On Strong Product of Two Fuzzy Graphs

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1 Inernaional Journal of Scienific and Research Publicaions, Volume 4, Issue 10, Ocober ISSN On Srong Produc of Two Fuzzy Graphs Dr. K. Radha* Mr.S. Arumugam** * P.G & Research Deparmen of Mahemaics, Periyar E.V.R. College, Tiruchirapalli-6003 ** Gov. High School, Thinnanur, Tiruchirapalli Absrac- In his paper, he srong produc of wo fuzzy graphs is defined. I is proved ha when wo fuzzy graphs are effecive hen heir srong produc is always effecive and i is proved ha he srong produc of wo complee fuzzy graphs is complee. Also i is proved ha he srong produc of wo conneced fuzzy graphs is always conneced. The lower and upper runcaions of he srong produc of wo fuzzy graphs are obained. The degree of a verex in he srong produc of wo fuzzy graphs is obained. A relaionship beween he direc sum and he srong produc of wo fuzzy graphs is obained. Index Terms- Fuzzy Graph, Direc Sum, Srong Produc, Effecive Fuzzy Graph, Connecedness, Upper and Lower Truncaions. 010 Mahemaics Subjec Classificaion: 03E7, 05C07. F I. INTRODUCTION uzzy graph heory was inroduced by Azriel Rosenfeld in The properies of fuzzy graphs have been sudied by Azriel Rosenfeld[9]. Laer on, Bhaacharya[7] gave some remarks on fuzzy graphs, and some operaions on fuzzy graphs were inroduced by Mordeson.J.N. and Peng.C.S.[3]. The conjuncion of wo fuzzy graphs was defined by Nagoor Gani.A and Radha.K.[4]. We defined he direc sum of wo fuzzy graphs and sudied he effeciveness, connecedness and regular properies of he direc sum of wo fuzzy graphs [8]. In his paper, he srong produc of wo fuzzy graphs is defined. I is proved ha when wo fuzzy graphs are effecive hen heir srong produc is always effecive and i is proved ha he srong produc of wo complee fuzzy graphs is complee. Also i is proved ha he srong produc of wo conneced fuzzy graphs is always conneced. The lower and upper runcaions of he srong produc of wo fuzzy graphs are obained. The degree of a verex in he srong produc of wo fuzzy graphs is obained. A relaionship beween he direc sum and he srong produc of wo fuzzy graphs is obained. Firs le us recall some preliminary definiions ha can be found in [1]-[9]. A fuzzy graph G is a pair of funcions (σ, μ) where σ is a fuzzy subse of a non empy se V and μ is a symmeric fuzzy relaion on σ. The underlying crisp graph of G:(σ, μ) is denoed by G*(V, E) where E V V. Le G:(σ, μ) be a fuzzy graph. The underlying crisp graph of G:(σ, μ) is denoed by G*:(V, E) where E V V. A fuzzy graph G is an effecive fuzzy graph if μ(u,v) = σ(u) σ(v) for all (u,v)e and G is a complee fuzzy graph if μ(u,v) = σ(u) σ(v) for all u,vv. Therefore G is a complee fuzzy graph if and only if G is an effecive fuzzy graph and G* is complee. (σ, μ ) is a spanning fuzzy subgraph of (σ,μ) if σ =σ and μ μ, ha is, if σ (u) = σ (u) for every uv and μ (e) μ(e) for every ee.. The degree of a verex u of a fuzzy graph G is defined as d G (u) (uv) (uv) uv uve The Caresian produc of wo fuzzy graphs G 1 :(σ 1,μ 1 ) and G :(σ,μ ) is defined as a fuzzy graph G= G 1 G : (σ 1 σ, μ 1 μ ) on G*:(V,E) where V = V 1 V and E = {((u 1, v 1 )(u, v )) / u 1 = u, v 1 v E or v 1 = v,u 1 u E 1 } wih (σ 1 σ )(u,v) = σ 1 (u) σ (v), for all (u, v) V 1 V and 1 (u 1) (v1v ),if u1 u, v1v E ( 1 ) u 1, v1 u, v (v 1) 1(u1 u ),if v 1 v,u1u E1 The conjuncion or he ensor produc of wo fuzzy graphs G 1 :(σ 1,μ 1 ) and G :(σ,μ ) is defined as a fuzzy graph G = G 1 G : (σ, μ) on G*:(V,E) where V = V 1 V and E = {((u 1, v 1 )(u, v )) / u 1 u E 1, v 1 v E } wih σ(u 1, v 1 ) = σ 1 (u 1 ) σ (v 1 ), for u,v u,v (u u ) (v v ),for all u,v u,v E. all (u 1, v 1 ) V 1 V and If G 1 :(σ 1,μ 1 ) and G :(σ,μ ) are wo fuzzy graphs such ha σ 1 μ hen σ μ 1 [6]. The lower and upper runcaions of σ a level, 0 < 1, are he fuzzy subses σ () and σ () defined respecively by, (u),if u,if u (u) and (u) 0,if u (u),if u.

