Generarlized Operations on Fuzzy Graphs

Transcription

1 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-5 (Onlne) ol3 No9 3-Specal ssue Internatonal Conference on Recent Trends n Appled Scences wth ngneerng Applcatons enerarlzed Operatons on Fuzzy raphs enugopalam Nagamurth Kumar Maya kumar wwwsteorg Abstract To dscuss the Cartesan Product Composton unon and on on Interal-alued fuzzy graphs We also ntroduce the noton of Interal-alued fuzzy complete graphs Some propertes of self complementary graph Key Words : Interal-alued fuzzy graph self complementary Interal alued fuzzy complete graphs Mathematcs Subect Classfcaton : 5C99 Introducton It s qute well known that graphs are smply model of relatons A graphs s a conenent way of representng nformaton nolng relatonshp between obects The obects are represented by ertces and realtons by edges When there s agueness n the descrptonof the obects or n ts relatonshps or n both t s natural that we need to desgn a Fuzy raph Model Applcaton of fuzzy relatons are wdespread and mportant; especally n the feld of clusterng analyss neural networks computer networks pattern recognton decson makng and expert systems In each of these the basc mathematcal structure s that of a fuzzy graph We know that a graphs s a symmetrc bnry relaton on a nonempty set Smlary a fuzzy graph s a symmetrc bnary fuzzy relaton on a fuzzy subset The frst defnton of a fuzzy graph was by Kaufmann [8] n 973 based on Zadeh s fuzzy relatons [46] But t was Azrel Rosenfeld [35] who consdered fuzzy relatons onf uzzy sets and deeloped the theory of fuzzy graphs n 975 urng the sam etme RTYeh and SYBang [44] hae also ntroduced arous connectedness concepts n fuzzy graphs Prelmnares efnton : Let be a nonempty set A fuzzy graphs s a par of functons : (σ ) where σ s a fuzzy subset of and s a symmetrc fuzzy relaton on σ e σ: [ ] and : x [ ] such that (u ) σ(u) Λ σ() for all u n We denot the underlyng (crsp) graph of : (σ ) by *:(σ* *) where σ* s referred to as the (nonempty) set of nodes and * x Note that the crsp graph s a specal case of a fuzzy graph wth each ertex and edge of hang degree of membershp We need not consder loops and we assume that s reflexe Also the underlygn set s assumed to be fnte and σ can be chosen n any manner so as to satsfy the defnton of a fuzzy graphs n all the examples efnton : The fuzzy graph H: (τ ) s called a partal fuzzy subgraph of : (σ ) f τ σ and In partcular we call H: (τ ) a fuzzy subgraph of : (σ ) f τ(u) σ(u) uτ* and (u) (u ) (u )*For any threshold t t σ {u : σ(u) t} and {(u ) x : (u ) t} Snce (u ) σ(u) Λ σ() for all u n we hae σ so that (σ ) s a graph wth ertex set σ and edge set for t[ ] Note: Let : (σ ) be a fuzzy graph If t t then (σ ) s a subgraph of (σ ) Note : Let H : (τ ) be a partal fuzzy subgraph of : (σ ) For any threshold t (τ ) s a subgraph of (σ ) efnton 3 : For any fuzzy subset τ of such that τ σ the partal fuzzy subgraph of (σ ) nduced by τ s the maxmal partal fuzzy subgraph of (σ ) that has fuzzy node set τ Ths s the partal fuzzy subgraph (τ ) where Τ(u ) τ(u) Λ (u ) for all u efnton 4 : The fuzzy graph H: (τ ) s called a fuzzy subgraph of : (σ ) nduced by P f P τ(u) σ(u) u P efnton 5 : A partal fuzy subgraph (τ ) spans the fuzzy graph (σ ) f σ τ In ths case (τ ) s called a aprtal fuzzy spannng subgraph of (σ ) Next we ntroduce the concept of a fuzzy spannng subgrph as a specal case of partal fuzzy spannng subgraph Operatons 6: raphs g ( ) are smple : no multple edges and no loops An unordered par {x y} s deonte by xy or x y Operatons 7: raphs g ( ) are smple: no multple edges and no loops An unordered par {x y} s denote by xy or x y An operaton s a permutaton on the set of graphs on : α : g h Operatons 8: raphs g ( ) are smple: no multple edges and no loops An unordered par {x y} s denote by xy or x y 5

