Cyclic Symmetry of Riemann Tensor in Fuzzy Graph Theory
U S Naveen Balaji, S Sivasankar, Sujan Kumar S, Vignesh Tamilmani
aa r X i v : . [ m a t h . G M ] A ug Cyclic Symmetry of Riemann Tensor in Fuzzy GraphTheory
U S Naveen Balaji , S Sivasankar , Sujan Kumar S , and VigneshTamilmani Department of Science and Humanities, PES University, Bangalore -560085, Karnataka, India. email(s): [email protected],[email protected], [email protected],[email protected] Department of Mathematics, R V Institute of Technology andManagement, Bangalore - 560059, Karnataka, India.
Abstract
In this paper, we define a graph-theoretic analog for the Riemann tensor and ana-lyze properties of the cyclic symmetry. We have developed a fuzzy graph theoreticanalog of the Riemann tensor and have analyzed its properties. We have alsoshown how the fuzzy analog satisfies the properties of the × matrix of the Rie-mann tensor by expressing it as a union of the fuzzy complete graph formed bythe permuting vertex set and a Levi-Civita graph analog. We have concluded thepaper with a brief discussion on the similarities between the properties of the fuzzygraphical analog and the Riemann tensor and how it can be a plausible analogousmodel for the Petrov-Penrose classification. Tensors and Differential geometry are central to General Relativity, they are the foun-dation to the seminal theory of Einstein. The Riemann curvature tensor named afterBernhard Riemann is a higher-dimensional analogue of the Gaussian curvature and isclosely related to tidal forces, it represents the tidal force experienced by a particlemoving along a geodesic. In -dimensions, the Riemann tensor has components andobservations reveal a variety of algebraic symmetries such as the first skew symmetry, thesecond skew symmetry, and the block symmetry all of which reduce the componentsto independent components. The last algebraic symmetry, called the cyclic symmetryis closely associated with Bianchi’s first identity. For values of n > in n ( n − algebraically independent components of the Riemann tensor the components are rep-resented by the Weyl tensor which also possesses all three algebraic symmetries and inaddition it can be thought of as that part of the curvature tensor such that all con-tractions vanish, i.e., a pseudo-Riemannian manifold is said to be conformally flat if itsWeyl tensor vanishes. It was A.Z. Petrov who classified the algebraic symmetries of theWeyl tensor, called the Petrov-Penrose classification. Generally, gravitational fields areclassified in accordance to the Petrov-Penrose classification of their corresponding Weyltensor.In this paper we develop a graph theoretic analog of the Riemann tensor which we thenuse to help develop a fuzzy graph analog of the Petrov-Penrose classification. We exploit1he cyclic symmetry of the Riemann tensor to help define the graphical analog and alsodiscuss, through theorems, the similarities and properties of the graphical analog to itstensor form. The paper is organized as follows. Section contains the preliminaries andin section fuzzy graphical analogs of the Riemann tensor and the Levi-Civita symbolare defined. Section deals with the fuzzy approach to Pentrov-Penrose classification. Definition1.1 [1]. Let R n denote the Euclidean space of n -dimensions, i.e., the set of all n -tuples (cid:0) x , x , ..., x n (cid:1) (cid:0) −∞ < x i < ∞ (cid:1) with the usual topology (open and closed setsare defined in the usual way), and let R n denote the lower half of R n , i.e. the region of R n for which x ≤ . A map φ of an open set O ⊂ R n (respectively R n ) to an open set O ′ ⊂ R n (respectively R n ) is said to be of class C r if the coordinates (cid:0) x ′ , x ′ , ..., x ′ m (cid:1) of the image point φ ( p ) in