A golden ratio inequality for vertex degrees of graphs

TL;DR

This paper establishes a new inequality relating vertex degrees in graphs, inspired by crossing number studies, involving the golden ratio, and generalizes it to graphs with bounded maximum average degree.

Contribution

It introduces a novel inequality linking vertex degrees and crossing number concepts, with the exponent involving the golden ratio, and extends it to broader classes of graphs.

Findings

The inequality is tight with the exponent  being optimal.

The inequality applies to trees and general graphs with bounded maximum average degree.

Provides a new perspective on graph degree relations inspired by geometric constants.

Abstract

Motivated by the study of the crossing number of graphs, it is shown that, for trees, the sum of the products of the degrees of the end-vertices of all edges has an upper bound in terms of the sum of all vertex degrees to the power of $\phi^2$, where $\phi$ is the golden ratio. The exponent $\phi^2$ is best possible. This inequality is generalized for all graphs with bounded maximum average degree.