Abstract
Apart from its applications in Chemistry, Biology, Physics, Social Sciences, Anthropology, etc., there are close relations between graph theory and other areas of Mathematics. Fibonacci numbers are of utmost interest due to their relation with the golden ratio and also due to many applications in different areas from Biology, Architecture, Anatomy to Finance. In this paper, we define Fibonacci graphs as graphs having degree sequence consisting of n consecutive Fibonacci numbers and use the invariant Ω to obtain some more information on these graphs. We give the necessary and sufficient conditions for the realizability of a set D of n successive Fibonacci numbers for every n and also list all possible realizations called Fibonacci graphs for 1≤n≤4.
Highlights
Graph theory is one of the most popular subjects in mathematics as it can be applied to any area of science
There is a close relation between algebraic number theory and spectral graph theory
We introduce graphs whose degree sequences consist of consecutive Fibonacci numbers and call them Fibonacci graphs
Summary
Graph theory is one of the most popular subjects in mathematics as it can be applied to any area of science. Several mathematicians studied interrelations between number theory and graph theory and connected several properties of numbers with graphs. Mathematician Leonardo Pisano, Fibonacci or Leonardo of Pisa lived between 1170–1240 in Italy This number sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, · · · is called Fibonacci sequence. A Fibonacci cube graph of order n is a graph on Fn+2 vertices labeled by the Zeckendorf representations of the numbers 0 to F( n + 2) − 1 so that two vertices are connected by an edge iff their labels differ by a single bit, i.e., if the Hamming distance between them is exactly 1. Note that our definition of Fibonacci graph of order n means a graph with n vertices with degrees equal to n successive. The number of loops which is equal to the number of faces of the graph is formulized in all cases by means of the Ω invariant
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