Fibonacci Sumsets and the Gonality of Strip Graphs

TL;DR

This paper explores the divisor theory of graphs through additive combinatorics, computes the gonality of strip graphs, and reveals Fibonacci number properties in their Jacobians, advancing understanding of graph invariants and Fibonacci sequences.

Contribution

It introduces a new combinatorial approach to graph divisor theory and calculates gonality for specific outerplanar graphs, linking graph invariants to Fibonacci numbers.

Findings

Jacobians of these graphs are cyclic of Fibonacci order

Gonality of certain outerplanar graphs is computed

Results on additive properties of Fibonacci numbers

Abstract

We provide a new perspective on the divisor theory of graphs, using additive combinatorics. As a test case for this perspective, we compute the gonality of certain families of outerplanar graphs, specifically the strip graphs. The Jacobians of such graphs are always cyclic of Fibonacci order. As a consequence, we obtain several results on the additive properties of Fibonacci numbers.