# Strongly asymmetric graphs

Asymmetric graphs are graphs that have a trivial automorphism group $\textrm{Aut}(G)$, i.e. the only graph isomorphism from $G$ to itself is the identity.

Let's call a graph $G$ strongly asymmetric if the only graph homomorphism $h: G\to G$ is the identity (in other words, the endomorphism monoid $\textrm{End}(G)$ is trivial).

Given a (finite or infinite) cardinal $\kappa > 0$, is there a strongly asymmetric graph on $\kappa$ vertices?

• – Chris Godsil Nov 25 '14 at 13:44