2 Inernaional Journal of Scienific and Research Publicaions ISSN Le G:(σ,μ) be a fuzzy graph wih underlying crisp graph G*:(V,E). Take V () = σ, E () = μ. Then G () :(σ (),μ () ) is a fuzzy graph wih underlying crisp graph G () *:(V (), E () ). This is called he lower runcaion of he fuzzy graph G a level. Here V () and E () may be proper subses of V and E respecively. Take V () = V, E () = E. Then G () :(σ (), μ () ) is a fuzzy graph wih underlying crisp graph G () *:(V (),E () ). This is called he upper runcaion of he fuzzy graph G a level [5]. Le G 1 :(σ 1,μ 1 ) and G :(σ,μ ) denoe wo fuzzy graphs wih underlying crisp graphs G 1 *:(V 1,E 1 ) and G *:(V,E ) respecively. Le V = V 1 V and le E = {uv / u,vv; uv E 1 or uv E bu no boh }. Define G:(σ, μ) by (u),if u V V (uv),if uv E (u) (u),if u V V and (uv) (uv),if uv E 1(u) (u),if u V1 V Then if uv E 1, μ(uv) = μ 1 (uv) σ 1 (u) σ 1 (v) σ(u) σ(v), if uv E, μ(uv) = μ (uv) σ (u) σ (v) σ(u) σ(v). Therefore (σ, μ) defines a fuzzy graph. This is called he direc sum of wo fuzzy graphs. II STRONG PRODUCT Definiion.1 Le G 1 :(σ 1,μ 1 ) and G :(σ,μ ) denoe wo fuzzy graphs wih underlying crisp graphs G 1 *:(V 1,E 1 ) and G *:(V,E ) respecively. The normal produc of G 1 * and G * is G* = G 1 * G *: (V, E) where V = V 1 V and E = {(u 1, v 1 )(u, v ) / u 1 = u, v 1 v E or v 1 =v,u 1 u E 1 or u 1 u E 1 and v 1 v E }. Define G:(σ, μ), where σ = σ 1 σ and μ = μ 1 μ by σ(u 1, v 1 ) = σ 1 (u 1 ) σ (v 1 ), for all (u 1, v 1 ) V 1 V and 1 (u 1) (v1v ),if u1 u, v1v E u 1, v1 u, v (v 1) 1(u1 u ),if v1 v,u1u E1 1(u1u ) (v1v ),if u1u E 1, v1v E If u 1 = u, v 1 v E, σ 1 (u 1 ) μ (v 1 v ) = σ 1 (u 1 ) σ 1 (u ) μ (v 1 v ) σ 1 (u 1 ) σ 1 (u ) σ (v 1 ) σ (v ) = σ 1 (u 1 ) σ (v 1 ) σ 1 (u ) σ (v ) = σ(u 1, v 1 ) σ(u, v ) Similarly if v 1 = v, u 1 u E 1, σ (v 1 ) μ 1 (u 1 u ) σ(u 1, v 1 ) σ(u, v ) If u 1 u E 1 and v 1 v E, μ 1 (u 1 u ) μ (v 1 v ) σ 1 (u 1 ) σ 1 (u ) σ (v 1 ) σ (v ) = σ(u 1, v 1 ) σ(u, v ) Hence μ((u 1, v 1 )(u, v )) σ(u 1, v 1 ) σ(u, v ). Therefore G:(σ, μ) is a fuzzy graph. This is called he normal produc or he srong produc of he fuzzy graphs G 1 and G and is denoed by G 1 G. Example. The following Figure1 gives an example of he srong produc of wo fuzzy graphs. Figure 1: The