Asymptotic normality of Laplacian coefficients of graphs

Abstract Let G be a simple graph with n vertices and let C ( G ; x ) = ∑ 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 Δ ( 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.