O ′ are r -times continuously differentiable functions of thecoordinates (cid:0) x ′ , x ′ , ..., x ′ m (cid:1) of p in O . Definition 1.2 [1]. A C r n -dimensional manifold M is a set M together with a C r atlas {U α , φ α } , where the U α are subsets of M and the φ α are one-one maps of thecorresponding U α to open sets in R n such that (1) The U α cover M , i.e. M = S α U α , (2) if U α ∪ U β is non-empty, then the map φ α ◦ φ − β : φ β ( U α ∪ U β ) → φ α ( U α ∪ U β ) is a C r map of an open subset of R n to an open subset of R n . Definition 1.3 [2,7]. The mapping f : V → U , where the open sets V, U ∈ R n , is calleda homeomorphism if it is bijective and if f and its inverse f − are continuous. Definition 1.4 [2,7]. A chart for a topological space M is a homeomorphism φ from anopen subset U of M to an open subset of a Euclidean space. The chart is traditionallyrecorded as the ordered pair ( U, φ ) . Definition 1.5 [2,7]. A tangent vector v to the differential manifold M at a point p ∈ M is defined as ( (cid:0) V ρ , z ρ , v z ρ (cid:1) , where (( V ρ , z ρ ) are charts which contain p and v z ρ = v jz ρ , j = 1 , , ..., n are vectors in R n . Definition 1.6 [2,7]. Let Λ be a p -form field defined by Λ = Λ αβ...ζ dx α ∧ dx β ∧ ... ∧ dx ζ where α, β, ..., ζ are arbitrary indices. The exterior derivative acts on this p -form fieldto produce a ( p + 1) -form field as follows d Λ = d Λ αβ...ζ dx α ∧ dx β ∧ ... ∧ dx ζ . (1)The exterior derivative of a ( p − -form produces a p -form defined as follows Ξ = 1 p ! Ξ i i ... i p dx i ∧ dx i ∧ ... ∧ dx i p (2)and the exterior derivative is defined as d Ξ = 1 p ! ∂ Ξ i i ... i p ∂x i dx i ∧ dx i ∧ ... ∧ dx i p . (3)2 efinition 1.7 [1]. A Cartesian product is defined as the ordered set of vectors andone-forms (cid:0) η , ..., η m , Y , ..., Y n (cid:1) , where the Y ’s and η ’s are arbitrary vectors and one-forms respectively. The Cartesian product is expressed as the product of the tangentspace T p of vectors at a point p and the tangent space’s dual or the cotangent space ∗ T of -forms at p written as follows Π nm = ∗ T p × ∗ T p × ... × ∗ T p × | {z } n factors × T p × T p × ... × T p | {z } m factors . (4) Definition 1.8 [1]. A tensor of rank (cid:18) nm (cid:19) at a point p is a function on Π nm which islinear in each argument, i.e., if T is a tensor of rank (cid:18) nm (cid:19) at p , the number into which T maps the element (cid:0) η , ..., η m , Y , ..., Y n (cid:1) of Π nm as T (cid:0) η , ..., η m , Y , ..., Y n (cid:1) , wherethe Y ’s and η ’s are arbitrary vectors and one-forms respectively. Definition 1.9 [2,3]. The Riemann tensor is a four-index tensor which has compo-nents in -dimensions. Making use of the symmetry relations, R iklm = − R ikml = − R kilm , (5)the number of independent components is reduced to . Using the condition R iklm = R lmik , (6)the number of coordinates reduces to . Finally, using R iklm + R ilmk + R imkl = 0 , (7) independent components are left. Definition 1.10 . A graph G = ( V, E ) is analogous to the Riemann tensor with a vertexset V = { v , v , v , v } , where v = i , v = k , v = l , v = m , if it satisfies the following:a. The vertex v is fixed and is connected to only one in V − { v } ,b. There are three vertices which span a K graph amongst themselves,c. The direction of the cycle in the K graph determines the overall sign assigned to thegraph, i.e., if the cycle is in the counter clockwise direction, we assign a positive signand if the cycle is in the clockwise direction, we assign a negative sign. Definition 1.11 . Let H ⊆ G denote the K graph in which the three positions arelabelled B, C and D , with B being adjacent to the fixed vertex position A . We notethat in position B , the vertices v , v