The term "asymmetric graph" is used with more than one meaning. In its principal meaning, an asymmetric graph is a graph having no nonidentity graph automorphism. Equivalently, the automorphism group of an asymmetric graph is the trivial group (Erdős and Rényi 1963). An asymmetric graph in this sense is also called an identity graph.
In a less-standard usage obtained by negating the Harary-Palmer convention for symmetric graph, an asymmetric graph is one that
is not both edge-transitive and vertex-transitive.
For example, the path graph 
satisfies this definition but is not an identity
graph, since it has two graph automorphisms.
In directed graph terminology, an asymmetric digraph contains no pair of oppositely directed arcs. Equivalently,
if 
is an arc, then 
is not an arc (Alspach et
al. 1974). This is distinct from the two notions for undirected
graphs described above.
