On the number of graphs with a given endomorphism monoid

For a given finite monoid M , let ς M ( n ) be the number of graphs on n vertices with endomorphism monoid isomorphic to M . For any nontrivial monoid M we prove that 2 1 2 ( n 2 − ( 1 + o ( 1 ) ) d ( M ) n ) ≤ ς M ( n ) ≤ 2 1 2 ( n 2 − ( 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 | 3 2 . ery k there exists a monoid M of size k with d ( M ) ≤ 3 , on the other hand if a group of unity of M has a size k > 2 then c ( M ) ≥ log k log log k + 1 .