Let us have a point A0 from which we could make a move of one step at a time. When we make the first step from A0-> A1 it will generate a simple graph of one edge and two vertices: A0 and A1. We could continue making further steps, one step at a time, from each generated vertex (except A0) until it gets three neighbors(third order). In the second instance there is only one move possible, A1->A2 and will get a linear graph of two edges and three vertices(A0, A1, A2).

In the third instance we could make steps from two vertices A1->A11 and A2->A3. Thus we will get a graph with five vertices (A0, A1, A11, A2, A3). In the forth instance we could make move from A11->A12, A2->A21, A3->A4, and will get a graph with eight vertices (A0, A1, A11, A12, A2, A21, A3, A4).

In the fifth instance we will generate a graph with 13 vertices, then in the sixth it will be 21, in the seventh it is 34, in the eight it is 55, in the ninth it is 89, and in the tenth instance the total number of vertices will be 144.

If we count the newly generated vertices only, in the first instance it is 1, in the second it is also 1, in the third it is 2, in the fourth it is 3, fifth-5, sixth-8, seventh-13, eight-21, ninth-34, and the tenth-55. We could notice that in both cases, the number of all vertices and the number of newly generated vertices in each particular graph are Fibonacci numbers: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… The total number or vertices in each graph is two steps ahead on the Fibonacci sequence from the number of newly generated vertices.

Goran Đorđević December 2014

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Appendex 1

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Appendix 2

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Appendix 3

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