*3.1. Methodology of HAC*5*C*7[*p*, *q*] *Nanotube Formulas*

In the Nanotube *HAC*5*C*7[*p*, *q*],(*p*, *q* ≥ <sup>1</sup>), we have that *<sup>V</sup>*(*HAC*5*C*7[*p*, *q*]) = 8*pq* + *p* and *<sup>E</sup>*(*HAC*5*C*7[*p*, *q*]) = 12*pq* − *p*. The cardinality of vertices of degree two and three are 2*p* + 2 and 8*pq* − *p* − 2, respectively. According to their sum the the degree over its neighbors of each vertex, the edge set can be partitioned to six disjoint sets as follows.

$$\begin{array}{l} E\_1(\operatorname{HAC\_5C\_7}[p,q]) = \{sr \in E(\operatorname{HAC\_5C\_7}[p,q]) \mid \operatorname{dgr}(s) = 6, \operatorname{dgr}(r) = 7\} \\ E\_2(\operatorname{HAC\_5C\_7}[p,q]) = \{sr \in E(\operatorname{HAC\_5C\_7}[p,q]) \mid \operatorname{dgr}(s) = 6, \operatorname{dgr}(r) = 8\} \\ E\_3(\operatorname{HAC\_5C\_7}[p,q]) = \{sr \in E(\operatorname{HAC\_5C\_7}[p,q]) \mid \operatorname{dgr}(s) = 7, \operatorname{dgr}(r) = 9\} \\ E\_4(\operatorname{HAC\_5C\_7}[p,q]) = \{sr \in E(\operatorname{HAC\_5C\_7}[p,q]) \mid \operatorname{dgr}(s) = 8, \operatorname{dgr}(r) = 8\} \\ E\_5(\operatorname{HAC\_5C\_7}[p,q]) = \{sr \in E(\operatorname{HAC\_5C\_7}[p,q]) \mid \operatorname{dgr}(s) = 8, \operatorname{dgr}(r) = 9\} \\ E\_6(\operatorname{HAC\_5C\_7}[p,q]) = \{sr \in E(\operatorname{HAC\_5C\_7}[p,q]) \mid \operatorname{dgr}(s) = 9, \operatorname{dgr}(r) = 9\} \end{array}$$

The cardinality of edges in *<sup>E</sup>*1*HAC*5*C*7[*p*, *<sup>q</sup>*], *<sup>E</sup>*2*HAC*5*C*7[*p*, *q*]and *<sup>E</sup>*5*HAC*5*C*7[*p*, *q*]are 2*p*. The cardinality of edges in *<sup>E</sup>*3*HAC*5*C*7[*p*, *q*] and *<sup>E</sup>*4*HAC*5*C*7[*p*, *q*] are *p*. The cardinality of edges in *<sup>E</sup>*6*HAC*5*C*7[*p*, *q*] is 12*pq* − 9*p*. Such a partition is shown in Figure 1 in which red, green, blue, yellow, brown and black edges are the edges belong to *<sup>E</sup>*1*HAC*5*C*7[*p*, *<sup>q</sup>*], *<sup>E</sup>*2*HAC*5*C*7[*p*, *<sup>q</sup>*], *<sup>E</sup>*3*HAC*5*C*7[*p*, *<sup>q</sup>*], *<sup>E</sup>*4*HAC*5*C*7[*p*, *<sup>q</sup>*], *<sup>E</sup>*5*HAC*5*C*7[*p*, *q*] and *<sup>E</sup>*6*HAC*5*C*7[*p*, *q*] respectively.