srong produc G 1 G of G 1 and G Theorem.3: If G 1 and G are wo effecive fuzzy graphs, hen G 1 G is an effecive fuzzy graph. Le G 1 and G be effecive fuzzy graphs. Then μ 1 (u 1 u ) = σ 1 (u 1 )σ 1 (u ) for any u 1 u E 1 and μ (v 1 v ) = σ (v 1 ) σ (v ) for any v 1 v E. Therefore proceeding as in he definiion, If u 1 = u, v 1 v E, μ((u 1, v 1 )(u, v )) = σ 1 (u 1 ) μ (v 1 v ) = σ 1 (u 1 )σ 1 (u ) σ (v 1 )σ (v ) = (σ 1 (u 1 )σ 1 (u )) (σ (v 1 )σ (v )) = σ(u 1, v 1 ) σ(u, v ). Similarly, If v 1 = v, u 1 u E 1, μ((u 1, v 1 )(u, v )) = σ(u 1, v 1 ) σ(u, v ) If u 1 u E 1 and v 1 v E, μ((u 1, v 1 )(u, v )) = σ(u 1, v 1 ) σ(u, v ).

3 Inernaional Journal of Scienific and Research Publicaions 3 ISSN Hence G 1 G is an effecive fuzzy graph. Theorem.4: If G 1 and G are wo complee fuzzy graphs, hen G 1 G is a complee fuzzy graph. Le G 1 and G be complee fuzzy graphs. Then G 1 and G are effecive fuzzy graphs and G 1 * andg * are complee graphs. Therefore G 1 G is an effecive fuzzy graph by Theorem. and G 1 * G * is a complee graph. Hence G 1 G is a complee fuzzy graph. Example.5: The following Figure gives an example of he srong produc of wo effecive fuzzy graphs. Figure : The srong produc G 1 G of wo effecive fuzzy graphs G 1 and G Example.6: The following Figure 3 gives an example of he srong produc of wo complee fuzzy graphs. Figure 3: The srong produc G 1 G of wo complee fuzzy graphs G 1 and G Theorem.7: The srong produc of wo conneced fuzzy graphs is always a conneced fuzzy graph. Le G 1 :(σ 1,μ 1 ) and G :(σ,μ ) be wo conneced fuzzy graphs wih underlying crisp graphs G 1 *:(V 1,E 1 ) and G *:(V,E ) respecively. Le V 1 = {u 1, u,..,u m } and V = {v 1, v,..,v n }. The srong produc of wo conneced fuzzy graphs G 1 :(σ 1,μ 1 ) and G :(σ,μ ) can be aken as G:(σ, μ) where σ = σ 1 σ and μ = μ 1 μ. Now consider he m sub graphs of G wih he verex ses {u i v 1, u i v,..,u i v n } for i=1,,,m. Each of hese sub graphs of G is conneced since he u i s are he same and since G is conneced, each v i is adjacen o a leas one of he verices in V. Also since G 1 is conneced, each u i is adjacen o a leas one of he verices in V 1. Therefore here exiss a leas one edge beween any pair of he above m sub graphs. Hence G is a conneced fuzzy graph.

