Weakly s-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 <= d. (3) A bipartite graph with girth g = 2s is always weakly t-arc transitive for all 0 <= t <= s, but not weakly (s + 2)-arc transitive. Further, a bipartite graph with girth g = 2s is weakly (s + 1)-arc transitive if and only if the graph has diameters. (c) 2007 Elsevier B.V. All rights reserved.