C*6*C*7[*p*, *q*] | *dgr*(*s*) = 6, *dgr*(*r*) = 7 *<sup>E</sup>*2*HAC*5*C*6*C*7[*p*, *q*] = *sr* ∈ *<sup>E</sup>HAC*5*C*6*C*7[*p*, *q*] | *dgr*(*s*) = 6, *dgr*(*r*) = 8 *<sup>E</sup>*3*HAC*5*C*6*C*7[*p*, *q*] = *sr* ∈ *<sup>E</sup>HAC*5*C*6*C*7[*p*, *q*] | *dgr*(*s*) = 7, *dgr*(*r*) = 8 *<sup>E</sup>*4*HAC*5*C*6*C*7[*p*, *q*] = *sr* ∈ *<sup>E</sup>HAC*5*C*6*C*7[*p*, *q*] | *dgr*(*s*) = 8, *dgr*(*r*) = 8 *<sup>E</sup>*5*HAC*5*C*6*C*7[*p*, *q*] = *sr* ∈ *<sup>E</sup>HAC*5*C*6*C*7[*p*, *q*] | *dgr*(*s*) = 8, *dgr*(*r*) = 9 *<sup>E</sup>*6*HAC*5*C*6*C*7[*p*, *q*] = *sr* ∈ *<sup>E</sup>HAC*5*C*6*C*7[*p*, *q*] | *dgr*(*s*) = 9, *dgr*(*r*) = 9 Thecardinalityofedgesin*<sup>E</sup>*1*HAC*5*C*6*C*7[*p*,*<sup>q</sup>*],*<sup>E</sup>*2*HAC*5*C*6*C*7[*p*,*q*] and*<sup>E</sup>*5*HAC*5*C*6*C*7[*p*,*q*]

 are 4*p*, the cardinality of edges in *<sup>E</sup>*3*HAC*5*C*6*C*7[*p*, *q*] and *<sup>E</sup>*4*HAC*5*C*6*C*7[*p*, *q*] are 2*p* while the cardinality of edges in *<sup>E</sup>*6*HAC*5*C*6*C*7[*p*, *q*] are 24*pq* − 18*p*. The representatives of these partitioned edge set are demonstrated in Figure 3, in which the edge set with color green, red, brown, blue, yellow and black are *<sup>E</sup>*1*HAC*5*C*6*C*7[*p*, *<sup>q</sup>*], *<sup>E</sup>*2*HAC*5*C*6*C*7[*p*, *<sup>q</sup>*], *<sup>E</sup>*3*HAC*5*C*6*C*7[*p*, *<sup>q</sup>*], *<sup>E</sup>*4*HAC*5*C*6*C*7[*p*, *<sup>q</sup>*], *<sup>E</sup>*5*HAC*5*C*6*C*7[*p*, *q*] and *<sup>E</sup>*6*HAC*5*C*6*C*7[*p*, *q*] respectively.
