Construction of Measurable Incidence and Adjacency Matrices from Product Measures

This study presents a new method of representing graphs and a new approach of constructing both incidence and adjacency matrices using the theory of product measures. It further shows that matrices constructed by this approach are measurable which a major advantage of this method is.


INTRODUCTION
In Computer Science and Mathematics, a graph is defined as a mathematical structure that is normally used to model a pairwise relation between elements from some collection.But, in the most common sense of the word, a graph can be defined as an ordered pair G = (V, E) where the set V consists of the vertices or nodes while the set E is made up of the edges or lines which connect the nodes (Balakrishnan and Ranganathan, 2000).There are various types of graphs such as bipartite graphs, hyper-graphs, directed graphs (digraphs), multiline graphs, networks, line graphs, planar graphs, undirected graphs, vertex-transitive graphs, etc (Balakrishnan and Ranganathan, 2000;Chen and Hwang, 2000;Diestel, 2006;Li et al., 2011;Mandl, 1979;Schrijver, 2009;Strayer, 1992;Xu, 2003Xu, , 2001)); and it is clear that their construction resulted from everyday life problems and hence the need to research about them.
Graphs are normally represented in two forms namely; Adjacency matrix form and Incidence matrix form and it is in either of these two forms that graphs are commonly stored in computers (Xu, 2003).Therefore, all other types of matrices namely: Toeplitz matrices, Circulant matrices, Null matrices, Triangular matrices, Diagonal matrices, etc (Agaian, 1985;Bhatia, 2007;Brauer and Gentry, 1968;Fiedler, 1971) are either adjacency matrices or incidence matrices in origin depending on the constraints subjected to the graph in question.
In this study, we would represent a graph as G = ﴾ ( V , A, µ ) ﴿ , ( E , β, v ^ where V is the vertex set of G, A is the sigma-algebra of V and µ is a measure defined on A while E is the edge set of G, β is the sigma-algebra of E and v is a measure defined on β. The goal of this study is to show that the product measure spaces (V×E, A⊗β, µ×v) and V×V, A⊗A, µ× µ) represent the incidence and adjacency matrices respectively and that they are measurable (Berberian, 1999;Friedman, 1982;Hewitt and Stromberg, 1965;Lieb and Loss, 2001).
This study presents a new method of graph representation.It also proposes a new approach of constructing both incidence and adjacency matrices based on the principles of product measures; and demonstrate that both the proposed graph representation and the proposed construction algorithm possess the characteristic or feature of measurability, which is a major advantage of both methods.Finally, some areas where the research can be found applicable are discussed.

Crucial definitions and theorems:
In this section, we will define the important terms and prove the theorems that would be used in the construction of the said matrices from the theory of product measures.We start by fixing some notations.Let (V, A, µ) and (E, β, v) be measurable spaces.A subset Ψ⊆ V×E is called a measurable rectangle if Ψ = Ω ×ɸ for some Ω ϵ A and ɸ ∈β.We let R to denote the class of all measurable rectangles with the fact in mind that, it is not a σalgebra but a semi-algebra of the subsets of V×E.The sigma-algebra of the subsets of V×E generated by the semialgebra R is called the product sigma-algebra and is denoted by A⊗B.
Definition 1: Let ( X , A 1 µ 1 be a measure space and let a mapping ϕ exists such that: 0: X → R* where, R* denotes the set of extended real numbers. Then the mapping ϕ is said to be measurable if f ϕ -1 (I) ϵA 1 for all intervals Definition 2: Given a set a power set Ψ a power of Ψ (A) and µ a measure defined on A; the triple (Ψ, A, µ) is called a measure space.If µ (Ψ) <∞ then (Ψ, A, µ) is called a finite measure space otherwise it is a non-finite measure space.
Theorem 1: Given two measure spaces (X, A 1 , µ 1 ) and (Y , A 2 , µ 2 ) together with projection maps of the form: then the following statements hold:

