A study on random permutation graphs.

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 $(i - j) (pi_n(i) - pi_n (j)) u003c 0$. In this paper, we study various statistics of random permutation graphs. In particular, we prove central limit theorems for the number $m$-cliques and cycles of size at least $m$. Other problems of interest are on the number of isolated vertices, the distribution of a given node (the mid-node as a special case) and extremal degree statistics. Besides, we introduce a directed version of random permutation graphs, and provide two distinct paths that provide variations/generalizations of the model discussed in this paper.