A study on random permutation graphs

TL;DR

This paper investigates various structural properties of random permutation graphs, providing explicit formulas for degrees, isolated vertices, cliques, and connected components, enhancing understanding of their probabilistic characteristics.

Contribution

It offers new explicit formulas and analyses for key graph statistics in random permutation graphs, advancing theoretical understanding of their structure.

Findings

Formulas for degree distribution and isolated vertices

Probabilities of specific numbers of connected components

Analysis of clique counts and graph connectivity

Abstract

For a given permutation $\pi_n$ in $S_n$, a random permutation graph is formed by including an edge between two vertices $i$ and $j$ if and only if $(i - j) (\pi_n(i) - \pi_n (j)) < 0$. In this paper, we study various statistics of random permutation graphs. In particular, the degree of a given node, the number of nodes with a given degree, the number of isolated vertices, and the number of cliques are analyzed. Further, explicit formulas for the probabilities of having a given number of connected components and isolated vertices are obtained.