An -route of a graph is a sequence of vertices of such that for , 1, ..., (where is the edge set of ) and
for , 2, ..., .
If a graph
contains an -route
with , then is said to be -transitive, -arc-transitive, or arc-transitive of order if the automorphism group of acts transitively on all -routes.
Note that some authors present other letters to , for example (Harary 1994) and .
Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 200, 1994.Holton,
D. A. and Sheehan, J. The
Petersen Graph. Cambridge, England: Cambridge University Press, pp. 202-210,
1993.
-route of a graph 
is a sequence of vertices 
of 
such that 
for 
, 1, ..., 
(where 
is the edge set of 
) and 
for 
, 2, ..., 
.
contains an 
-route
with 
, then 
is said to be 
-transitive, 
-arc-transitive, or arc-transitive of order 
if the automorphism group of 
acts transitively on all 
-routes.
, for example 
(Harary 1994) and 
.