Proof (i):
Let the projection maps ϕ X : X ×Y→ X (ϕ Y : X ×Y ) be defined as: ∀x < ∈E X and ∀y∈Y.We consider the map ϕ X and make the following analysis: We have seen from above that for any set A ∈ A 1 the inverse projection map • Here we also claim that ϕ y is A 1 ⊗A 2 measurable and consider the projection map: which is defined as: ϕ y (x, y) = y ∀ (x, y) ∈ X ×Y.Now we would show that ∀H ∈A 2 the inverse projection map ϕ y -1 (M) is also in sigma-algebra A 1 ⊗A 2 we would proceed as follows.Given any set M which is any element of the sigma-algebra A 2 then the inverse map ϕ y -1 (M) becomes: Theorem 2: Let (X, A 1 , µ 1 ) and (Y, A 2 , µ 2 ) be two measurable spaces such that theorem holds.Then the σalgebra A 1 ∈A 2 is a member of the a-algebra of the subsets of X×Y.
Proof: Let Z be any σ-algebra of the sub-sets of X ×Y such that ϕx and ϕy are both Z-measurable (Theorem 1).Our goal is to show that Z⊇A 1 ⊗ ˓ 2 and we proceed by supposing that the set A∈A 1 and the set B ∈A 2 , then from theorem 1 it is clear enough that the following equations: are true.Since A×B∈R, we can without fear evaluate the product A×B as follows: which implies that R⊆Z and hence the following result: For more clarity of Eq. ( 1) we let X and Y to be non-empty sets and supposed C and D to be families of subsets of X and Y respectively.Thus we can analyze as follows: and hence Eq. (1) becomes: Construction from product measures: In this section, we would describe the construction of both incidence and adjacency matrices using the concepts of product measures.
We let (V, A 1 , µ 1 ) and (E, A 2 , µ 2 ) to be spaces where V and E are sets, A 1 and A 2 are the sigmaalgebras of the sets V and E respectively and µ 1 and µ 2 are measures defined on A 1 and A 2 respectively.We want to show that the spaces (V×E, A 1 ⊗A 2 , µ 1 ×µ 2 ) and (V×V, A 1 ⊗A 2 , µ 1 ×µ 2 ) represent both incidence and adjacency matrices and they are measurable.For the construction to be meaningful, we only need to show that there exists measure Q such that: for every A∈A 1 , B∈A 2 .It is a clear fact that: where S is any sigma-algebra of the subsets of V×E and R denotes the semi-algebra of the subsets of V×E and hence the following equation is true: where, Step I: We let: defined by: For every A∈V and for every B∈E and show that Ω is a measure on R. We proceed as follows.It is obvious that Q satisfies the null-empty set property, i.e: Ω (ϕ) = 0 Next, we would show that Ω is countable additive i.e: where, A 1 , A t e A 1 , B 1 , B t ∈ A 2 and (A t ×B t )⋂ (A r ×B r ) = ϕ for t≠r.We begin by letting A and B to be two pair-wise disjoint sets which means that: We now fix x∈A, vary y∈B and maintain the condition (x, y) ∈A×B.The above operation implies that there exists a t such that (x, y) ∈ A t ×B t which in-turn means that x∈A t and such that y∈B t .Thus y∈B implies that y∈B t where x∈A t and we have: where, 2) can be further written in the form If x ∉A, then z ∉A t ∀ t and we can write X A t ( x ) = 0 where χ At (x ) is the character is-tic function of A t with respect to x .Also, if x ∈A then x ∈ A t for all t in F(x ) and hence χ At (x ) = 1.Thus Eq. ( 3) can be modified as follow: Therefore, Applying the monotone convergence theorem (MCT) on (V, A 1 , µ 1 ) Eq. ( 4) becomes: Hence Ω is countable additive.Therefore, Such that: Is a measure on the semi-algebra A 1 × A 2 Step II: We use the general extension theory to extend Ω to a unique measure Ω on the sigma-algebra generated by R and claim that μ # and μ $ are σfinite i.e., given: such that: then Ω is a measure.We proceed as follows.
For our claim above to be meaningful the equations below must be true: and μ # (V i )<+∞ for all i: 6) and μ $ (E j )<+∞ for all j .Thus: is a partition of V×E by elements of R such that: Therefore, isσ-finite.The above described construction goes for incidence matrices while that of adjacency matrices follows the same procedures as above.

DISCUSSION
The concept of measurability is very important in various disciplines such as, analysis, statistics, economics, computer science, etc.
In statistics related areas the presence of the power set in this representation, which is equivalent to the universal set makes it possible for the probabilities to be calculated and hence represented in a matrix form given rise to matrices like, doubly stochastic matrices, right stochastic matrices, left stochastic matrices, etc.
In the computer science disciplines, a knowledge of the sigma algebra backed by measurability makes computation of any matrix represented in this form to be faster than other matrices of different representation as the algorithm is knowledgeable about the power set.
In Graph Theory, the power set (sigma algebra) of this representation can be used to store all perfect matchings of a graph (bipartite graph) and hence doubly stochastic matrices can be generated since perfect matchings are associated with the graphs of doubly stochastic matrices.At the same time permutation matrices can be obtain because of the fact that any convex combination of permutation matrices gives rise to a doubly stochastic matrix (Ando, 1989;Bhatia, 1997;Borwein and Lewis, 2000).

CONCLUSION
We have presented a new approach of graph representation and at the same time a new method of constructing measurable incidence and adjacency matrices.The advantage of this noble work is simply measurability as both the representation and the construction are measurable.