PROPERTIES OF BIPOLAR FUZZY GRAPHS

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1 Internatonal ournal of Mechancal Engneerng and Technology (IMET) Volume 9, Issue 1, December 018, pp , rtcle ID: IMET_09_1_056 valable onlne at aeme.com/jmet/ssues.asp?typeimet&vtype 9&IType1 ISS rnt: and ISS Onlne: IEME ublcaton Scopus Indexed ROERTIES OF IOR FZZY SOFT GRHS M.Rajeshwar Research scholar Reva nversty, ssstant rofessor n Mathematcs, School of Engneerng, resdency nversty, angalore R. Murugesan rofessor, Department of Mathematcs, Reva nversty, angalore K.. Venkatesh rofessor of Mathematcs and Comp. Scence, Myanmar Insttute of Informaton Technology, Myanmar STRCT polar Fuzzy sets and polar soft sets are two unalke soft computng procedure for depct vagueness and uncertanty. We relate ths soft computng models n fuson to study vagueness and uncertanty n bpolar graphs. We present the syllabary of bpolar fuzzy soft graphs, strong bpolar fuzzy soft graphs, complete bpolar fuzzy soft graphs, regular bpolar fuzzy soft graphs, and explore some of ther propertes. Keywords: polar Fuzzy Soft Graphs, Strong polar Fuzzy Soft Graphs, Complete polar Fuzzy Soft Graphs, Regular polar Fuzzy Soft Graphs. Cte ths rtcle: M.Rajeshwar, R. Murugesan and K.. Venkatesh, ropertes of polar Fuzzy Soft Graphs, Internatonal ournal of Mechancal Engneerng and Technology, 9(1), 018, pp et/ssues.asp?typeimet&vtype9&itype e1 1. ITRODCTIO In 1975[1] Rosenfeld ntroduced fuzzy graph theory. Durng the same tme varous concepts n connectedness wth fuzzy graphs was ntroduced by Yeh and ang [8]. agoorgan and Radha [9] ntroduced the degree of a vertex n some fuzzy graphs. The noton of soft sets was adopted by Molodtsov,[10].for descrbng the uncertanty n mathematcs the new tool s found and ts called as soft set In 199, Zhang [11] ntated the concept of bpolar fuzzy sets as a generalzaton of fuzzy sets. polar fuzzy sets are extenson of fuzzy sets whose range of membershp degree s [1,1].The concept of bpolar fuzzy graphs was ntroduced by kramn 011,also defned dfferent operatons. IMET/ndex.asp 57 edtor@aeme.com

2 ropertes of polar Fuzzy Soft Graphs In ths paper the bpolar fuzzy soft graph s explaned and also the strong, complete, regular and totally regular bpolar fuzzy soft graph.. REIMIRIES Defnton.1[]: et be an ntal unverse set and be the set of parameters. et () denotes the power set of. par (, ) s called a soft set over where s a mappng gven by (). Defnton. []: et be an ntal unverse set and E be the set of parameters. et. par (, ) s called fuzzy soft set over where s a mappng gven by F : I, where I denotes the collecton of all fuzzy subsets of. Defnton.[]: et be a unverse, Ea set of parameters and E. Defne F : F, where F sthe collecton of all bpolar fuzzy subsets of. Then (F, ) s sad to be a bpolar fuzzy soft set over a unverse. It s defned by ( F, ) F F e { + ( e ) c, ( c ), µ ( c ): ( c ), ( e ) µ. Defnton.[]: n ntersecton of two bpolar fuzzysoft sets (F, )and (G, )s a bpolar fuzzy soft set(h, C), where C φ and H : C F s defned by H(e)F(e) G(e), e Cand denoted by (H, C)(F, ) (G, ) Defnton.5[]: non of two bpolar fuzzy soft sets over a common unverse s a bpolar fuzzy soft set (H, C), where C and H : C F s defned by H(e)F(e)f e \ G(e)f e \ F(e) G(e)f e F, G, H, C Denoted as Defnton.6[1]: fuzzy graph wth V as the underlyng set s a par of functons G (σ, μ) where σ : V [0,1] s a fuzzy subset and μ : V x V [0,1] s a symmetrc fuzzy relaton on the fuzzy subset σ for all u,v V such that μ(u,v) σ(u) Λ σ(v). The underlyng crsp graph G ( σ, µ ) of s denoted by G (V, E) where E V x V. fuzzy relaton can also be expressed by a matrx called fuzzy relaton matrx M a ] where a µ u, u ) [ j j ( j Defnton.7[5]: bpolar fuzzy graph G(V,,) s a nonempty set V together wth a par of functons ( µ, µ ) : V [0,1] [ 1,0] and ( µ, µ ) : V V [0,1] [ 1,0] such that for all x, y V, µ µ ( x, y) mn µ ( x), µ ( y) ( x, y) max µ ( x), µ ( y) Defnton.8: et G(V,,) be the bpolar fuzzy graph, then the order of bpolar fuzzy graph s defned as ( G) µ, O µ v V v V IMET/ndex.asp 58 edtor@aeme.com

