Fibonacci and Lucas graphs

Cangül et al. (2020) introduced the definition of Fibonacci graphs formed by Fibonacci numbers and used the Ω invariant to gain more insight into these graphs [1]. They provided the necessary and sufficient conditions for the realizability of the degree set 𝐷 of Fibonacci graphs formed by 𝑛 consecutive Fibonacci numbers, for 1≤ 𝑛 ≤ 4 . In this thesis, the definition of Fibonacci graphs whose degree sequences consist of Fibonacci numbers is presented, along with the illustration of Fibonacci graphs with 1, 2, 3, and 4 vertices using loops to demonstrate their realizability. Furthermore, various findings related to the Ω invariant existing in the literature have been compiled and presented together [1–4]. Lucas graphs are also studied in a similar manner to Fibonacci graphs. The first chapter provides information about the subject, purpose, and structure of the thesis. The second chapter defines some basic concepts, particularly focusing on the Ω invariant, and provides the necessary background for the rest of the thesis. The third chapter offers detailed information about Fibonacci numbers and Fibonacci graphs, including the definitions and illustrations necessary to demonstrate the realizability of Fibonacci graphs with 1, 2, 3, and 4 vertices. The fourth chapter discusses Lucas graphs. Finally, the fifth and last chapter presents the conclusions and recommendations of the thesis.