4 Inernaional Journal of Scienific and Research Publicaions 4 ISSN III TRUNCATIONS OF THE STRONG PRODUCT OF TWO FUZZY GRAPHS Theorem 3.1: (G 1 G ) () = G 1() G () and (G 1 G ) () = G () 1 G (). 1 u, v,if 1 u, v We have 1 u, v () 0,if 1 u, v Now (σ 1() σ () ) (u,v) = σ 1() (u) σ () (v) If (σ 1 σ )(u, v), hen σ 1 (u) σ (v) σ 1 (u) and σ (v) σ 1() (u) = σ 1 (u), σ () (v) = σ (v) σ 1() (u) σ () (v) = σ 1 (u)σ (v)= (σ 1 σ )(u, v). If (σ 1 σ )(u, v) <, hen σ 1 (u) σ (v) < eiher σ 1 (u) <, σ (v) or σ 1 (u), σ (v) < or σ 1 (u) <, σ (v) < σ 1() (u) =0, σ () (v) = σ (v) or σ 1() (u) = σ 1 (u), σ () (v) = 0 or σ 1() (u) = 0, σ () (v) = 0 σ 1() (u)σ () (v) = 0. 1 u, v,if 1 u, v Therefore 1() () u, v 1() u () v 0,if 1 u, v Hence (σ 1 σ ) () (u,v) = (σ 1() σ () ) (u,v) for every (u,v) V 1 V. Now 1 u 1, v1 u, v u, v u, v,if u, v u, v () 0,if u, v u, v If (σ 1 σ )(u, v), hen σ 1 (u 1 ) μ (v 1 v ) or σ (v 1 ) μ 1 (u 1 u ) or μ 1 (u 1 u ) μ (v 1 v ) Proceeding as above, we can show ha 1 u1 v 1, v,if 1 u1 v 1, v u1 1 u 1,u,if v1 1 u 1,u 1() () u 1, v1 u, v 1 u 1,u v 1, v,if 1 u 1,u v 1, v 0,if 1 u 1,u v 1, v u, v u, v,if u, v u, v 0,if u, v u, v Therefore (μ 1 μ ) () ( (u 1, v 1 )(u, v )) = (μ 1() μ () )((u 1, v 1 )(u, v )) for every edge (u 1,v 1 )(u, v ) in G 1 G. Hence (G 1 G ) () = G 1() G (). Proceeding in he same way, we can show ha (G 1 G ) () = G () 1 G (). IV. DEGREE OF A VERTEX IN THE STRONG PRODUCT OF TWO FUZZY GRAPHS The degree of any verex in he srong produc G 1 G of wo fuzzy graphs G 1 :(σ 1,μ 1 ) and G :(σ,μ ) is given by, d G G (u i, v j) 1 (u i ) (v jv ) 1(u iu k ) 1(v j) 1(u iu k ) (v jv ). This expression can be 1 ui u k,vjve uiuke 1,v j = v uiuke 1,v jve simplified using he erms of he degrees of verices in G 1 and G wih some consrains. Theorem 4.1: If G 1 :(σ 1,μ 1 ) and G :(σ,μ ) are wo fuzzy graphs such ha σ 1 μ and σ μ 1 and μ 1 μ = c (a consan), hen he degree of a verex in he srong produc of he wo fuzzy graphs G 1 :(σ 1,μ 1 ) and G :(σ,μ ) is given by, d (u,v ) d (v ) d (u ) [d (u )d (v )]c. G * * 1G i j G j G1 i G i j 1 G Le G 1 :(σ 1,μ 1 ) and G :(σ,μ ) be wo fuzzy graphs wih underlying crisp graphs G 1 *:(V 1,E 1 ) and G *:(V,E ) respecively. Suppose ha σ 1 μ and σ μ 1 and μ 1 μ = c (a consan), hen and d (u, v ) (u ) (v v ) (u u ) (v ) (u u ) (v v ) G1G i j 1 i j 1 i k j 1 i k j. ui u k,vjve uiuke 1,v j = v uiuke 1,v jve

