New Modularity Bounds for Graphs G(n, r, s) and Gp (n, r, s)

We analyze the behavior of the modularity of G(n, r, s) graphs in the case of r = o(root n) and n -> infinity and also that of G(p) (n, r, s) graphs for fixed r and s as n -> infinity. For G(n, r, s) graphs with r >= cs(2), we obtain substantial improvements of previously known upper bounds. Upper and lower bounds previously obtained for G(n, r, s) graphs are extended to the family of G(p)(n, r, s) graphs with p = p(n) = omega(n(-r-s-1/2)) and fixed r and s.