  Regarding the adjacency matrices of n-vertex graphs and related graph
Laplacian, we introduce two families of discrete matrix models constructed both
with the help of the Erdos-Renyi ensemble of random graphs. Corresponding
matrix sums represent the characteristic functions of the average number of
walks and closed walks over the random graph. These sums can be considered as
discrete analogs of the matrix integrals of random matrix theory.
  We study the diagram structure of the cumulant expansions of logarithms of
these matrix sums and analyze the limiting expressions in the cases of constant
and vanishing edge probabilities as n tends to infinity.
