图的拉普拉斯系数的渐近分布

Asymptotic Distributions of Laplacian Coefficients of Graphs

设G是n个顶点的简单图,它的拉普拉斯矩阵为L(G)=D(G)-A(G),其中D(G)和A(G)分别是图G的度对角矩阵和邻接矩阵.设图G的拉普拉斯特征多项式为C(G;x)=det(xI-L(G))=∑_(k=0)^(n)(-1)^(n-k)c(G,k)x^(k),其中c(G,k)称为图G的拉普拉斯系数.研究了树和单圈图的拉普拉斯系数的渐近正态分布问题,明确了树和单圈图的拉普拉斯系数是渐近正态分布的,同时,当与拉普拉斯系数相关的均值μ_(n)趋于有限值时,利用拉普拉斯系数的发生函数,明确了完全图的拉普拉斯系数是渐近泊松分布的.此外,也明确了树,单圈图以及完全图的无符号拉普拉斯系数的渐近分布特性.

Let G be a simple graph with n vertices,its Laplacian matrix is defined as L(G)=D(G)-A(G),where D(G)is the degree diagonal matrix and A(G)is the adjacency matrix of the graph.The Laplacian characteristic polynomial of G is denoted by C(G;x)=det(xI-L(G))=∑_(k=0)^(n)(-1)^(n-k)c(G,k)x^(k),where c(G,k)are called Laplacian coefficients of the graph G.In this paper,we show that the Laplacian coeficients of trees and unicyclic graphs are approximately normally distributed respectively.When the mean μ_(n) associated to Laplacian coefficients tends a finite limit,by the generating function of Laplacian coefficients,we demonstrate Laplacian coefficients of complete graphs are approximately Possion distributed.We also get similar results about signless Laplacian coefficients of trees and unicyclic graphs and complete graphs.