and v are equally likely for occupation and oncea particular vertex is occupied, the others vertices occupy positions C and D in cyclicorder (See Figure 2).Suppose v occupies the vertex position B then, v would occupy C and v would oc-cupy D . For each arrangement of the vertices we have two variants which differ from3igure 1: The vertex points that occupy the vertex positions B , C , and D for thesubgraph H (= K ) , where H ⊂ G and the occupation of v , v , and v in each of thesepositions is equally likely.each other by a negative sign which is determined by the direction of the cycle assignedto the subgraph H . Thus, for each combination of the adjacent vertices we have threevariants listed below (see Figure 2):1. With { v , v , v , v } we have graph G for clock wise cycle and G for a counterclock wise cycle which are related as, G = − G .2. With { v , v , v , v } we have graph G for clock wise cycle and G for a counter clockwise cycle which are related as, G = − G .3. With { v , v , v , v } we have graph G for clock wise cycle and G for a counter clockwise cycle which are related as, G = − G . Definition 1.12 . Let T ( α, β, γ, δ ) = T αβγδ denote a function which takes the sequenceof the vertices as an input and displays the vertices as indices of a particular graph T as the output. With this formulation, we can express a graph G i as follows:a. G ≡ G ( v , v , v , v ) = G v v v v = G iklm ,b. G ≡ G ( v , v , v , v ) = G v v v v = G ikml ,c. G ≡ G ( v , v , v , v ) = G v v v v = G ilmk ,d. G ≡ G ( v , v , v , v ) = G v v v v = G ilkm ,e. G ≡ G ( v , v , v , v ) = G v v v v = G imkl , &f. G ≡ G ( v , v , v , v ) = G v v v v = G imlk . Definition 1.13 . Let G ik ( lm ) = ( G iklm − G ikml ) such that (cid:2) G ik ( lm ) + G il ( mk ) + G im ( kl ) (cid:3) = 0 (8) Theorem 1.1 . Three permuting indices, by definition . , condenses six graphs tothree graphs and thus, G i ( klm ) = 0 . (9) Proof
To condense the number of graphs we can make use of the fact that k, l and m permute cyclically as follows G i ( klm ) = 13! [ G iklm + G ilmk + G imkl ] , (10)4igure 2: From the above figure, G = − G , G = − G , and G = − G and thus,
3! [ G iklm + G ilmk + G imkl ] = 0 or G i ( klm ) = 0 . Theorem 1.2 . The antisymmetry in each pair of indices (vertices) of a graph G con-structed from definition . implies that there are P = n ( n − ways of choosingindependent pairs of indices. Proof
In our discussion we have n = 4 indices and thus there are P = (4)(3) = 6 waysof choosing pairs which resulted in graphs G , G , G , G , G , and G . Theorem 1.3 . A graph G with one fixed vertex and α number of permuting verticespossesses ξ ( K α ) number of edges, where ξ ( K α ) is the number of edges of a K α complete graph. Proof
Since every graph has ξ ≤ γ ( γ − number of edges where γ is the number ofvertices, the number of edges of a graph constructed from definition . will be thenumber of edges of the subgraph H and the edge connecting the fixed vertex to one ofthe vertices of the permuting vertex set. Since the subgraph H is a K graph, we have ξ ( K ) number of edges. Definition 1.14 . Let u = ik, u = il, u = im, u = kl, u = km, u = lm denote theantisymmetric pairs of indices such that we obtain a × matrix representation of the5ossible combinations of the graphs formed by the antisymmetric pairs given by G = G ikik G ikil G ikim G ikkl G ikkm G iklm G ilik G ilil G ilim G ilkl G ilkm G illm G imik G imil G imim G imkl G imkm G imlm G klik G klil G klim G klkl G klkm G kllm G kmik G kmil G kmim G kmkl G kmkm G kmlm G lmik G lmil G lmim G lmkl G lmkm G lmlm (11) Theorem 1.4 . The matrix of the graphs formed by the different combinations of anti-symmetric indices is symmetric due to the property of union of graphs.
