On the number of graphs with a given endomorphism monoid

For a given finite monoid M, let ςM(n) be the number of graphs on nn vertices with endomorphism monoid isomorphic to M. For any nontrivial monoid M we prove that 212(n2−(1+o(1))d(M)n)≤ςM(n)≤212(n2−(1+o(1))c(M)n) where c(M) and d(M) are constants depending only on M with 1.83≤c(M)≤d(M)≤3|M|+1+8|M|32.For every kk there exists a monoid M of size kk with d(M)≤3, on the other hand if a group of unity of M has a size k>2k>2 then c(M)≥logkloglogk+1.