Diameter and connectivity of 3-arc graphs

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple ( v , u , x , y ) of vertices such that both ( v , u , x ) and ( u , x , y ) are paths of length two. The 3-arc graph of a given graph G , X ( G ) , is defined to have vertices the arcs of G . Two arcs u v , x y are adjacent in X ( G ) if and only if ( v , u , x , y ) is a 3-arc of G . This notion was introduced in recent studies of arc-transitive graphs. In this paper we study diameter and connectivity of 3-arc graphs. In particular, we obtain sharp bounds for the diameter and connectivity of X ( G ) in terms of the corresponding invariant of G .