Proof
Consider the elements G = G ( i, k, i, l ) = G ikil and G = G ( i, l, i, k ) = G ilik ofthe matrix G . Graphically, the graphs can be expressed as a union of two subgraphs,i.e, G ( i, k, i, l ) = G ( i, k ) ∪ G ( i, l ) and G ( i, l, i, k ) = G ( i, l ) ∪ G ( i, k ) . Thus, we observethat G ( i, l, i, k ) = G ( i, k, i, l ) or G ikil = G ilik and similarly, the other graphs along theprinciple diagonal of the matrix G are equal making the matrix a symmetric one. Theorem 1.5 . The number of independent components of the subgraphs of G is givenby P ( P + 1) − n !( n − n (cid:0) n − (cid:1) . (12) Proof
We note that for the graphs of the form G ( v , v , v , v ) , there are (cid:0) n (cid:1) possiblechoices for v and v , for graphs of the form G ( v , v , v , v ) , there are (cid:0) n (cid:1) ways tochoose different v , v and v and v ways to choose the index that is used twice fromthat which results in a total of (cid:0) n (cid:1) choices, and for graphs of the form G ( v , v , v , v ) ,there are (cid:0) n (cid:1) choices. Thus, in total we have (cid:18) n (cid:19) + 3 (cid:18) n (cid:19) + 2 (cid:18) n (cid:19) = n ( n − Here, we have n = 4 indices and (4 − /
12 = 20 subgraphs each accounting for thefollowing symmetries:1. G iklm = G lmik ,2. G iklm = − G kilm = − ( − G kiml ) = − ( − ( G ikml )) = ... , &3. G i ( klm ) = 0 . Theorem 1.6 . The graph formed by the elements of
G − { v P D } is a K graph, where v P D = { u u , u u , ..., u u } is the vertex points of the principle diagonal of G . Note
The K graph has a vertex set, V = { u , u , u , u , u , u } and a edge set, E = { e , e , e , e , e , e , e , e , e , e , e , e , e , e } , where e ij = e ji , i.e., e = G ikil = G ilik = e , ..., e = G iklm = G lmik = e , ... Theorem 1.7 . The number of independent components for r number of permutingindices of a graph constructed from definition . is given by: P ( P + 1) − n ! r !( n − r )! = 18 ( n − n [( n − n + 2] − n ! r !( n − r )! , (13)and since the indices , , , ..., r permute cyclically, G ...r ) = 1 r ! [ G ...r + G ...r + ... + G r... ] (14)6 heorem 1.8 . The elements of the principle diagonal of the matrix G is a set of ( r +1) C number of vertices. Proof
For G ( v , v , v , v ) = G iklm , we have four indices amongst which one is fixed(which is i ) and two slots available for the formation of a pair. Thus, to choose a pair,i.e., two indices out of four there are C = 6 available combinations. Thus, for ( r + 1) number of vertices there are ( r +1) C available combinations and since each combinationwas labelled as a vertex u i , there are ( r +1) C number of vertices. Note
An important property of cyclic symmetry is that this theorem holds if and onlyif ( r + 1) is a number divisible by . This arises due to the condition that we require aneven number of indices for formation of pairs. Definition 2.1 [4,5]. Let V be a non empty set. A fuzzy graph is a pair of functions G = ( σ, µ ) , where σ is a fuzzy subset of V and µ is a symmetric fuzzy relation on σ ,i.e., σ : V → [0 , and µ : V × V → [0 , such that µ ( u, v ) ≤ σ ( u ) ∧ σ ( v ) , for all u, v ∈ V . Definition 2.2 . A fuzzy graph G with the vertex set V = { v , v , v , v } , where v = i, v = k, v = l, v = m , is analogous to the Riemann tensor if it satisfies thefollowing:a. The vertex v is fixed and is connected to only one other vertex,b. The fixed