On Chaotic Cartesian Product of Graphs and Their Retractions

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1 ISSN (print), (online) International Journal of Nonlinear Siene Vol.5(03) No.,pp On Chaoti Cartesian Produt of Graphs and Their Retrations M. Abu-Saleem Department of Mathematis, Al-Laith University ollege for girls,umm AL-Qura University, Saudi Arabia (Reeived 8 Otober 0, aepted 0 Deember 0) Abstrat: Our aim in the present artile is to introdue and study a new type of Cartesian produt of graphs, namely haoti Cartesian produt of graphs.chaoti graphs whih represented as non trivial subgraphs of haoti Cartesian produt haoti graphs are haraterized.the haoti inidene matries and haoti adjaeny matries representing the haoti graphs indued from haoti Cartesian produt of graphs are obtained.the effet of retrations on a finite number of produt haoti subgarphs are dedued. Keywords: haoti Cartesian produt; retration; graph Bakground and definitions Chaos theory is applied in many sientifi disiplines: mathematis, programming, mirobiology, biology, omputer, siene, eonomis, engineering, finane, philosophy, physis, politis, population dynamis, psyhology, and robotis. Chaoti behavior has been observed in the laboratory in a variety of systems inluding eletrial iruits, lasers, osillating hemial reations, fluid dynamis, and mehanial and magneto-mehanial devies, as well as omputer models of haoti proesses. Observations of haoti behavior in nature inlude hanges in weather, the dynamis of satellites in the solar system, the time evolution of the magneti field of elestial bodies, population growth in eology, the dynamis of the ation potentials in neurons, and moleular vibrations. There is some ontroversy over the existene of haoti dynamis in plate tetonis and in eonomis. One of the most suessful appliations of haos theory has been in eology, where dynamial systems suh as the Riker model have been used to show how population growth under density dependene an lead to haoti dynamis. Chaos theory is also urrently being applied to medial studies of epilepsy, speifially to the predition of seemingly random seizures by observing initial onditions. A related field of physis alled quantum haos theory investigates the relationship between haos and quantum mehanis. The orrespondene priniple states that lassial mehanis is a speial ase of quantum mehanis, the lassial limit. If quantum mehanis does not demonstrate an exponential sensitivity to initial onditions, it is unlear how exponential sensitivity to initial onditions an arise in pratie in lassial haos. Reently, another field, alled relativisti haos, has emerged to desribe systems that follow the laws of general relativity. The initial onditions of three or more bodies interating through gravitational attration (see the n-body problem) an be arranged to produe haoti motion[-5,8]. An abstrat graph G is a diagram onsisting of a finite non -empty set of elements alled verties denoted byv (G) together with a set of unordered pairs of these elements, alled edges denoted by E(G) [0,]. A simple graph is a graph with no loops or multiple edges [0,]. A onneted graph is a graph in one piee.[0,].let G be a graph without loops,with n-verties labeled,, 3,... n.the adjaeny matrix A(G) is the n n matrix in whih the entry in row i and olumn j is the number of edges joining the verties i and j [0,].The inidene matrix of a graph gives the (0, )-matrix whih has a row for eah vertex and olumn for eah edge, and (v, e) = iff vertex v is inident e upon edge[0,]. A graph H is said to be a subgraph of a graph G if V (H) V (G) and E(H) E(G) [6,9]. A subset A of a topologial spae X is alled a retrat of X if there exists a ontinuous map r : X A (alled a retration) suh that r(a) = a a A [7]. Main results Aiming to our study we introdue the following Corresponding author. address: mohammedabusaleem005@yahoo.om Copyright World Aademi Press, World Aademi Union IJNS /70

2 M. Abu-Saleem: On Chaoti Cartesian Produt of Graphs and 87 Let H and H be two onneted simple graphs of two vertex sets a, b,, x, y. Then their Cartesian produt is simple graph as in Figure.. Figure : Cartesian produt of two onneted graphs. Both inidene and adjaeny matries I, A, I, A respetively of the graphs H and H take the forms I = 0 0 A = 0 0, I = [ ] [ ] 0, A =. On the other hand, both inidene and adjaeny matries of the Cartesian produt H H has of the following form Then we arrive to the following theorem: I = 0, A = Theorem The matrix of both inidene and adjaeny represented the Cartesian produt of two onneted simple graphs an be obtained by the inidene and adjaeny representing eah of the given onneted simple graphs onstituting suh a Cartesian produt. Now we will disuss the Cartesian produt of haoti graphs.the haoti graph is a geometri graph arry many other physial haraters suh that the haoti graphs H, H, H as in Figure. H Where H = (V ( H ), E( H )) and H = (V ( H ), E( H ), V h = a ih, b ih, ih, i =,,..., E h = a ih b ih, a ih ih, b ih ih, i =,,..., V h = x ih, y ih, i =,,..., and IJNS homepage:

