Some Properties on Nano Topology Induced by Graphs

Transcription

1 AASCIT Journal of anosiene 2017; 3(4): ISS: (Print); ISS: (Online) Some Properties on ano Topology Indued by Graphs Arafa asef 1 Abd El Fattah El-Atik 2 1 Department of Physis and Engineering Mathematis Faulty of Engineering Kafrelsheikh University Kafrelsheikh Egypt 2 Department of Mathematis Faulty of Siene Tanta University Tanta Egypt address nasefa50@yahooom (A asef) aelatik55@yahooom (A El F El-Atik) Keywords Graphs ough Sets elation ano Topologial Graphs eeived: June Aepted: July Published: September Citation Arafa asef Abd El Fattah El-Atik Some Properties on ano Topology Indued by Graphs AASCIT Journal of anosiene Vol 3 o pp Abstrat A nano topologial spae beame a new type of modern topology in terms of rough sets The paper aims to analyze some real life problems using nano topology We point out that some examples and results as proposed by Lellis Thivagar et al (2016) are not true The orretions will improve further extensions of the results in [1] Some new forms of topologial strutures on a simple direted graph and give more generalized nano topology indued by graphs will be established 1 Introdution Graph theory has reently emerged as a subjet in its own right as well as being an important mathematial tool an suh diverse subjets as operational researh hemistry soiology and genetis A graph G = ( V E ) [2] is an ordered pair of disjoint sets ( V E ) where V is a nonempty set and E is a subset of unordered pairs of V The verties and edges of a graph G are the elements V = V ( G ) and E = E( G ) respetively We say that a graph G is finite (resp infinite) if the set V ( G ) is finite (resp infinite) The degree of a vertex u V( G) is the number of edges ontaining u If there is no edge in a graph G ontains a vertex u then u is alled an isolated point and so the degree of u is zero An edge whih has the same vertex to ends is alled a loop and the edge with distint end is alled a link A graph is simple if it has no loop and no two of its links join the same pair of verties A graph whih has no edge is alled a null graph A simple graph is alled omplete graph if any two distint verties are joined by an edge If the vertex set of a graph G an be split into two disjoint sets A and B so that eah edge of a graph G joins a vertex of A and a vertex of B then a graph G is a bipartite graph A omplete bipartite graph is a bipartite graph in whih eah vertex in A is joined to eah vertex in B by just one edge Given a graph G a walk in G is a finite sequene of edges of the form v0v1 vv 1 2 vn 1vn We all v 0 the initial vertex and v n the final vertex of the walk A walk in whih all the edges are distint is a trial If in addition the verties v0 v1 v2 vn are distint (exept possibly v0 = v n ) then the trial is a path A path or trial is losed if v0 = v n ) and a losed path ontaining at least one edge is a yle We an ombine two graphs to make a larger graph If the two graphs are G1 = ( V( G1) E( G 1)) and G2 = ( V ( G2) E( G 2)) where V ( G 1) and V ( G 2) are disjoint then their union G1 G2 is the graph with the vertex set V ( G1 ) V ( G2) and edge family E( G ) E( G ) A graph that is in one piee so that any two verties are onneted by a 1 2

2 20 Arafa asef and Abd El Fattah El-Atik: Some Properties on ano Topology Indued by Graphs path is a onneted graph and disonneted otherwise Clearly any disonneted graph G an be expressed as the union of onneted graphs eah of whih is a omponent of G [2-4] In many appliations it is neessary to assoiate with eah edge of a graph an orientation or diretion In some situations the orientation of the edges is a "true" orientation in the sense that the system represented by the graph exhibits some unilateral property For example the diretion of the non-way streets of a ity and the orientations representing the unilateral property of a ommuniation network are true orientation of the physial system In other situations the orientations used a "pseudo"-orientation used in lieu of an elaborate referene system For example in eletrial network theory the edges of a graph are assigned arbitrary orientations to represent the referenes of the branh