Distance Degree Regular Graphs and Distance Degree Injective Graphs: An Overview

Abstract

The distance d(v,u) from a vertex v of G to a vertex u is the length of shortest v to u path. The eccentricity ev of v is the distance to a farthest vertex from v. If d(v,u) = e(v), (u ≠ v), we say that u is an eccentric vertex of v. The radius rad(G) is the minimum eccentricity of the vertices, whereas the diameter diam(G) is the maximum eccentricity. A vertex v is a central vertex if e(v) = rad(G), and a vertex is a peripheral vertex if e(v)  = diam(G). A graph is self-centered if every vertex has the same eccentricity; that is, rad(G) = diam(G). The distance degree sequence (dds) of a vertex v in a graph G = (V, E) is a list of the number of vertices at distance 1, 2, ... . , e(v) in that order, where e(v) denotes the eccentricity of v in G. Thus, the sequence (di0,di1,di2,  …,  dij,…) is the distance degree sequence of the vertex vi in G where dij denotes the number of vertices at distance j from vi. The concept of distance degree regular (DDR) graphs was introduced by Bloom et al., as the graphs for which all vertices have the same distance degree sequence. By definition, a DDR graph must be a regular graph, but a regular graph may not be DDR. A graph is distance degree injective (DDI) graph if no two vertices have the same distance degree sequence. DDI graphs are highly irregular, in comparison with the DDR graphs. In this paper we present an exhaustive review of the two concepts of DDR and DDI graphs. The paper starts with an insight into all distance related sequences and their applications. All the related open problems are listed.