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)(v,u,x,y) of vertices such that both (v,u,x)(v,u,x) and (u,x,y)(u,x,y) are paths of length two. The 3-arc graph of a given graph GG, X(G)X(G), is defined to have vertices the arcs of GG. Two arcs uv,xyuv,xy are adjacent in X(G)X(G) if and only if (v,u,x,y)(v,u,x,y) is a 3-arc of GG. 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)X(G) in terms of the corresponding invariant of GG.