Asymptotic normality of Laplacian coefficients of graphs

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 $\mathrm{\mid }E(G)\mathrm{\mid }$ is large compared to the maximum degree $\mathrm{\Delta }(G)$, 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.