5 Inernaional Journal of Scienific and Research Publicaions 5 ISSN d (u, v ) (v v ) (u u ) c G1G i j j 1 i k ui u k,vjve uiuke 1,v j = v uiuk E 1,v jve d (v ) d (u ) [d (u )d (v )]c. G * * j G1 i G i j 1 G Theorem 4.: If G 1 :(σ 1,μ 1 ) and G :(σ,μ ) are wo fuzzy graphs such ha σ 1 μ and σ μ 1 and μ 1 μ = C (a consan), hen he degree of a verex in he srong produc of he wo fuzzy graphs G 1 :(σ 1,μ 1 ) and G :(σ,μ ) is given by, d (u,v ) [1 d (v )]d (u ) [1 d (u )]d (v ) [d (u )d (v )]C. G * * * * 1G i j G j G 1 i G i G 1 j G i j 1 G Le G 1 :(σ 1,μ 1 ) and G :(σ,μ ) be wo fuzzy graphs wih underlying crisp graphs G 1 *:(V 1,E 1 ) and G *:(V,E ) respecively. Suppose ha σ 1 μ and σ μ 1 and μ 1 μ = C (a consan), hen and d (u, v ) (u ) (v v ) (u u ) (v ) (u u ) (v v ) G1G i j 1 i j 1 i k j 1 i k j ui u k,v jve uiuk E 1,v j = v uiuk E 1,v jve (v v ) (u u ) [ (u u ) (v v ) (u u ) (v v )] j 1 i k 1 i k j 1 i k j ui u k,v jve uiuk E 1,v j = v uiuk E1,v jve d (v ) d (u ) (u u ) (v v ) [ (u u ) (v v )] d (v ) d (u ) d (v )d (u ) d (u )d (v ) C G * * j G1 i G j G 1 i G i G 1 j * G j G1 i uiuk E 1,v jve [1 d (v )]d (u ) [1 d (u )]d (v ) [d (u )d (v )]C. G j G1 i 1 i k j 1 i k j uiuk E 1,v jve uiuk E 1,v jve uiuk E 1,v jve * * * G i G 1 j G i j 1 G Theorem 4.3: If G 1 :(σ 1,μ 1 ) and G :(σ,μ ) are wo fuzzy graphs such ha σ 1 µ and μ 1 μ = c (a consan), hen he degree of a verex in he srong produc is given by, d (u,v ) d (v ) (u ) d (u ) [d (u )d (v )]c. G * * * 1G i j G j 1 i G1 i G i j 1 G Le G 1 :(σ 1,μ 1 ) and G :(σ,μ ) be wo fuzzy graphs wih underlying crisp graphs G 1 *:(V 1,E 1 ) and G *:(V,E ) respecively. Suppose ha σ 1 μ. Then σ μ 1. This implies ha σ 1 σ. Also μ 1 μ = c (a consan). d (u, v ) (u ) (v v ) (u u ) (v ) (u u ) (v v ) G1G i j 1 i j 1 i k j 1 i k j ui u k,v jve uiuk E 1,v j = v uiuk E 1,v jve (u ) (u u ) (u u ) (v v ) d (v ) (u ) d (u ) [d (u )d (v )]c. * G 1 i 1 i k 1 i k j ui u k,vjve uiuk E 1,v j = v uiuk E 1,v jve j 1 i G * * 1 i G i j 1 G Example 4.4: If G 1 :(σ 1, μ 1 ) and G :(σ, μ ) are wo fuzzy graphs such ha σ 1 μ and μ 1 μ = c (a consan), hen heir srong produc G G : (, ) is given in he following example. 1 Figure 4: The Srong Produc of wo fuzzy graphs such ha σ 1 μ and μ 1 μ =0.3. d (u,v ) d (v ) (u ) d (u ) [d (u )d (v )]c G * * * 1G 1 1 G G1 1 G G

6 Inernaional Journal of Scienific and Research Publicaions 6 ISSN d (u,v ) d (v ) (u ) d (u ) [d (u )d (v )]c G * * * 1G G 1 G1 G 1 G V. RELATIONSHIP BETWEEN THE DIRECT SUM AND THE STRONG PRODUCT Theorem 5.1: The srong produc of wo fuzzy graphs G 1 and G is he direc sum of he Caresian produc of G 1 and G and he conjuncion of G 1 and G. From he definiions, (σ 1 σ )(u,v) = (σ 1 σ )(u,v) = (σ 1 σ )(u,v) = σ 1 (u) σ (v) for every (u,v) V 1 V. So ((σ 1 σ ) (σ 1 σ ))(u, v) = σ 1 (u) σ (v) for every (u,v) V 1 V. Hence (σ 1 σ )(u,v) = ((σ 1 σ ) ( σ 1 σ ))(u, v) for every (u,v) V 1 V. From he definiions of Caresian produc and he conjuncion, 1 u1 v 1, v,if u1 u, v1v E 1 1 u 1, v1 u, v u1 1 u 1,u,if v1 v,u1u E1 1 u 1,u v1, v,if u1u E1 and v1v E Hence G 1 G = (G 1 G ) (G 1 G ). 