vertex has a vertex membership, σ = 1 , since the probability of finding v in that vertex position is definite,c. There are three vertices which span a K graph amongst themselves,d. The permuting vertices each have a vertex membership, σ = 1 / . since the proba-bility of finding one of the permuting vertices at a particular vertex position of the K is equal to / ,e. The antisymmetry of the index pairs which reflect on the type of graph is expressedvia the number of cycles traversed in the K graph. Let P v , P v and P v represent theprobabilities of finding the vertices v , v and v in the vertex positions of K . Now, thepermuting combination { v , v , v } yields:1. G iklm if the number of cycles traversed is even in number, &2. G ikml (= − G iklm ) if the number of cycles traversed is odd in number.Thus, in general we can express the graph of a permuting combination with m numberof cycles as follows, G ( i, k, l, m ) = ( − m G iklm = (cid:26) G iklm , for m = even − G iklm = G ikml , for m = odd (cid:27) . (15) Definition 2.3 [11,12]. A fuzzy graph G is said to be complete if µ ( u, v ) = σ ( u ) ∧ σ ( v ) ,for all u, v ∈ V . Theorem 2.1 . The graph formed by the α number of permuting vertices of the fuzzygraph G , with one other fixed vertex, is a complete graph whose domination set consistsof only one vertex which is connected to both the fixed vertex and the other ( α − permuting vertices. Proof
Consider the fuzzy graph constructed based on definition . with a permutingvertex set V P = { v , v , v } which yields G iklm . Here, the vertex v is the only one whichis adjacent to both the fixed vertex v and the other vertices from V p − { v } . Thus, the7omination set of G is D G = { v } . Definition 2.4 [6,13]. An arc ( u, v ) of a fuzzy graph G is called a strong arc if µ ( u, v ) = σ ( u ) ∧ σ ( v ) , for all u, v ∈ V . Theorem 2.2 . The graph formed by the permuting vertices of the fuzzy graph con-structed based on definition . is a complete graph with all it’s arcs being strong. Proof
For the permuting vertex set V P = { v , v , v } with vertex memberships / eachwe have, in accordance to definition . , µ ( v , v ) = σ ( v ) ∧ σ ( v ) = 1 / µ ( v , v ) = µ ( v , v ) . Definition 2.5 . Let the
Levi-Civita graph analogue ǫ ( v , x, y ) = ǫ ixy be a graph suchthat ( − m ǫ ( i, x, y ) = P n =1 P n , if ( i, x, y ) is ( v , x, y ) , ( x, y, v ) , or ( y, v , x ); m = even − P n =1 P n , if ( i, x, y ) is ( y, x, v ) , ( v , y, x ) , or ( x, v , y ); m = odd , if v = x, or x = y, or y = v ; m = 0 for loops (16)where ( x, y ) is one of the permuting index pairs, i.e, { ( v , v ) , ( v , v ) , ( v , v ) } , and P n is the probability of finding the index in a particular position. Since, we have threeindices and the probability of finding an index in a particular position is equally likely, P = P = P = 1 / , ( − m ǫ ( i, x, y ) = +1 , if ( i, x, y ) is ( v , x, y ) , ( x, y, v ) , or ( y, v , x ); m = even − , if ( i, x, y ) is ( y, x, v ) , ( v , y, x ) , or ( x, v , y ); m = odd , if v = x, or x = y, or y = v ; m = 0 for loops . (17) Theorem 2.3 . The union of ǫ ( v , x, y ) and the complete graph formed of permutingindices yields a loop at the common vertex, i.e., the vertex both adjacent to the fixedvertex and the other permuting ones. When the fuzzy graph G = ( σ, µ ) is expressed asthe union of the complete fuzzy graph (formed of the permuting indices) and ǫ ( v , x, x ) ,where x is the common vertex, the vertex membership of the common vertex is reducedby σ ′ ( v ) = σ ( v ) α , (18)where α is the number of permuting vertices. Proof
