Journal of Discrete Mathematics

Volume 2014 (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.

#### 1. Introduction

The study of sequences in Graph Theory is not new. A* sequence* for a graph acts as an invariant that contains a list of numbers rather than a single number. A sequence can be handled and studied as easily as a single numerical invariant, but a sequence carries more information about the graph it represents. There are many sequences representing a graph in literature, namely, the degree sequence, the eccentric sequence, the distance degree sequence, the status sequence, the path degree sequence, and so forth. A sequence* S* is said to be* graphical* if there is a graph which realizes* S*.* Degree sequences* of graphs were the first ones to be studied, as the question of realizability of any sequence for a graph was a fundamental one. An existential characterization was given by Erdos and Gallai [1]. Then the constructive characterization was found independently by Havel [2] and later by Hakimi [3] that is now referred to as the Havel and Hakimi algorithm. The* eccentric sequences* were the next and the first in the class of distance related sequences to be studied for undirected graphs. Some fundamental results in this direction are due to Lesniak-Foster [4], Ostrand [5], Behzad, and Simpson [6], which primarily deal with the conditions for graphical eccentric sequences. The minimal eccentric sequences were mainly studied by Nandakumar [7]. But, for digraphs, the eccentric sequences were studied quite late. We can find some papers in 2008 due to Gimbert and Lopez [8].

Next distance based sequences were the* path degree sequences* and* distance degree sequences.* These were studied by Randic [9] for the purpose of distinguishing chemical isomers by their graph structure. Path degree sequence of a graph has its application in describing atomic environments and in various classification schemes for molecules.

We will now define all the terms that give graph theoretic expressions for the above discussed cases.

For all undefined terms we refer to [10].

Let denote a graph with the set of vertices , whose cardinality is the* order * and two element subsets of , known as* edges* forming , whose cardinality is* size *. Unless mentioned otherwise, in this article, by a graph we mean an undirected, finite graph without multiple edges and self-loops.

The distance from a vertex of to a vertex is the length of shortest to path. The* degree* of a vertex is the number of vertices at distance one.

The sequence of numbers of vertices having degree is called the* degree sequence*, which is the list of the degrees of vertices of in nondecreasing order.

The* eccentricity * of is the distance to a farthest vertex from .

The minimum of the eccentricities is the and the maximum* diameter* of .

A graph is said to be* self-centered* if all vertices have the same eccentricity.

If , we say that is an* eccentric vertex *of .

The* eccentric sequence* of a connected graph is a list of the eccentricities of its vertices in nondecreasing order.

The distance and path degree sequences of a vertex are generalizations of the degree of a vertex. 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 -tuple of distance degree sequences of the vertices of with entries arranged in lexicographic order is the distance degree sequence (DDS) of . Similarly, we define the* path degree sequence* (pds) of as the sequence where denotes the number of paths in of length having as the initial vertex. The ordered set of all such sequences arranged in lexicographic order is called the path degree sequence (PDS) of . Clearly, for any graph .

The* distance distribution* of is the sequence , where is the number of pairs of vertices in that are at distance apart.

The* status * of a vertex in is the sum of the distances from to all other vertices in . This concept was introduced by Harary [11]. Thus, using the distance degree sequence of , we have .

The* median * of is the set of vertices with* minimum status*. A graph is called a* self-median* graph if all of its vertices have the same status.

The status of a vertex is also called the* distance* at that vertex. The* mean distance* at the vertex , denoted by , is defined as .

The* mean distance* for the graph, denoted by , is defined as the mean of the distances between all pairs of vertices of ; that is, , where and are vertices of . In terms of status, we get .

As an illustration consider the graph as shown in Figure 1.