一个有向图d的有向P_k-路图P_k(d)是通过把d中的所有有向k长路作为点集;两点u=X_1X_2…X_(k+1),V=y_1y_2…y_(k+1)之间有弧uV当X_I=y_(I-1),I=2,3,…,k+1.明显地,当k=1时P_k(d)就是通常的有向线图l(d).在[1,2]中,P_2-路图得到完整刻画。在[3]中,brOErSMA等人研究了有向P_2-路图的一些性质,特别是在相似性和传递性方面。在他们的文章中,描述了所有与自身的有向P_2-路图同构的有向图d,证明了对于任意的有向图d_1和d_2,若P_2(d_1)(？)P_2(d_2),“几乎总是“暗示d_1(？)d_2,并描述了所有这样的有向图:其有向P_2-路图是欧拉图或哈密顿图。另外,对于任意一个不包含有向2长圈且至少包含一个有向2长路的有向图d,有P_2(d)(？)l--2(d).在这篇文章中我们刻画了有向P_2-路图。同时,我们考虑了有向P_2-路图的直径问题,并对于正则有向图,给出了其有向P_2-路图的独立数的一个上界。The directed P_k-path graph P_k(D) of a digraph D is obtained by representing the directed paths of length k of D by vertices.Two vertices u = x_1x_2…x_(k+1),v=y_1y_2…y_(k+1) are joined by arc uv if x_i=y_(i-1),i=2,3,…,k+1.Clearly,the directed path graph P_k(D) is the line digraph L(D) when k=1.In[1,2],the P_2-path graph is characterized.In[3],several properties of P_2(D) were studied,in particular with respect to isomorphism and traversability.Broersma et al.characterized all digraphs D with P_2(D)≌D,show that P_2(D_1)≌P_2(D_2) "almost always" implies D_1≌D_2.and characterized all digraphs D with P_2(D) is Eulerian or Hamiltonian.Furthermore,for any digraph D with at least one directed path of length 2 and no directed cycle of length 2,P_2(D)≌L--2(D).In this note,we obtain a characterization of P_2(D),and give the upper bound and the lower bound of the diameter of the directed P_2-path graph of a strongly connected digraph,and also a upper bound of the independence number of the directed P_2-path graph of a regular digraph.supportedbyNSFC(10831001)
