VGR: Virtual Graphene


 Graphene is an allotrope of carbon in the form of a single layer of atoms in a two-dimensional hexagonal lattice in which one atom forms each vertex. This paper introduce new concept namely virtual graphene. For constructing this type of graphene, we use division graph by two concepts. VGR is not belonging to any chemical structures, but it able to generate the graphene structure.


Introduction
Graphene is a crystalline allotrope of carbon with 2-dimensional properties. Its carbon atoms are densely packed in a regular atomic-scale chicken wire (hexagonal) pa ern [1]. As we know that chemical graphene has chemical formula and there exist physically, while VGR does not use any chemical formulas. VGR use two concepts; DGBT[2] and virtually material decomposition for constructing virtual graphene. e remaining sections of the paper are organized as follows: Next section describe, de nitions and related terminologies. Section 3 introduce virtual graphene structure construction. Conclusion and future works described in section 4.

De nitions
We need to de ne some basic concepts before we get into topics related to virtual graphene. Please pay a ention to these de nitions are provided for the convenience of the VGR constructing, it is possible this de nition are not true in real world, so we assume that our denitions are also virtual.
Decomposition: Equivalent to dividing by two. Combination: Equivalent to adding two numbers. Material: Equivalent to number. Peak: It is a place where a decomposable material cannot remain forever without decomposition, eventually the material splits into several parts and moves to the lower peaks.
Slope: A place where material is not stable and ow to nearest peak or valley. ere are two types of slopes.  Valley: It is a place where two material combine with together and stay for a while the next decomposition occur. Important point: A valley for lower peaks, is itself a peak. please see Fig. 3.  3 Virtual graphene construction e VGR theory is based on this idea that, the decomposable material cannot remain indenitely integrated, eventually decomposing into several parts over time. In decomposition step we use DGBT concepts. For convince, we introduce this section with an example. We want to construct graphene of number 3 by using DGBT(3). So we explain step by step how to DGBT and material composition able to construct the graphene of number 3.
In rst step: According DGBT(3), root number is 3, and number 1, 5 assigned as le and right child of root respectively. Now for constructing one level of graphene, number 3 assigned to peak, then its decomposed into two section; for decomposition we use DGBT, so number 3 decomposed into 1, 5, number 1 ows to the le peak and number 5 ows to the right peak and remain there until to decomposed again. Fig.4 show decomposing of number 3. Figure 4: decomposition step as showed in this picture, we see that number 3 decomposed to the numbers 1 and 5.
In second step: If we note the structure of DGBT we see that number 1, and 5 itself divided by two again; So for constructing VGR, number 1 and 5 decomposed separately. Fig. 5 shows that DGBT and its related graphene construction. : is picture show that a er a time slice numbers 1, 5 that are in peak, decomposed to 0, 1 and 2, 5 respectively.
In third step: In this step, the value of slopes that leads to valley combined with together, and other slopes that are not lead to valley, ow to neighbor peak. If we look at DGBT(3) nodes 5, 2 added with together and generate node 7 as a common node. but nodes 0(i.e right child of node 1 ) and node 5( i.e le child of node 5) directly transmited to the next level of graph. According to DGBT for constructing graphene, number 0, 5 over the slop( represented with orange color) owed to next peaks, and numbers 5, 2 on slop (with blue color) combined with together and generate valley with value of 7. Please see gure 6. According Fig. 6, we have two peaks and one valley, of course valley(7) itself as peak for the next valleys.
As we see, this tree steps could generate one level of graphene structures, We can repeat these steps in nitely to reach in nite graphene structure; of course, rst step execute one time but steps 2, 3 could be repeated. For instance if we continue steps 2, 3 once again, we reach to the following structures of graphene. ese structures as shown in Fig 7, 8.  Figure 6: is picture show that, combination (adding) of number 5 and 2 in valley, and owing 0 and 5 to next peak.
If we repeat step 2 again, we have the following picture. Fig 7. shows this operation results. We see that if we repeat this steps we can get the bigger size of graphene. Note: Now that we are familiar with the basic concepts and how graphene is formed, we can eliminate DGBT structure and produce directly graphene from decomposition and combination concepts.
For example we have peaks 0, 3, 7, 5 from Fig. 8, if we perform step 2, peak 0 decomposed(divided by two) int 0, 0 peak 3 into 1, 5 peak 7 into 3, 5 and nally 5 decomposed into 2, 5. Please see Fig. 9 part a; if we run step three slopes 0, 1 combined(added) with together, slopes 5, 3 with together and slopes 5, 2 combined with together then generate valleys 1, 8, 7 respectively. Fig. 9 part b show that valleys 1, 8, 7 and peaks 0, 5. is section show that how we can generate the graphene structure virtually by using DGBT theory and virtual material decomposition concepts. To keep the paper simplicity, more examples are provided in the appendix.

Conclusion
In this article, the virtual graphene structure introduced for the rst time. Also, we showed that how DGBT could generate graphene structure without any chemical formula. VGR used two concept; one is virtually decomposition of material and others, DGBT concepts. According to the DGBT, each distinct number has in nite unique DGBT, so each distinct number could generate unique in nite virtual graphene.
In this section some of virtual graphene structure shown in Fig 10, 11, 12 and 13. We can continue this structures in nitely. As we see that unique number generate unique VGR.