Graphs and Degree Equitability

Copyright © 2013 Ahmad N. Al-kenani et al. 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 Let be a graph. If  , G V E    is a function from the vertex set   V G to the set of positive integers. Then two vertices are   G , u v V  -equitable if     1 u v    . By the degree, equitable adjacency between vertices can be redefine almost all of the variants of the graphs. In this paper we study the degree equitability of the graph by defining equitable connectivity, equitable regularity, equitable connected graph and equitable complete graph. Some new families of graphs and some interesting results are obtained.


Introduction
All the graphs considered here are finite and undirected with no loops and multiple edges.As usual p V  and q E  denote the number of vertices and edges of a graph , respectively.In general, we use G X to denote the subgraph induced by the set of vertices X and and denote the open and closed neighbourhoods of a vertex , respectively.The degree of the vertex in is denoted by or . For graph theoretic terminology, we refer to Harary [1].The degree equitable domination has been studied in [2].A subset of V G is called an equitable dominating set of a graph if for every , there exists a vertex such that uv and The minimum cardinality of such a dominating set is denoted by e  and is called the equitable domination number of .The set is minimal if for any vertex , is not an equitable dominating set of .The equitable neighbourhood of denoted by is defined as then is in the equitable dominating set.Such vertices are called equitable isolated vertices.u Let be a simple graph with n vertices .Then its adjacency matrix  matrix whose entries are given by ij a 1, if and are adjacent; 0, otherwise.
In the same way, the degree equitable adjacency matrix denoted by e 1, if and are equitable adjacent ; 0, otherwise.
where The equitable adjacency between any two vertices in V is defined as follows: the vertex v is equitable adjacent to if and only if is adjacent to Degree equitable adjacency has interesting applications in the context of social networks.In a network, nodes with nearly equal capacity may interact with each other in a better way.In society, persons with nearly equal status, tend to be friendly.In industry, employees with nearly equal powers form associations and move closely.Equitability among citizens in terms of wealth, health, status, etc is the goal of a democratic nation.These ideas motivated us in this paper to study the degree equitability of a graph by defining and studying some basic properties of degree equitable connectivity, degree equitable regularity, and degree equitable completeness of a graph.Some new families of graphs and some interesting results are obtained.In this paper for brevity we use equitable instead of degree equitable.

Elementary Results
Let be a graph.An equitable walk is defined as a finite alternating sequence of vertices and equitable edges, beginning and ending with vertices, such that each equitable edge is incident with the vertices preceding and following it.No equitable edge appears (is covered or traversed) more than once in the equitable walk.A vertex, however, may appear more than once.An equitable walk which begin and end at the same vertex called closed eq- uitable walk.An equitable walk is not closed if the terminal vertices are distinct.An equitable walk in which no vertex appears more than once is called an eq- uitable path.The number of edges in an equitable path is called the length of the equitable path.A closed equitable walk in which no vertex (except the initial and the final vertex) appears more than once is called an equitable circuit.
Now, we prove some results representing the relations between the sum of the equitable degree of the vertices, the number of edges and the number of equitable edges.
Theorem 2.1.For any graph with and q edges, .
  and we have: For any graph , the number of equitable edge denoted by e is called the equitable size.The vertex is odd number (even number).Theorem 2.2.The sum of the equitable degrees of a graph is twice the number of equitable edges in it, that is Proof.Let be a graph.Then any equitable edge contributes to the equitable degrees of two distinct vertices.Thus when the equitable degrees of the vertices are added, each equitable edge is counted exactly two times.Thus the sum of the equitable degrees is twice the equita-G ble size of the graph, that is .
  Every graph has an even number of equitable odd vertices.
Proof.Suppose that the sum of the equitable degrees of the equitable odd vertices is x and the sum of the equitable degrees of the equitable even vertices is .The number is even, and the number Hence x is even.If there are equitable odd vertices, the even number t x is the sum of t odd numbers.So t even. is We can define the equitable complete graph as a connected graph which all its edges are equitable edges.and analogous to the equitable complete graph we can defined the equitable complement graph of a graph as following: . Then