Consider the fuzzy graph G i,k,l,m , where the vertex v is the commonly adjacentto both the fixed and the permuting edges. Now, we can define ǫ ( v , x, x ) = ǫ ( i, k, k ) and since the graph is to be expressed as the union of ǫ ( i, k, k ) and G ( i, k, l, m ) , wehave ǫ ( i, k, k ) ∪ G k,l,m = G i,k,l,m in which σ ( v ) = σ ( v ) = 1 / , σ ( v ) = 1 , and σ ′ ( v ) = σ ( v )3 = 1 / . 8 On Route to a Fuzzy Petrov-Penrose Classification
Generally, gravitational fields are classified in accordance to the
Petrov-Penrose clas-sification of their corresponding Weyl tensor. This is an algebraic classification basedon the idea that the curvature tensor can be thought of as a × matrix and thereduction of these matrix naturally results in general categories of curvature tensors.In this section, we present the relations between the fuzzy graphical analog of the Rie-mann tensor and the Riemann tensor and more specifically, study the similarities of theproperties between the matrices as defined in equations 11 and 19. From the symme-tries of the Riemann curvature tensor , we can write it as a R αβγδ and associate an index I = 1 , , ..., with each pair , , , , , of the independent values that αβ and γδ can take. The curvature tensor can be expressed as a × , M IK matrix as given below R R R | R R R R R R | R R R R R R | R R R − − −− − − −− − − −− − − − − | − − − − − − −− − − −− − − −− R R R | R R R R R R | R R R R R R | R R R (19)The matrix M IK can alternatively be expressed as follows M IK = (cid:18) A B−B T C (cid:19) , (20)where A , B , C , are × matrices. Notice that the matrix B is null. This can be shownby first lowering the index and making use of the property of the Levi-Civita symbol asfollows
T r B = R + R + R = ǫ R + ǫ R + ǫ R . (21)Comparing the matrix of equation 19 to that of equation 11, we can prove that the traceof matrix B is null in the graph theoretical case by expressing the fuzzy graph as theunion of the complete fuzzy graph and ǫ ( v , x, x ) as follows, T r B = G ( i, k, k, l ) + G ( i, l, k, m ) + G ( i, m, l, m )= ǫ ( i, k, k ) ∪ G ( k, m, l ) + ǫ ( i, l, l ) ∪ G ( l, k, m ) + ǫ ( i, m, m ) ∪ G ( m, l, m )= 0 . (22)We also observe that the matrices A and C are equal to their transposes, i.e., A = A T and C = C T . The structure of the matrix represented in equation 19 is based on sepa-rating the components of the Riemann curvature tensor into three distinct sets, R α α , R βγδ , and R γδµν . Observe that the first set is a × matrix in the indices α and β andas for the other two, they are to be fixed by removal of antisymmetry that they possess.Thus, we introduce the following × matrices along with their fuzzy counterpart Ψ αβ = R α β = G ( i, α, i, β ) , Σ αβ = ǫ αγδ R γδ β = ǫ ( α, v , v ) ∪ G ( v , v , i, β )Λ αβ = ǫ αγδ ǫ βµν R γδµν = ǫ ( α, v , v ) ∪ ǫ ( β, v , v ) ∪ G ( v , v , v , v ) , (23)where ǫ abc is a three-dimensional Levi-Civita tensor. These matrices yield the following9elations under the Ricci flatness condition, R XY = 0Ψ αα = 0 , Σ αβ = Σ βα , Ψ αβ = − Λ αβ . (24)According to the definitions given above we have the matrix Ψ αβ to have the followingform Ψ = R , Ψ = R , Ψ = R , ... = ⇒ Ψ αβ = R R R R R R R R R = G ikik G ikil G ikim G ilik G ilil G ilim G imik G imil G imim = Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ (25)Comparing this matrix to the × form obtained previously, we find that Ψ αβ is com-prised of the components of first quarter of the matrix (after lowering their index). Thus, Ψ αβ = A . Now, to the matrix Σ αβ . Observe that in the components of the Σ αβ Σ = 12 ǫ R , Σ = 12 ǫ R , Σ = 12 ǫ R , ..., (26)the factor / is removed by the symmetry of the matrix, i.e., since Σ αβ = Σ βα , Σ = Σ , ... , and hence Σ (12) = 2Σ = Σ + Σ = ǫ |{z} =1 R = R . (27)Similarly, we can calculate the other components to obtain the following matrix Σ αβ = R R R R R R R R R = G lmik G lmil G lmim G mkik G mkil G mkim G klik G klil R klim = Σ Σ Σ Σ Σ Σ Σ Σ Σ . (28)Comparing this matrix to the × form obtained previously, we find that Σ αβ is com-prised of the components of third quarter of the matrix (, i.e. the first half of the secondrow). Thus, Σ αβ = −B T . In matrix Λ αβ , notice that there is symmetry in the indicesand also among matrix components due to the block symmetry of the curvature tensor.the following are the components of the matrix Λ αβ Λ = ǫ ǫ R , Λ = ǫ ǫ R , Λ = ǫ ǫ R , Λ = ǫ ǫ R , ... (29)We know that Λ αβ is a symmetric matrix thus, components such as Λ = Λ = ⇒ Λ (12) = 2Λ , and this eliminates the factor (1 / . Now, to account for the remaining (1 / , consider the matrix components a = Σ and a = Σ (using index a to avoidconfusion), in which there exists a block symmetry between Riemann curvature tensor R αβγδ = R δγαβ = R βαγδ R = R . This implies that a = Σ = ǫ ǫ R = ǫ ǫ R = Σ = a = ⇒ a (12) = a + a = ǫ |{z} =1 ǫ |{z} =1 R = R . (30)Similarly, we can calculate the other components to obtain the following matrix Λ αβ = R R R R R R R R R = G lmlm G lmmk G lmkl G mklm G mkmk G mkkl G kllm G klmk G klkl = Λ Λ Λ Λ Λ Λ Λ Λ Λ (31)Comparing this matrix to the × form obtained previously, we find that Λ αβ is com-prised of the components of fourth quarter of the matrix (, i.e. the second half of thesecond row). Thus, Λ αβ = C .Let Ω αβ be a symmetric complex tensor defined as follows Ω αβ = (Ψ αβ + 2 i Σ αβ − Λ αβ ) = (Ψ αβ + 2 i Σ αβ + Ψ αβ ) = Ψ αβ + i Σ αβ Ω αβ = G ( i, α, i, α ) + i ǫ ( α, v , v ) ∪ G ( v , v , i, β ) . (32)It is well known that classification of the Riemann curvature tensor can be reduced to asimple eigen value problem where we consider the eigen value equation Ω αβ k β = λk α , inwhich the complex eigenvalues λ = λ R + iλ I satisfy the condition λ (1) + λ (2) + λ (3) = 0 since Ω αα = 0 . The matrix’s classification is now dependent on the number of independenteigenvectors and leads to six different cases, called Petrov Types I , II , D , III , N , and O . In this paper we have introduced a graph theoretic and fuzzy graph theoretic analogof the Riemann tensor and have shown how the latter satisfies the properties of the × matrix of the Riemann tensor. We then use the definitions and the propertiesdiscussed in sections and to develop a fuzzy graphical analog of the Petrov-Penroseclassification. We hope to study the detailed fuzzy graphical connections among thevarious Petrov types in future papers where we would explore connections between fuzzygraphs and exterior algebra and study fuzzy analogs of the Weyl tensor , the
Hodge staroperator , and the