The Study Of Weakly Arc-transitive Graphs

In this paper, weakly arc-transitive graphs are studied. A graph is called weakly s-arc transitive if the endomorphism monoid acts transitively on the s-arc(s>1). On the basis of weakly vertex-transitive graphs and weakly edge-transitive graphs, some properties of weakly 1-arc transitive graphs are obtained and all weakly 1-arc transitive graphs whose order is less than 7 are given; weakly 1/2-transitive graphs are also studied. A minimum weakly 1/2-transitive graph is obtained and the relationship of weakly edge-transitive graphs and weakly 1-arc transitive graphs is discussed. Moreover, weakly s-arc transitive graphs are studied. For the non-bipartite graphs, weakly s-arc transitivity is equivalent to s-arc transitivity; for the bipartite graphs, weakly s-arc transitivity depends on girth g and diameter d.