Equitable Connectivity and Equitable Regularity
A graph is said to be equitable connected if there is at least one equitable path between every pair of vertices in .Otherwise, G is equitable disconnected.It is easy to see that an equitable disconnected graph consists of two or more equitable connected graphs.Each of these equitable connected subgraphs is called equitable com- ponent.Clearly any equitable connected graph is connected but the converse is not true equitable in general.For example, the graph 1,m G G K , where is connected but not equitable connected.Proof.Proof.Suppose that we have the partition of V G into disjoint subsets 1 and 2 V such that there exists no equitable edge in whose one end vertex is in subset 1 and the other in subset .Consider two arbitrary vertices and of Theorem 3.3.If a graph (equitable connected or equitable disconnected) has exactly two vertices of odd equitable degree, then there exists an equitable path joining these two vertices.
Proof.Let be a graph with all even vertices except vertices 1 and 2 , which are odd.From Theorem 2.3, no graph can have an odd number of equitable odd vertices.Therefore, in graph , 1 and 2 v must belong to the same equitable component, and hence must have equitable path between them.

G v v G v
Copyright © 2013 SciRes.AM Definition 3.4.Let be a graph on vertices.An equitable disconnecting set of edges is a subset such that G is equitable disconnected.The edge equitable connectivity, e , is the smallest number of edges in any equitable disconnecting set.
We adopt the convention that .Thus if and only if is a graph, then .That is, the edge equitable connectivity of can be no larger than the minimum equitable degree of .
let G be an equitable connected graph on vertices and suppose is a vertex of equitable degree The following result is straightforward.
, is the smallest number of vertices in any equitable vertex cut of .A vertex whose removal increases the number of equitable components of is called equitable cut-vertex (or point of equitable articulation).For the graph in Fig- ure 1, but The maximal equitable connected subgraph of that has no equitable cut-vertex is called an equitable block of .
, then is equitable disconnected and .If , then this graph is equitable connected with equitable bridge x , that means 2 G K  or one of the vertices which incident with x is equitable cut vertex.Therefore,

An Equitable Line Graph and an
Equitable Total Graph .Remark.The equitable line graph of equitable connected graph is connected but not equitable connected in general.In Figure 3, the equitable line graph of equitable connected graph is connected but not equitable connected.
The vertex set of e corresponds to the vertices and equitable edges of and .
If v is a vertex and x uw  be equitable edge in a graph , then the degree of the vertex is 2) e if and only if is an equitable complete graph.
G 3) If 1 2 , then but the converse is not true.

E q
 whose vertices have equitable degree  

 
L G e q  q G Theorem 4.8.A connected graph is isomorphic to its equitable line graph if and only if it is a cycle.e T e e i q q q d     .

1 .
The isomorphism between the graphs preserve the number of equitable component.

Figure 1 .
Figure 1.Equitable cut edge but not cut edge.
not, x is equitable bridge of this subgraph and hence the removal of or v results in an equitable disconnected.Hence, 10.Every -regular graph is -equitable regular graph but the converse is not true, in general.kExample.The graph in the Figure2is 2-equitable regular graph but not regular.

Definition 4. 1 .
Given a graph , its equitable line graph G and only if their corresponding equitable edges share a common endpoint (are adjacent) in .Proposition 4.2.The line graph of equitable connected graph is connected.Proof.If is equitable connected, it contains equitable path connecting any two of its edges, which translates into a path in G

T G 2 )
Two vertices are adjacent in e if and only if their corresponding elements are either adjacent or incident in .T G The following result is immediate.Proposition 4.5.For any graph with at least one equitable edge, the following hold.GObservation 4.10.For any graph the equitable total graph is a subgraph of the total graph of a graph .
have the same number of vertices that means .Hence is an equitable complete graph.
If is a graph with vertices andedges and its vertices have equitable degree T .The number of vertices of the graph   e T G is the sum of number of equitable edge and number of verthe sum of edges in G and the number of edges in the equitable line graph of G and twice the number equitable edges in , that is; then removal edges results in equitable disconnected graph, that means the removal