3 M.Rajeshwar, R. Murugesan and K.. Venkatesh Defnton.9: et G(V,,) be the bpolar fuzzy graph, then the sze of bpolar fuzzy graph s defned as S, u v u v ( G) µ ( u, v), µ ( u v) Defnton.10: et G(V,,) be the bpolar fuzzy graph, then the strong bpolar fuzzy graph s defned as µ µ ( x, y) mn µ ( x), µ ( y) ( x, y) max µ ( x), µ ( y).man result Defnton.1: n bpolar fuzzy soft graph G ( G,,, ) s such that,, ξ a) G (, ) s a smple graph b) s a nonempty set of parameters c) (, ), d) (, ) s a bpolar fuzzy soft set over s a bpolar fuzzy soft set over,ξ e) (, ),, ξ ζ p Example.: s a bpolar fuzzy (sub)graph of { ( a)( xy) mn ( a)( x), ( a)( y) ( a)( xy) max ( a)( x), ( a)( y) { a ; x y V, The bpolar fuzzy soft graph s denoted by ( a) σ G for all. That s Consder the bpolar fuzzy graph e 1, e, e, (, ) and (, ) s a bpolar fuzzy soft set over and E respectvely, wth bpolar,,ξ fuzzy approxmaton functon F : and : F s defned as,,ξ e a / 0., 0., a / 0.5, 0., a / 0.6, 0.5 follows { ( e ) { a /( 0., 0. ), a /( 0., 0.), a /( 0., 0.), 1 ( e ) { a /( 0., 0.1 ), a /( 0., 0.), a /( 0., 0.), 1 ( e ) { a a /( 0., 0.1 ), a a /( 0., 0.), a a /( 0., 0.) 1 ( e ) { a a /( 0.1, 0.), a a /( 0., 0.), a a /( 0., 0.1), 1 1. G.The parameter set s denoted by { IMET/ndex.asp 59 edtor@aeme.com

4 ropertes of polar Fuzzy Soft Graphs ( e ) { a a /( 0.1, 0.01 ), a a /( 0., 0.1 ), a a /( 0., 0.1), 1 1 Thus, ( e ) ( e ), ( e ) Fgure 1: polar fuzzy subgraphs, ( e ) ( ( e ), ( e )) and ( e ) ( ( e ), ( e )) are σ 1, 1, ξ 1 σ,, ξ σ bpolar fuzzy soft graphs of G. Defnton.: n bpolar fuzzy soft graph G ( G,,, ) bpolar fuzzy soft graph s defned as O ( G) ( e )( a) e a V,.,, ξ Defnton.: n bpolar fuzzy soft graph G ( G,,, ) fuzzy soft graph s defned as S ( G) ( e )( ab) e Example.5: a V,.,, ξ, then the order of, then the sze of bpolar,, ξ Consder the bpolar fuzzy graph G.The parameter set s denoted by { e 1,e,, and (, ) s a bpolar fuzzy soft set over and E respectvely, wth bpolar fuzzy,ξ approxmaton functon F : and : F s defned as follows,,ξ e a / 0., 0., a / 0,0, a / 0., 0. { ( e ) { a /( 0.1, 0. ), a /( 0., 0.), a /( 0., 0.), 1 ( e ) { a a /( 0,0), a a /( 0,0), a a /( 0., 0.) 1 ( e ) { a a /( 0.1, 0.), a a /( 0., 0.1 ), a a /( 0.1, 0.1), 1 1, IMET/ndex.asp 50 edtor@aeme.com

