Eigenvalue distribution of large weighted bipartite random graphs

Abstract

We study eigenvalue distribution of the adjacency matrix $A^{(N,p, \alpha)}$ of weighted random bipartite graphs $\Gamma= \Gamma_{N,p}$. We assume that the graphs have $N$ vertices, the ratio of parts is $\frac{\alpha}{1-\alpha}$ and the average number of edges attached to one vertex is $\alpha\cdot p$ or $(1-\alpha)\cdot p$. To each edge of the graph $e_{ij}$ we assign a weight given by a random variable $a_{ij}$ with all moments finite. We consider the moments of normalized eigenvalue counting measure $\sigma_{N,p, \alpha}$ of $A^{(N,p, \alpha)}$. The weak convergence in probability of normalized eigenvalue counting measures is proved.