1 u 1,v1 u,v VI. CONCLUSION In his paper, he srong produc of wo fuzzy graphs is defined. I is proved ha when wo fuzzy graphs are effecive hen heir srong produc is always effecive and i is proved ha he srong produc of wo complee fuzzy graphs is complee. Also i is proved ha he srong produc of wo conneced fuzzy graphs is always conneced. The lower and upper runcaions of he srong produc of wo fuzzy graphs are obained. The degree of a verex in he srong produc of wo fuzzy graphs is obained. A relaionship beween he direc sum and he srong produc of wo fuzzy graphs is obained. Operaion on fuzzy graph is a grea ool o consider large fuzzy graph as a combinaion of small fuzzy graphs and o derive is properies from hose of he small ones. Through his paper, a sep in ha direcion is made. REFERENCES [1] Frank Harary, Graph Thoery, Narosa / Addison Wesley, Indian Suden Ediion, [] John N. Modeson and Premchand S.Nair, Fuzzy Graphs and Fuzzy Hypergraphs, Physica-verlag Heidelberg, 000. [3] J.N.Mordeson and C.S. Peng, Operaions on fuzzy graphs, Informaion Sciences 79 (1994), [4] Nagoorgani. A and Radha. K, Conjuncion of Two Fuzzy Graphs, Inernaional Review of Fuzzy Mahemaics, 008, Vol. 3, [5] Nagoorgani. A and Radha. K, Some Properies of Truncaions of Fuzzy Graphs, Advances in Fuzzy Ses and Sysems, 009, Vol.4, No., [6] Nagoorgani. A and Radha. K, Regular Propery of Fuzzy Graphs, Bullein of Pure and Applied Sciences, Vol.7E ( No.)008, [7] P. Bhaacharya, Some Remarks on Fuzzy Graphs, Paern Recogniion Leer 6 (1987), [8] Radha.K and Arumugam. S, On Direc Sum of Two Fuzzy Graphs, Inernaional Journal of Scienific and Research Publicaions, Volume 3, Issue 5, May 013, ISSN [9] Rosenfeld. A, (1975) "Fuzzy graphs". In: Zadeh, L.A., Fu, K.S., Tanaka, K., Shimura, M. (eds.), Fuzzy Ses and heir Applicaions o Cogniive and Decision Processes, Academic Press, New York, ISBN , pp AUTHORS Firs Auhor Dr. K. Radha, M.Sc.,M.Phil.,Ph.D., P.G & Research Deparmen of Mahemaics, Periyar E.V.R. College, Tiruchirapalli radhagac@yahoo.com Second Auhor Mr.S. Arumugam, M.Sc.,M.Phil.,B.Ed.,(Ph.D.), Gov. High School, Thinnanur, Tiruchirapalli anbu.saam@gmail.com Correspondence Auhor Dr. K. Radha, M.Sc.,M.Phil.,Ph.D., P.G & Research Deparmen of Mahemaics, Periyar E.V.R. College, Tiruchirapalli radhagac@yahoo.com