Title: Half-arc-transitive graphs with small number of alternets

Abstract: A graph is said to be half-arc-transitive if the automorphism group of the graph acts transitively on the vertex set and the edge set of the graph, and not transitively on the set of all arcs of the graph. In this talk we will describe the notion of the alternets in the half-arc-transitive graphs, and give some results in the case when the number of alternets is small.