Subgraph distributions in dense random regular graphs

TL;DR

This paper proves that the count of a fixed connected subgraph in dense random regular graphs follows an asymptotic Gaussian distribution, revealing intricate dependencies on specific subgraph structures.

Contribution

It establishes the Gaussian nature of subgraph counts in dense regular graphs and analyzes how variance depends on specific cycles and paths, extending previous understanding.

Findings

Subgraph counts are asymptotically Gaussian in dense regular graphs.

Variance depends on cycles of length 3, 4, 5, and paths of length 3 in the subgraph.

Provides control over distribution of spectral moments in random regular graphs.

Abstract

Given connected graph $H$ which is not a star, we show that the number of copies of $H$ in a dense uniformly random regular graph is asymptotically Gaussian, which was not known even for $H$ being a triangle. This addresses a question of McKay from the 2010 International Congress of Mathematicians. In fact, we prove that the behavior of the variance of the number of copies of $H$ depends in a delicate manner on the occurrence and number of cycles of length $3,4,5$ as well as paths of length $3$ in $H$. More generally, we provide control of the asymptotic distribution of certain statistics of bounded degree which are invariant under vertex permutations, including moments of the spectrum of a random regular graph. Our techniques are based on combining complex-analytic methods due to McKay and Wormald used to enumerate regular graphs with the notion of graph factors developed by Janson in the context of studying subgraph counts in $\mathbb{G}(n,p)$.