Some problems concerning distance and path degree sequences

Abstract

Let G denote a graph with point set {v ₁, v ₂, ..., v _|G|} d _ij the number of points in G that are at distance j from v _i . Then, the sequence (d _i0, d _i1, d _i2, ..., 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 (p _i0, p _i1, p _i2, ..., 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.

A substantial amount of information about a graph is contained in these sequences and there is considerable interest in them due to their applications in chemistry and operations research. References to these applications are made and open problems concerning the existence and extremal properties of graphs having certain types of Distance and Path Degree Sequences are presented together with some partial results on these problems.