3 88 International Journal of Nonlinear Siene, Vol.5(03), No., pp Figure : Chaoti Cartesian produt of two onneted graphs. E h = x ih y ih, i =,,..., where eah pure haoti vertex overlapped on the geometry.both of the haoti inidene 0 0 and adjaeny matries of the of haoti graphs H, H are I h = 0 A h = 0, [ ] I h =, 0 0 A h = [ 0 ] 0 where, = 0...,and 0 = On the other hand, both haoti inidene and adjaeny matries of the haoti Cartesian produt H has of the following forms I h = 0,A h = From the above disussion,we formulate the following theorem Theorem The matrix of both inidene and adjaeny represented the haoti Cartesian produt of two onneted simple graphs an be obtained by the inidene and adjaeny representing eah of the given onneted simple graphs onstituting suh haoti Cartesian produt. Definition The haoti Cartesian produt H of haoti onneted simple graphs H = (V ( H ), E( H )) and H = (V ( H ), E( H )) is the haoti graph with vertex set (V ( H ), E( H )) where the haoti vertex (a ih, x ih ) is adjaent to vertex (b ih, y ih ), IJNS for ontribution: editor@nonlinearsiene.org.uk

4 M. Abu-Saleem: On Chaoti Cartesian Produt of Graphs and 89 i =,,..., whenever a ih b ih E( H ) and x ih = y ih or a ih = b ih and x ih y ih E( H ).For fixed a haoti vertex a ih of H the verties (a ih, x ih ) : x ih V ( H ) indue haoti subgraph of H is isomorphi to H. Theorem 3 Let G be a onneted simple graph x ih its haoti vertex.thenthere exist a olletion of haoti subgraphs H, H,..., H m of G, x ih V ( H j ), j =,,... m, i =,,... suh that G = H... H m indue a retration r(g ) = H... r( H s )... H m, for s =,,..., m or = H... r( H s )... r( H t )... H m,for s, t =,,..., m, s < t or = m Π k= r( k H )). Proof. Consider the representation G = H... H m in whih all haoti subgraphs H j are indeomoposable and x ih V ( H j ) for all j =,,... m, i =,,.... We prove this result by mathematial indution on m,the number of haoti subgraphs. When m =, r( H ) = r (V ( H ), E( H )) (V ( H ), E( H )) is the retration of haoti subgraphs with vertex set is (V ( H ), E( H )) where the haoti vertex (a ih, x ih ) is adjaent to vertex (b ih, y ih ), i =,,..., whenever a ih b ih E( H ) and x ih = y ih or a ih = b ih and x ih y ih E( H ).Then learly we get (V ( H ), E( H )) (V ( H ), E( H )) = r or = (V ( H ), E( H )) r (V ( H ), E( H )) or = r (V ( H ), E( H )) r (V ( H ), E( H )). r( H H ) = r (V ( H ), E( H )) (V ( H ), E( H )) Now suppose that the result is true for (m )haoti subgraphs,thus r( H... m H H m ) = r( H... m H ) r( H m ) or = r( H... m H ) H m or = ( H... m H ) r( H m ).In ase r( H... m H H m ) = r( H... m H ) r( H m ),we obtain from the indution assumption r( H... m H ) r( H m ) = ( H... r( H s )... m H ) r( H m ) for s =,,..., m or = (H... r( H s )... r( H t )... m H ) r( H m ),for s, t =,,..., m, s < t. or = m Π r( H k ) r( H m ) = Π m r( H k ).Similarly for other two ases. This ompletes the proof. k= k= Theorem 4 There are two haoti onneted simple subgraphs H and H suh that their limit retration of a haoti Cartesian produt is not isomorphi to the haoti Cartesian produt of limit retration.ie. lim r n( H ) lim r n ( H ) lim r n( H ). n n n Proof. Let H and H be two haoti onneted simple subgraphs as inf igure.3, then lim r n( H ) is either n H or homeomorphi to H. But, lim r n( H ) =haoti point. Therefore, lim r n( H ) lim r n ( H ) lim r n( H ). n n n n 3 Conlusion The present paper gives important results related to the mathematial theory of operations on haoti graphs and their retrations. IJNS homepage:

5 90 International Journal of Nonlinear Siene, Vol.5(03), No., pp Referenes Figure 3: The limit retrations of Chaoti Cartesian produt of two graphs. [] M. Abu-Saleem. Folding on the haoti Cartesian produt of manifolds and their fundamental group. Tamkang journal of mathematis, 39(008): [] M. Abu-Saleem. Folding on the wedge sum of graphs and their fundamental group. Applied seines, (00):4-9. [3] M. Abu-Saleem. Dynamial manifold and their fundamental group. Adv. Stud. Contemp. Math, 0()(00):5-3. [4] M. El-Ghoul, A. E. El-Ahmady, T.Homoda. On haoti graphs and appliations in physis and bilogy.chaos, Soliton and fratals, 7(006): [5] M. El-Ghoul, A. E. El-Ahmady, M. Abu-Saleem. On haoti fundamental group and their foldings. J. Fuzzy Math, 9()(009):3-3. [6] P.J.Giblin. Graphs, Surfaes and homology: An introdution to algebrai topology. Chapman and Hall Ltd., London, (977). [7] W. S. Massey. Algebrai topology: An introdution. Harourt Brae and world, New York, (967). [8] S.I.Nada, E.Hamouda. On the folding of graphs-theory and appliation. Chaos, Soliton and fratals,4(009): [9] A.T.White. Graphs, groups and surfaes.amsterdam, North-Holland Publishing Company, (973). [0] R.J.Wilson. Introdution to graph theory.edinburgh, Oliver &Boyed, (97). [] R.J.Wilson, J.J.Watkins. Graphs, An introdutory approuh, A first ourse in disrete mahematis.canada John Wiley &sons, In, (990). IJNS for ontribution: editor@nonlinearsiene.org.uk