THE MAXIMUM AND MINIMUM DEGREES OF RANDOM BIPARTITE MULTIGRAPHS  

作　　者：陈爱莲 张福基 李皓 

机构地区：[1]College of Mathematics and Computer Science Fuzhou University [2]School of Mathematical Sciences Xiamen University [3]Laboratoire de Recherche en Informatique UMR 8623 C. N. R. S.-Université de Paris-sud

出　　处：《Acta Mathematica Scientia》2011年第3期1155-1166,共12页数学物理学报（B辑英文版）

基　　金：supported by NSFC (10671162;10831001;10871046)

摘　　要：In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows：let m = m（n） be a positive integer-valued function on n and ζ（n,m;{pk}） the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A ={a_1,a_2,...,a_n} and B = {b_1,b_2,...,b_m}, in which the numbers t_（ai）,b_j of the edges between any two vertices a_i∈A and b_j∈ B are identically distributed independent random variables with distribution P{t_（ai）,b_j=k}=pk,k=0,1,2,...,where pk ≥0 and ∞Σk=0 pk=1. They obtain that X_（c,d,A）, the number of vertices in A with degree between c and d of G_（n,m）∈ζ（n, m;{pk}） has asymptotically Poisson distribution, and answer the following two questions about the space ζ（n,m;{pk}） with {pk} having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {pk} can there be a function D（n） such that almost every random multigraph G_（n,m）∈ζ（n,m;{pk}） has maximum degree D（n）in A？ under which condition for {pk} has almost every multigraph G（n,m）∈ζ（n,m;{pk}） a unique vertex of maximum degree in A？In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows：let m = m（n） be a positive integer-valued function on n and ζ（n,m;{pk}） the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A ={a_1,a_2,...,a_n} and B = {b_1,b_2,...,b_m}, in which the numbers t_（ai）,b_j of the edges between any two vertices a_i∈A and b_j∈ B are identically distributed independent random variables with distribution P{t_（ai）,b_j=k}=pk,k=0,1,2,...,where pk ≥0 and ∞Σk=0 pk=1. They obtain that X_（c,d,A）, the number of vertices in A with degree between c and d of G_（n,m）∈ζ（n, m;{pk}） has asymptotically Poisson distribution, and answer the following two questions about the space ζ（n,m;{pk}） with {pk} having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {pk} can there be a function D（n） such that almost every random multigraph G_（n,m）∈ζ（n,m;{pk}） has maximum degree D（n）in A？ under which condition for {pk} has almost every multigraph G（n,m）∈ζ（n,m;{pk}） a unique vertex of maximum degree in A？

关 键 词：maximum degree minimum degree degree distribution random bipartite multigraphs 

分 类 号：O157.5[理学—数学]