Some relations between special vertex values of suborbital graphs and special number sequences

In this study, by using Lorentz matrix multiplication, nth powers of some special matrices are obtained, their quadratic equations and characteristic roots are investigated. Especially by finding nth power of matrix M under Lorentz matrix multiplication, some identities obtained under classical matrix multiplication have been reached again by using Lorentz matrix multiplication. Information was given about the development of Graph theory in the process. Suborbital graphs, G_(u,N), F_(u,N) and Farey graphs were examined. In the F_(u,N) suborbital graph, the Lorentz matrix, which gives the vertices obtained under the classical matrix multiplication under Lorentz matrix multiplication, was obtained,. It was seen that the Lorentz matrix is not a member of the Modular group. In the matrix A^n obtained for k=3, the relevant matrix was written in the type of Lucas numbers using the identity F_n≅α^n/√5. From the relation between matrices and continuous fractions, the vertices of suborbital graph were written with Lucas numbers. The vertices of suborbital graph obtained for (u,N)=(3,4), n=15 written in the form of the Fibonacci and Lucas number sequences types were compared and it was observed that the values of vertices are very close to each other. However, new identities were obtained from the equation F_2n/F_(2n+2) =(-p_n)/p_(n+1) ≅L_n/(αL_(n+1) ) and proved. Dijkstra algorithm was applied to the Farey graph and the minimum length from a source vertex to the other vertices and a tree were obtained.