5 M.Rajeshwar, R. Murugesan and K.. Venkatesh Thus, ( e ) ( e ), ( e ) Fgure : polar fuzzy subgraphs and ( e ) ( ( e ), ( e )) are bpolar σ 1, 1, ξ 1 σ fuzzy soft graphs of G. O. The order of bpolar fuzzy soft graph s ( G) ( 1., 1.) S. The sze of bpolar fuzzy soft graph s ( G) ( 0.6, 0.8) Defnton.6: n bpolar fuzzy soft graph G ĩs a strong bpolar fuzzy soft graph f ( e) s a strong bpolar fuzzy soft graph ).e., p ( e)( ab) mn ( e)( a), ( e)( b), ( e)( ab) max ( e)( a), ( e)( b) Example.7: ξ,, ξ σ ( ab) E. Consder the bpolar fuzzy graph G.The parameter set s denoted by { e 1,e,, and (, ) s a bpolar fuzzy soft set over and E respectvely, wth bpolar fuzzy,ξ approxmaton functon F : and : F s defned as follows,,ξ e a / 0., 0., a / 0.5, 0., a / 0.6, 0.5, a / 0.7, 0.6 { ( e ) { a /( 0., 0. ), a /( 0., 0.), a /( 0., 0.), a /( 0.5, 0.), 1 ( e ) { a a /( 0., 0.1 ), a a /( 0., 0.), a a /( 0., 0. ), a a /( 0., 0.1) 1 ( e ) { a a /( 0.1, 0.), a a /( 0., 0.), a a /( 0., 0.1 ), a a /( 0., 0.), 1 1, IMET/ndex.asp 51 edtor@aeme.com

6 ropertes of polar Fuzzy Soft Graphs Thus, ( e ) ( e ), ( e ) Fgure : Strong bpolar fuzzy soft graph and ( e ) ( ( e ), ( e )) are strong σ 1, 1, ξ 1 σ bpolar fuzzy soft graphs of G. Defnton.8: n bpolar fuzzy soft graph G ĩs a complete bpolar fuzzy soft graph f σ ( e) s a complete bpolar fuzzy soft graph ).e., p ( e)( ab) mn ( e)( a), ( e)( b), ( e)( ab) max ( e)( a), ( e)( b) Example.9: ξ,, ξ a, b V. Consder the bpolar fuzzy graph G.The parameter set s denoted by { e 1,e,, and (, ) s a bpolar fuzzy soft set over and E respectvely, wth bpolar fuzzy,ξ approxmaton functon F : and : F s defned as follows,,ξ e a / 0., 0., a / 0.5, 0., a / 0.6, 0.5, a / 0.7, 0.6 { ( e ) { a /( 0., 0. ), a /( 0., 0.), a /( 0., 0.), a /( 0.5, 0.), 1 ( e ) { aa /( 0., 0.1, ) a a /( 0., 0., ) a a /( 0., 0., ) aa /( 0., 0.1, ) aa /( 0.5, 0., ) a a /( 0., 0. ), ξ, ( e ) { a a /( 0.1, 0. ), a a /( 0., 0. ), a a /( 0., 0.1, ) a a /( 0., 0. ), a a /( 0., 0.1, ) a a /( 0., 0. ), ξ IMET/ndex.asp 5 edtor@aeme.com

