Journal of Discrete Mathematics

Volume 2014, Article ID 358792, 12 pages

http://dx.doi.org/10.1155/2014/358792

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

Department of Mathematics, Bangalore University, Central College Campus, Bangalore 560001, India

Received 29 June 2014; Revised 25 October 2014; Accepted 28 October 2014; Published 8 December 2014

Academic Editor: Luisa Gargano

Copyright © 2014 Medha Itagi Huilgol. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The distance from a vertex of to a vertex is the length of shortest to path. The *eccentricity * of is the distance to a farthest vertex from . If , we say that is an *eccentric vertex* of . The *radius * is the *minimum eccentricity* of the vertices, whereas the *diameter * is the *maximum eccentricity*. A vertex is a *central vertex* if , and a vertex is a *peripheral vertex* if . A graph is *self-centered* if every vertex has the same eccentricity; that is, . The *distance degree sequence (dds)* of a vertex in a graph is a list of the number of vertices at distance in that order, where denotes the eccentricity of in . Thus, the sequence is the distance degree sequence of the vertex in where denotes the number of vertices at distance from . 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.