Some problems concerning distance and path degree sequences

Let G denote a graph with point set {v1, v2, ..., v|G|} d ij the number of points in G that are at distance j from v i . Then, the sequence (di0, di1, di2, ..., d ij , ...) is called the distance degree sequence of v i in G. The |G|-tuple of distance degree sequences of the points of G with entries arranged in lexicographic order is the Distance Degree Sequence of G. Similarly, we define the path degree sequence of v i in G as the sequence (pi0, pi1, pi2, ..., p ij , ...) where p ij is the number of paths in G with initial point v i and which have length j. The ordered set of all such sequences arranged in lexicographic order is called the Path Degree Sequence of G.