Student:	ÜMMÜGÜLSÜN AKBABA
Supervisor:	DOÇ. DR. ALİ HİKMET DEĞER
Department:	Matematik
Institution:	Graduate School of Natural and Applied Sciences
University:	Karadeniz Technical University Turkey

Title of the Thesis:	SPECIAL VERTICES OF TREES ON SUBORBITAL GRAPHS WITH RECURRENCE RELATIONS OF CONTINUED FRACTIONS
Level:	Ph.D.
Acceptance Date:	11/11/2022
Number of Pages:	73
Registration Number:	Di1534

Summary:

The main purpose of this thesis; from the notion of minimal length path (trees) of

suborbital graph 𝐅௨,ே, is to examine the vertices in this path by associating them with

continued fraction structures.

The thesis consists of two parts. In the first part, the preliminary information to be

used in the study is under the headings of Fibonacci, Lucas, Pell, Pell-Lucas number

sequences, generating functions, continued fractions, recurrence relations, Γ modular

group, Farey, 𝐅௨,ே, 𝐆௨,ே graphs are given. In the second part, after obtaining the

transformation that gives the vertices on the minimal-length path in the 𝐅௨,ே suborbital

graph with the 𝑘௜

values on 𝑢

ଶ + (−1)௜𝑘௜𝑢 + 1 ≡ 0 (𝑚𝑜𝑑𝑁), 𝑖 = 1,2 ,1 < 𝑘௜ ≤ 𝑁

congruence equation, the continuous fraction structure that emerges here is discussed.

Then, the vertices are associated with the Fibonacci and Pell integer sequences in the

literature for special 𝑘௜ values of that continuous fraction structure with the help of

recurrence relations. From here, the matrix relations of these integer sequences are

obtained. Then, the generating functions are examined with the help of another recurrence

relation obtained from this continued fraction structure.

Key Words: Suborbital graphs, Continued fractions, Minimal length paths, Fibonacci

sequences, Lucas sequences, Pell sequences, Pell-Lucas sequences.