7 M.Rajeshwar, R. Murugesan and K.. Venkatesh Thus, ( e ) ( e ), ( e ) Fgure : Complete bpolar fuzzy soft graph and ( e ) ( ( e ), ( e )) are complete σ 1, 1, ξ 1 σ bpolar fuzzy soft graphs of G., Defnton.10: n bpolar fuzzy soft graph G ( G,,, ) σ ( e),, ξ, ξ on G (, ). If s a regular bpolar fuzzy graph of degree t for all e, then G s a t-regular bpolar fuzzy soft graph. Example.11: Consder the bpolar fuzzy graph e 1, e, e, e, (, ) and (, ) s a bpolar fuzzy soft set over and E respectvely, wth bpolar,,ξ fuzzy approxmaton functon F : and : F s defned as,,ξ e a / 0., 0., a / 0.5, 0., a / 0.6, 0.5, a / 0.7, 0.6 G.The parameter set s denoted by { follows { ( e ) { a /( 0., 0. ), a /( 0., 0.), a /( 0., 0.), a /( 0.5, 0.), 1 ( e ) { a /( 0.1, 0.), a /( 0., 0. ), a /( 0., 0.), a /( 0., 0.1), 1 ( e ) { a /( 0.5, 0.), a /( 0., 0.1 ), a /( 0., 0.1 ), a /( 0., 0.), 1 ( e ) { a a /( 0., 0.1 ), a a /( 0., 0.), a a /( 0., 0.1 ), a a /( 0., 0.) 1 ( e ) { a a /( 0.1, 0.), a a /( 0., 0.), a a /( 0.1, 0.), a a /( 0., 0.), 1 1 ( e ) { a a /( 0.1, 0.1 ), a a /( 0., 0.1 ), a a /( 0.1, 0.1 ), a a /( 0., 0.1), 1 1 ( e ) { a a /( 0., 0.1 ), a a /( 0.1, 0.), a a /( 0., 0.1 ), a a /( 0.1, 0.), IMET/ndex.asp 5 edtor@aeme.com

8 ropertes of polar Fuzzy Soft Graphs Thus, σ σ ( e ) ( e ), ( e ) 1 Fgure 5: Regular bpolar fuzzy soft graph ( 1 1 ), ( e ) ( ( e ), ( e )),,, ξ σ,, ξ ( e, e ) and ( e ) ( ( e ), ( e )) ( e ),, ξ σ,, ξ are t-regular bpolar fuzzy soft graphs of G. Defnton.1: n bpolar fuzzy soft graph G ( G,,, ) σ ( e) on,, ξ G (, ). If s a totally regular bpolar fuzzy graph of degree t for all e, then G s a t-totally regular bpolar fuzzy soft graph. Example.1: Consder the bpolar fuzzy graph G.The parameter set s denoted by { e 1,e,, and (, ) s a bpolar fuzzy soft set over and E respectvely, wth bpolar fuzzy,ξ approxmaton functon F : and : F s defned as follows,,ξ e a / 0.5, 0., a / 0., 0., a / 0.1, 0., a / 0., 0. { ( e ) { a /( 0., 0.), a /( 0., 0.), a /( 0., 0.), a /( 0., 0.5), 1 ( e ) { a a /( 0.1, 0.1 ), a a /( 0., 0.), a a /( 0., 0.1 ), a a /( 0., 0.) 1, IMET/ndex.asp 5 edtor@aeme.com

