Linking number of monotonic cycles in random book embeddings of complete graphs

TL;DR

This paper investigates the linking numbers of disjoint cycles in random book embeddings of complete graphs, revealing their distribution relates to Eulerian numbers and providing formulas for average linking measures.

Contribution

It introduces a novel analysis connecting linking numbers in random graph embeddings to Eulerian numbers, with explicit formulas for their distributions and averages.

Findings

Distribution of linking numbers described by Eulerian numbers

Mean squared linking number is proportional to the number of interior edges

Average squared linking number in complete graphs grows linearly with n

Abstract

A book embedding of a complete graph is a spatial embedding whose planar projection has the vertices located along a circle, consecutive vertices are connected by arcs of the circle, and the projections of the remaining "interior" edges in the graph are straight line segments between the points on the circle representing the appropriate vertices. A random embedding of a complete graph can be generated by randomly assigning relative heights to these interior edges. We study a family of two-component links that arise as the realizations of pairs of disjoint cycles in these random embeddings of graphs. In particular, we show that the distribution of linking numbers can be described in terms of Eulerian numbers. Consequently, the mean of the squared linking number over all random embeddings is $\frac{i}{6}$, where $i$ is the number of interior edges in the cycles. We also show that the mean of the squared linking number over all pairs of $n$-cycles in $K_{2n}$ grows linearly in $n$.