urrents and voltages The study of direted graphs arises from making the roads into one way streets For more details of a graph theory one an show [2-5] In many appliations it is neessary to assoiate with eah edge of a graph an orientation or diretion In some situations the orientation of the edges is a "true" orientation in the sense that the system represented by the graph exhibits some unilateral property For example the diretion of the non-way streets of a ity and the orientations representing the unilateral property of a ommuniation network are true orientation of the physial system In other situations the orientations used a "pseudo"-orientation used in lieu of an elaborate referene system For example in eletrial network theory the edges of a graph are assigned arbitrary orientations to represent the referenes of the branh urrents and voltages The study of direted graphs arises from making the roads into one way streets Lellis Thivagar and ihard proposed the theory of a nano topology [6 7] Also Lellis Thivagar et al [1] defined and studied nano topology indued by a diret simple graph The present work aims to orret some examples in [1] Also some new forms of topologial strutures on a simple direted graph and give more generalized nano topology indued by graphs will be establish Finally the approximations of some strutues by a relation whih may have possible appliations in quantum physis and superstring theory will be studied 2 Preliminaries In this setion we give some fundamental notions related to nano topology and graph theory Definition 21 [2 4] If G( V E ) is a direted graph and u v V ( G) then (i) u is invertex of v if uv E( G) (ii) u is outvertex of v if vu E( G) (iii) The indegree of a vertex v is the number of verties u suh that uv E( G) (iv) The outdegree of a vertex v is the number of verties u suh that uv E( G) Definition 22 [8] Let U be a nonempty finite set of objets alled the universe and be an equivalene relation on U named as the indisernibility relation Elements belonging to the same equivalene lass are said to be indisernible with one another The pair ( U ) is said to be the approximation spae Let X U (i) The lower approximation of X with respet to is the set of all objets whih an be for ertain lassified as X with respet to and it is denoted by L( X ) that is L ( X ) = { ( x) : ( x) X} x U where ( x ) denotes the equivalene lass determined by x (ii) The upper approximation of X with respet to is the set of all objets whih an be possibly lassified as X with respet to and it is denoted by H ( X ) that is U( X ) = { ( x) : ( x) X ϕ} x U (iii) The boundary region of X with respet to is the set of all objets whih an be lassified neither as X nor as not X with respet to and it is denoted by B ( X ) that is B( X ) = U( X ) L( X ) Aording to Pawlak s definition X is alled a rough set if U( X ) L ( X ) Definition 22 [6 7] Let U be the universe be an equivalene relation on U and τ ( X ) = { U ϕ L ( X ) U ( X ) B ( X )} where X U and ( X ) τ satisfies the following axioms: (i) Uand ϕ τ( X ) ; (ii) The union of elements of any subolletion of τ ( ) X is in τ ( X ) (iii) The intersetion of the elements of any finite subolletion of τ ( ) in τ ( ) That is τ ( ) X X X forms a topology on U ( U τ ( X )) is alled a nano topologial spae The elements of τ ( X ) are alled nano open sets Definition 23 [1] Let G( V E ) be a graph and v V ( G) The neighbourhood of v is ( v) = { v} { u V ( G) : uv E( G)} Definition 24 [1] Let G( V E ) be a graph and H be a subgraph of G Then (i) The lower approximation L : P( V ( G)) P( V ( G)) is L ( V( H)) = { v : ( v) V ( H)} ; v V ( G) (ii) The upper approximation U : P( V ( G)) P( V( G)) is U ( V ( H)) = { ( v) : v V ( H)} ; (iii) The boundary region is B ( V ( H)) = U ( V ( H)) L ( V ( H))