9 M.Rajeshwar, R. Murugesan and K.. Venkatesh ( e ) { a a /( 0.1, 0.), a a /( 0., 0.5), a a /( 0,0), a a /( 0., 0.), 1 1 Hence t deg a 0.8, 0.6, t deg a 0.8, 0.6, t deg a 0.8, 0.6 and t deg a Therefore ( e1 ) s totally regular bpolar fuzzy soft graph. lso ς σ ( 0.8, 0.6) 1, t deg( a1 ) ( 0.7, 0.9), t deg( a ) ( 0.7, 0.9), t deg( a ) ( 0.7, 0.9) and t deg( a ) ( 0.7, 0.9) and ( e ) ς, σ s totally regular bpolar fuzzy soft graph. Thus ( e ) ( e ), ( e ) Fgure 6: Totally regular bpolar fuzzy soft graph, ( e ) ( ( e ), ( e )). σ 1, 1, ξ 1 σ ut ( e1 ) and ς e, σ ς, σ s not regular bpolar fuzzy soft graph. So G s not regular bpolar fuzzy soft graph. Theorem.1: et G be a bpolar fuzzy soft graph of graph and C,, ξ s a constant functon n bpolar fuzzy graph.then G s a totally regular bpolar fuzzy soft graph. roof. Suppose that G s a regular bpolar fuzzy soft graph and C Then C ( e )( a) ( c, c ) where c ( 0,1] [ 1,0) and c a t, t deg n bpolar fuzzy graphs t deg( a) deg( a) + C ϑ,λ ( e )( a) ( t, t ) + ( c, c ) ( t + c ), ( t + c ) Hence G s a totally regular bpolar fuzzy soft graph. σ e G. If G s a regular bpolar fuzzy soft e ς, σ of G for all e s a constant functon. a V, e and, e and a V. Snce Theorem.15: et G be a bpolar fuzzy soft graph of G. If G s both regular and totally regular bpolar fuzzy soft graph and C s a constant functon n bpolar fuzzy graph IMET/ndex.asp 55 edtor@aeme.com

10 ropertes of polar Fuzzy Soft Graphs σ graph. e of G for all e.then G s both regular and totally regular bpolar fuzzy soft roof. et G be both regular and totally regular bpolar fuzzy soft graph. Then deg ( a ) ( t, t ) and t deg ( a) ( s, s ) n bpolar fuzzy subgraphs ( e ) for all e ς, σ ( s, s ) deg( a) + C ( e )( a) ( s, s ) ( t, t ) + C ( e )( a) and for all a V. Then ( s, s ) ( t, t ) C ( e )( a) s t, s t C e a n σ e for all e and for all a V.. COCSIO In ths paper the bpolar fuzzy soft graph s explaned and also the order, sze, strong, complete, regular and totally regular bpolar fuzzy soft graph was explaned wth examples and t has a broad mplementaton n the area n data processor. REFERECE [1]. Rosenfeld, Fuzzy graphs, n:.. Zadeh, K.S. Fu, K. Tanaka and M. Shmura,(edtors), Fuzzy sets and ts applcaton to cogntve and decson process, cademc press, ew York (1975) pp []. ygünoglu, H. ygün, Introducton to fuzzy soft groups, Computers and Mathematcs wth pplcatons 58(009), [].K.Maj,.R.Roy, R.swas, On ntutonstc fuzzy soft sets, The journal of fuzzy mathematcs 1(00), [] Saleem bdullah, Muhammad slam and Kfayat llah,polar Fuzzy Soft sets and ts applcatons n decson makng problem,ournal of Intellgent and Fuzzy, March 01 [5] T. athnathan and. esntha Roslne. 01. Matrx Representaton of Double layered fuzzy graph and ts propertes, nnals of ure and ppled Mathematcs, Vol. 8, o., pp [6] Muhammad kram, Sara awaz, on fuzzy soft graphs, Italan journal of pure and appled mathematcs, (015), [7] Xu, W., Ma,., Wang, S., Hao, G., Vague soft sets and ther propertes, Computers and Mathematcs wth pplcatons 59(010), [8] R.T.Yeh and S.Y.ang, Fuzzy relatons, fuzzy graphs and ther applcatons to clusterng analyss, n:.. Zadeh, K.S. Fu, K. Tanaka and M. Shmura,(edtors), Fuzzy sets and ts applcaton to cogntve and decson process, cademc press, ew York (1975) [9].agoorgan and K.Radha, The degree of a vertex n some fuzzy graphs, Inter. ournal of lgorthms, Computng and Mathematcs, () (009) [10] D.Molodtsov, Soft sett heory frst results, Computers and Math- 577 ematcs wth pplcatons 7 (1999), IMET/ndex.asp 56 edtor@aeme.com