Graphs Generated by Measures

Copyright © 2016 A. Assari and M. Rahimi. 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. In this paper, a graph is assigned to any probability measure on the σ-algebra of Borel sets of a topological space. Using this construction, it is proved that given any number n (finite or infinite) there exists a nonregular graph such that its clique, chromatic, and dominating number equals n.


Introduction
The distance between two vertices in a graph is the number of edges in a shortest path connecting them.Diameter of a graph is the longest path between two vertices of the graph.
A clique is a subset of vertices of an undirected graph, such that its induced subgraph is complete; that is, every two distinct vertices in the clique are adjacent.Clique number of a graph Γ is the number of vertices of maximum clique in the graph and denoted by (Γ).The chromatic polynomial counts the number of ways a graph can be colored using no more than a given number of colors and denoted by (Γ).
A set  ⊆  of vertices in a graph Γ = (, ) is a dominating set if every vertex V ∈  is an element of  or adjacent to an element of .The domination number (Γ) of a graph Γ is the minimum cardinality of a dominating set of Γ.
An independent set or stable set is a set of vertices in a graph, with no two of which being adjacent.A maximum independent set is an independent set of largest possible size for a given graph Γ.This size is called the independence number of Γ and denoted (Γ).
Lots of graphs are constructed from algebraic objects such as semigroups ( [1]), groups ( [2,3]), and rings ( [4,5]).Recently some graphs have been constructed from Hilbert and topological spaces ( [6,7]) which have nice graph theoretical properties.This motivates us to construct a bigger category of graphs which has more properties.In this paper, we introduce a class of graphs generated by measures.Using this construction, we conclude that, given any number  (finite or infinite), there exists a nonregular graph such that its clique, chromatic, and dominating number coincides and is equal to .
Given a measure space (, Ω, ), a set  ∈ Ω is called an atom for , if  has a positive measure and for every  ∈ Ω, either ( ∩ ) or ( \ ) is zero.It is easily obtained by definition that if  is an atom for  and ( ∩ ) > 0, then  ∩  is also an atom for . is an atomic measure iff every measurable set of positive measure contains an atom.We may say that  is nonatomic if there are no atoms for .Therefore, in a nonatomic measure , every nonatomic measurable set of positive measure can split into two disjoint measurable sets, where both of them have positive measure.One can easily see that the zero measure is the only measure which is atomic as well as nonatomic measure.
Let (, Ω) be a measurable space and  and ] be two measures on it.We say that ] is absolutely continuous with respect to , denoted by ] ≪  iff, for every measurable set , ]() = 0 whenever () = 0.
We say that  is singular with respect to ] and denoted by  ⊥ ] if, given any  ∈ Ω, there exists some  ⊆ , such that ]() = ]() and () = 0.One can easily see that in this case ]() = sup{]( ∩ ) : () = 0}.It is easily seen that the following lemma holds.Johnson in [8] proved the following useful results about atomic and nonatomic measures.
Result 1.If  is an atom for  and ] ≪ , then either ]() = 0 or  is an atom for ].

Definitions and Examples
Let  be a topological space and B() be the -algebra of Borel subsets of .Let also  be a probability measure on B().Set In other words, C() is the collection of all absolutely continuous Borel measures with respect to .Define the following relation on C(): in other words, ] 1 ∼ ] 2 if and only if for any measurable set  we have Clearly, ∼ is an equivalence relation on C().Denoting the equivalence class of ] by []], let For simplicity, we write ] for []].

The Graphs of Finitely Atomic Measures
In the rest of the paper, by an atomic measure, we mean a finitely atomic measure with finite number of atoms.In this section, we study the graph corresponding to atomic measures and state some of its properties.The case of atomic measures is of special interest since the corresponding graph is finite and may be visualized and enumerated.
The following theorem satisfies for atomic measures.
Theorem 6.For an atomic measure  = ∑  =1      one has the following.
To prove part (2), let V ∈   and then V = ∑ which completes the proof.
(2) Since each vertex of Γ() is adjacent to at least an element of the set  1 = {  1 ,   2 , . . .,    } and all vertices in  1 are adjacent, then (, V) ≤ 3 for all , V ∈ ().On the other hand, set  1 fl   1  2 ⋅⋅⋅ −1 and  2 fl   2  3 ⋅⋅⋅  .Clearly, the path The following theorem determines the automorphism group of Γ(), where  is an atomic measure.Theorem 8.For any atomic measure  = ∑  =1      one has where   is the group of permutations on .
Proof.Let  ∈   .Define the mapping by Clearly  * is a bijection; so  * ∈ Aut(Γ()).Now, define by It is easily seen that Φ is an injective group homomorphism.
To complete the proof, it is enough to show that Φ is surjective.
Let  ∈   be defined by () = .So We show that  =  * .Let    1 ⋅⋅⋅   be any given element of ().Let also We should show that In the next theorem, we consider the domination number as well.
Theorem 13.If the nonatomic part of  is nonzero then the domination number of Γ = Γ() cannot be finite.
Proof.Let  =  1 +  2 , where  1 is the atomic and  2 is the nonatomic part of .Suppose that D = {V 1 , . . ., V  } is a dominating set of Γ.
For every 1 ⩽  ⩽ , we can choose a nonzero measurable set   with V  (  ) = 0 and   ∩  = 0. Let  = (∪  =1   ) ∪ .Now define the measure  by () = ( ∩ ) which is a measure different from all of V  's and adjacent to none of them.A contradiction with the hypothesis is that D is a dominating set.Therefore, every dominating set is infinite.Theorem 14.If the nonatomic part of  is nonzero, then (Γ) = ∞.
Proof.By the hypothesis of the theorem, there is some measurable set  of positive measure which does not contain any atom and hence there is a sequence of measurable sets such that  () =  ( 1 ) >  ( 2 ) >  ( 3 ) > ⋅ ⋅ ⋅ > 0. (37) Now define the sequence of measures ]  by ]  () = (  ∩ ) for  ∈ N. By definition of ]  's we have ]  ≪ , ( ∈ N).Now suppose that for some  <  we have ]  ⊥ ]  .Thus for   , there is some  ⊆   , where ]  (  ) = ]  () and ]  () = 0.But by definition ]  ≪ ]  which is a contradiction.Therefore, we conclude that the set {]  :  ∈ N} is an independent set of Γ, which is not finite and the theorem is proved.
Applying the material presented in this paper, we will have the following corollary.(38)
Also, each    is adjacent to    for  ̸ = .     is adjacent to    if ,  ̸ = .No other vertices are adjacent.Example 5. Let  = ∑  =1      be a finitely atomic measure with finite number of atoms where   's are distinct points in  and By Lemma 1, two vertices ] 1 =    1   2 ⋅⋅⋅   and ] 2 =    1   2 ⋅⋅⋅   are adjacent if and only if    ̸ =    for all 1 ≤  ≤  and 1 ≤  ≤ .