On Graphs With Same Distance Distribution

In the present paper we investigate the relationship between Wiener number W , hyper-Wiener number WW , Wiener vectors WV , hyper-Wiener vectors HWV , Wiener polynomial H , hyper-Wiener polynomial HH and distance distribution DD。It is showed that, for connected graphs G and G~* , the following conclusions are equivalent :( 1 ) DD(G)=DD(G~*)( 2 ) WV(G)=WV(G~*)( 3 ) HWV(G) = HWV(G~*)( 4 ) H(G) = H(G~*)( 5 ) HH(G) =HH(G~*)。and if G and G~* have same distance distribution DD , then they have same W and WW。Therefore , we consider the graphs with same distance distribution。Several construction methods for finding a class of graphs with same distance distribution are given。Furthermore , it is proved that, for all connected graphs with n vertices and m edges , if (?) < m < (?) , then they have the same distance distribution。We also give the smallest graphs with same distance distribution。...