Asymptotic normality of Laplacian coefficients of graphs

TL;DR

This paper demonstrates that Laplacian coefficients of various classes of graphs tend to follow a normal distribution when the graph size is large, with specific exceptions like complete graphs which follow a Poisson distribution.

Contribution

It establishes the asymptotic normality of Laplacian coefficients for several graph families and identifies cases with different limiting distributions.

Findings

Laplacian coefficients of paths, cycles, stars, wheels, and regular graphs are approximately normal.

Laplacian coefficients of complete and bipartite graphs are approximately Poisson.

Normality holds when the number of edges is large relative to maximum degree.

Abstract

Let $G$ be a simple graph with $n$ vertices and let $$C(G;x)=\sum_{k=0}^n(-1)^{n-k}c(G,k)x^k$$ denote the Laplacian characteristic polynomial of $G$. Then if the size $|E(G)|$ is large compared to the maximum degree $\Delta(G)$, Laplacian coefficients $c(G,k)$ are approximately normally distributed (by central and local limit theorems). We show that Laplacian coefficients of the paths, the cycles, the stars, the wheels and regular graphs of degree $d$ are approximately normally distributed respectively. We also point out that Laplacian coefficients of the complete graphs and the complete bipartite graphs are approximately Poisson distributed respectively.