Properties of the Fibonacci-sum graph

TL;DR

This paper investigates the properties of Fibonacci-sum graphs, revealing their automorphisms, degree sequences, treewidth, bipartition, and outerplanarity, thus deepening understanding of their structural characteristics.

Contribution

It explicitly characterizes the automorphisms of Fibonacci-sum graphs and analyzes their structural properties, including degree sequence, treewidth, bipartition, and outerplanarity.

Findings

Each $G_n$ has at most one non-trivial automorphism.

$G_n$ is bipartite and outerplanar.

The degree sequence and treewidth of $G_n$ are determined.

Abstract

For each positive integer $n$, the Fibonacci-sum graph $G_n$ on vertices $1,2,\ldots,n$ is defined by two vertices forming an edge if and only if they sum to a Fibonacci number. It is known that each $G_n$ is bipartite, and all Hamiltonian paths in each $G_n$ have been classified. In this paper, it is shown that each $G_n$ has at most one non-trivial automorphism, which is given explicitly. Other properties of $G_n$ are also found, including the degree sequence, the treewidth, the nature of the bipartition, and that $G_n$ is outerplanar.