Weakly ss-arc transitive graphs

Weakly s-arc transitive graphs are introduced and determined. A graph is said to be weakly s-arc transitive if its endomorphism monoid acts transitively on the set of s-arcs. The main results are: (1) A nonbipartite graph is weakly s-arc transitive if and only if it is s-arc transitive. (2) A tree with diameter d is weakly s  -arc transitive for all 0⩽s⩽d0⩽s⩽d. (3) A bipartite graph with girth g=2sg=2s is always weakly t  -arc transitive for all 0⩽t⩽s0⩽t⩽s, but not weakly (s+2)(s+2)-arc transitive. Further, a bipartite graph with girth g=2sg=2s is weakly (s+1)(s+1)-arc transitive if and only if the graph has diameter s.