3 AASCIT Journal of anosiene 2017; 3(4): Definition 25 [1] Let G be a graph ( v ) be a neighbourhood of v in V and H be a subgraph of G τ ( V ( H)) = { V ( G) ϕ L ( V ( H)) U ( V ( H)) B ( V ( H))} forms a topology on V ( G ) alled the nano topology on V ( G ) with respet to V ( H ) ( V ( G) τ ( V ( H))) is a nano topologial spae indued by a graph G Proposition 26 ([7] [9]) If ( U ) is an approximation spae and X Y U then we have the following properties of Pawlak s rough sets: (i) L ( X ) X U ( X ) (Contration and Extension) (ii) L ( ϕ) = U ( ϕ) = ϕ (ormality) and L ( U) = U ( U) = U (Co-normality) (iii) U ( X Y) = U ( X ) U ( Y) (Addition) (iv) U ( X Y) U ( X ) U ( Y) (v) L ( X Y) L ( X ) L ( Y) (vi) L ( X Y) = L ( X ) L ( Y) (Multipliation) (vii) L ( X ) L ( Y) and U( X ) U( Y) whenever X Y (Monotone) (viii) U ( X ) = [ U ( X )] and L ( X ) = [ U ( X )] where X denotes the omplement of X in U (Duality) (ix) U( U( X )) = L ( U( X )) = U( X ) (Idempoteny) (x) L ( L ( X )) = U( L ( X )) = L( X ) (Idempoteny) Figure 1 Graph for Observation 32 the set of verties of a diret simple graph G is V ( G) = { v1 v2 v3 v 4} and ( v1) = ( v2) = { v1 v2 v 3} ( v3) = { v 3} and ( v4) = { v3 v 4} Let H be a subgraph of G where V ( H) = { v1 v 2} Then L ( V ( H )) = { v 3 } U ( V ( H )) = V ( G ) and B ( V ( H )) = { v1 v2 v 4 } Therefore the nano topologial spae indued by a graph G is τ ( V ( H )) = { V ( G ) ϕ { v 3}{ v 1 v 2 v 4}} Aording to our definition in Observation 31 we modify emark 38 in [7] as follows: Observation 33 Consider Figure 2 3 Corretion of Some Lellis Thivagar's esults In this setion we orret some errors in [1] in the following observations Observation 31 Lellis Thivagar defined in Definition 32 in [7] the notion of upper approximation operation as follows: U : P( V ( G)) P( V ( G)) is U ( V ( H)) = { ( v) : v V ( H)} It is not suitable with the results in [1] So we orret it by the following U ( V ( H )) = { v : ( v ) V ( H ) ϕ } v V ( H ) Aording to our definition in Observation 31 we write Definition 32 in [7] as follows: Observation 32 Consider Figure 1 Table 1 ano topology for possible subgraphs of G( V E ) V( G ) L ( V( H )) U ( V( H )) B ( V( H )) τ ( V ( H )) { a } ϕ { a } { a } { V ( G) ϕ { a }} Figure 2 Graph for Observation 33 the set of verties of a diret simple graph G is V ( G) = { a b d e } and ( a) = { a } ( b) = { a b } ( ) = { e } ( d) = { d } and ( e) = { a d e } Table 1 shows the nano topology of any subgraph H of a graph G { b } ϕ { a b } { a b } { V ( G) ϕ { a b }} { } ϕ { e } { e } { V ( G) ϕ { e}} { d } ϕ { d } { d } { V ( G) ϕ { d}} { e } ϕ { a d e } { a d e } { V ( G) ϕ { a d e}} { a b } ϕ { a b } { a b } { V ( G) ϕ { a b }} { a } { a } { a e } { e } { V ( G) ϕ { a}{ a e}{ e}} { a d } ϕ { a d } { a d } { V ( G) ϕ { a d}} { a e } ϕ { a d e } { a d e } { V ( G) ϕ { a d e}}

4 22 Arafa asef and Abd El Fattah El-Atik: Some Properties on ano Topology Indued by Graphs V( G ) L ( V( H )) U ( V( H )) B ( V( H )) τ ( V ( H )) { b } ϕ { a b e } { a b e } { V ( G) ϕ { a b e}} { b d } ϕ { a b d } { a b d } { V ( G) ϕ { a b d}} { b e } { V ( G) ϕ } { d } { d } { d e } { e } { V ( G) ϕ { d}{ d e}{ e}} { e } { } { a d e } { a d e } { V ( G) ϕ { }{ a d e}{ a d e}} { d e } ϕ { a d e } { a d e } { V ( G) ϕ { a d e}} { a b } { a b } { a b e } { e } { V ( G) ϕ { a b}{ a b e}{ e}} { a b d } ϕ { a b d } { a b d } { V ( G) ϕ { a b d}} { a b e } { V ( G) ϕ } { a d } { a d } { a d e } { e } { V ( G) ϕ { a d}{ a d e}{ e}} { a e } { a } { a d e } { d e } { V ( G) ϕ { a }{ a d e}{ d e}} { a d e } { e } { a d e } { a d } { V ( G) ϕ { e}{ a d e}{ a d}} { b d } { d } V( G ) { a b e } { V ( G) ϕ { d}{ a b e}} { b e } { } V( G ) { a b d e } { V ( G) ϕ { }{ a b d e}} { b d e } { V ( G) ϕ } { d e } { d } { a d e } { a e } { V ( G) ϕ { d}{ a d e}{ a e}} { a b d } { a b d } V( G ) { e } { V ( G) ϕ { a b d}{ e}} { a b e } { a b } V( G ) { d e } { V ( G) ϕ { a b }{ d e}} { a b d e } { e } V( G ) { a b e } { V ( G) ϕ { e}{ a b e}} { a d e } { a d e } { a d e } ϕ { V ( G) ϕ { a d e}} { b d e } { d } V( G ) { a b e } { V ( G) ϕ { d}{ a b e}} V( G ) V( G ) V( G ) ϕ { V ( G) ϕ } ϕ ϕ ϕ ϕ { V ( G) ϕ } 4 More Properties on ano Topologial Graphs In this setion we dedue some new properties nano topologial graphs Observation 41 Let G( V E ) be a graph S 1 and S 2 are two subgraphs of G Then from Example 33 we state some properties as follows: L [ V ( S ) V ( S )] L ( V ( S ) L ( V( S ) Beause a of in Observation 33 V( S1) V ( S2) = { a b } L ({ a b }) = { a b } But L ({ a b}) = ϕ and L ({ }) = ϕ Then L [({ a b}) ({ })] = ϕ b U V S1 U V S2 U V S1 V S2 V ( S1) = { e } and 2 [ ( )] [ ( )] [ ( ) ( )] Take V ( S ) = { d e } U[ V ( S1) V ( S2)] = U ({ e}) = { a d e} and U [ V ( S )] U [ V ( S )] = { a d e} { a d e} = { a d e} 1 2 L [ V( S)] [ U ( V( S))] Beause of we take V( S) = { a } then [ V ( S)] = { b d e } and L ({ b d e}) = { d } Also U ( V ( S)) = { a } and { a } = { b d e } d U [ V( S)] [ L ( V( S))] Beause of we take V ( S) = { a b } then [ V ( S)] = { d e } and U ({ d e}) = { a d e } Also L ( V ( S)) = ϕ and [ ϕ] = ϕ e U [ U ( V ( S))] U ( V( S)) Take V( S) = { a } then U ( V ( S )) = { a } and U[ U ( V ( S))] = { a e } f U ( V ( S)) L [ U ( V ( S))] Take V ( S) = { a } then U ( V ( S )) = { a e } and L[ U ( V ( S))] = { a } g L [ U ( V( S))] V ( S) Take V ( S) = { b } then U ( V ( S )) = { a b } and L[ U ( V( S))] = { a b } h V( S) U [ L ( V( S))] Take V( S) = { a } then L ( V ( S )) = ϕ and so U[ L ( V ( S))] = ϕ i U [ L ( V ( S))] L ( V( S)) Take V ( S) = { a } then L ( V ( S )) = { a } and so U[ L ( V( S))] = { a } j L ( V ( S)) L [ L ( V ( S))] Take V ( S) = { a } then L ( V ( S )) = { a } and so L[ L ( V ( S))] = ϕ Theorem 39 Let G( V E ) be a graph S is a subgraphs of G Then the following are hold: (i) U [ L ( V ( S))] V ( S) L [ U ( V ( S))] (ii) L [ U [ L ( V ( S))]] = L ( V ( S ))] (iii) U [ L [ U ( V ( S))]] = U ( V ( S ))] Proof (i) Let x U[ L ( V ( S))] then there exists v L ( V ( S )) suh that vx E( G) Therefore x ( v) V ( S) ow sine x V ( S) then ( v) L ( V ( S)) implies x L [ U ( V ( S))] (ii) Let x L ( V ( S )) then ( x) U[ L ( V( S))] So x L[ U[ L ( V ( S))]] We know that U[ L ( V( S))] V( S) implies L[ U[ L ( V ( S))]] = L ( V ( S ))](iiii) We know that U[ L ( V ( S))] V( S) implies U[ L[ U ( V ( S))]] = U ( V ( S ))] Also V( S) L [ U ( V( S))] implies U [ L [ U ( V ( S))]] = U ( V ( S ))]

5 AASCIT Journal of anosiene 2017; 3(4): Appliation A tree [1] is a simple onneted graph that ontain no yles A fratal struture onsists of piees whih are similar to eah other and similar to the whole struture Eah of whih is a ompat metri spae say A onsists of piees A i whih are similar to eah other In other words A A A A assigned with a homeomorphism = 1 2 m funtion fi : A Ai = fi( A) This setting leads to smaller and smaller parts so the same maps an be applied for suh smaller piees f : A f ( A ) = f f ( A ) = A where Ai A and Aji Aj j i j i j i i ji name of topology is onerned with all questions diretly or indiretly related to ontinuity Therefore the theory of nano topology is one of the most important subjet in topology On the other hand topology plays a signi_ant rule in quantum physis hight energy physis and superstring theory [10] Thus we study the approximations of some strutues by a relation whih may have possible appliations in quantum physis and superstring theory eferenes [1] Lellis Thivagar M; Paul Manuel and Sutha Devi V: A detetion for patent infringement suit via nanotopology indued by graph Cogent Mathematis 3 (2016) 1-10 [2] Wilson J: Introdution to graph theory Fourth Edition Longmon Maleysia (1996) [3] Bondy J A and Murty U S : Graph theory with appliations Elsevier Siene Publishing Co In (1975) [4] Diestel : Graph theory II Heidelberg Springer-Verlag IV (2010) [5] Boninowski Z; Bryniarski E and Wybranie U: Extensions and intensions in the rough set theory Information Sienes 107 (1998) [6] Lellis Thivagar M and ihard C: On nano forms of weakly open sets International Journal of Mathematis and Statistis 1 (2013) Figure 3 Fratal strutres with one onneted point There are some fratal strutures with one onneted point In the following we approximate the fratal struture as a tree and give the possible nano topologial spaes for the tree 6 Conlusion The field of mathematial siene whih goes under the [7] Lellis Thivagar M and ihard C: On nano ontinuity Journal of Mathematial Theory and Modelling 3 (2013) [8] Pawlak Z: ough sets Theoretial Aspets of easoning About Data Kluwer Aadmi Publishers Dordreht 1991 [9] Bondy J A and Murty U S : Graph theory with appliations Elsevier Siene Publishing Co In (1975) [10] M S El ashie Topis in the mathematial physis of E-infinity theory Chaos Solitons Fratals 30 (2006)