2 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-5 (Onlne) ol3 No9 3-Specal ssue Internatonal Conference on Recent Trends n Appled Scences wth ngneerng Applcatons An operaton s a permutaton on the set of graphs on : α : g h Let Γ α α α n Be the subgroup of the symmetrc group generated by Γ {α α α n } Transtty 8 : The problem settng : en operatons Γ {α α α n } And any two graphs g h on oes there exst a composton Complement 9 : Let dges nonedges α Γ α α k α k- α such that α(g) h be the set of all -subsets {x y} ( g) C \ Neghbours : Neghbours of x N g (x) {y xy } N ( x) \ N ( x) { x} Nonneghbours of x Subgraphs : The symmetrc dfference : A B (A \ B) (B \ A) The sub graph of g nduced by : Complementng Subgraphs : enote by g 3Man Results Theorem 3 Let Proof A [ ] g A g A A I A g A and be two Interal alued Fuzzy raphs Then () () Consder the dentty map I : To proe () t s enough to proe (a) () ( ) ( ) () (b) () ( ) ( ) () ( ) ( ) (a) () ( ) ( ) by efnton 4 f f f f ( )( ) () ( ) ( ) g wwwsteorg 6

3 Mathematcal Theory and Modelng wwwsteorg ISSN (Paper) ISSN 5-5 (Onlne) ol3 No9 3-Specal ssue Internatonal Conference on Recent Trends n Appled Scences wth ngneerng Applcatons 7 f f f f (b) () f f f f f f f f (b) () f f f f f f f f To proe () t s enough to proe (a) () () (b) () () Consder the dentty map : I

4 Mathematcal Theory and Modelng wwwsteorg ISSN (Paper) ISSN 5-5 (Onlne) ol3 No9 3-Specal ssue Internatonal Conference on Recent Trends n Appled Scences wth ngneerng Applcatons 8 (a) () f f f f (a) () f f f f (b) () f f f f f f f or f (b) () f f f f f f f or f

5 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-5 (Onlne) ol3 No9 3-Specal ssue Internatonal Conference on Recent Trends n Appled Scences wth ngneerng Applcatons Theorem 3 ( ) Let Interal alued Fuzzy raphs Proof Let o where x and wwwsteorg and be two Interal alued Fuzzy raphs Then o s a strong {( u u )( u ): u u } {( u w) :( w) : w u } {( u u )( ): u u } () ( u u)( u ) ( u) ( u) ( u ) ( u) snce s strong ( u ) ( u ) ( u) ( ) ( o )( u u)( o )( u ) ( u u)( u ) ( u) ( u) ( u) ( u) ( u) ( u) ( u) ( o)( u u)( o)( u ) ( u w w ) ( w) ( u ) ( w ) ( u) ( w ) ( u) ( w) ( o )( u w)( o )( w) (( u w)( w) ) ( w) ( u ) ( w) ( u) ( w) ( w) ( o)( u w)( o)( w) u u u u ( u ) ( u) ( u ) ( u) o u u o ( u u)( ) ( u ) ( u ) ( u) ( u) ( u) ( u) ( o)( u u)( o)( ) () snce s strong snce s strong snce s strong () snce s strong snce s strong From () () () s a strong Interal alued Fuzzy raphs References [] A Nagoorgan K Radha Isomorphsm on fuzzy graphs Internatonal J Computatonal Math Sc (8) 9-96 [] A Perchant I Bloch Fuzzy morphsms between graphs Fuzzy Sets Syst 8 () [3] A Rosenfeld Fuzzy graphs Fuzzy Sets and ther Applcatons( LAZadeh KSFu MShmura ds) Academc Press New York (975) [4] AAlaou On fuzzfcaton of some concepts of graphs Fuzzy Sets Syst (999) [5] F Harary raph Theory 3rd dton Addson-Wesley Readng MA 97 9

6 Mathematcal Theory and Modelng ISSN (Paper) ISSN 5-5 (Onlne) ol3 No9 3-Specal ssue Internatonal Conference on Recent Trends n Appled Scences wth ngneerng Applcatons wwwsteorg [6] IB Turksen Interal alued fuzzy sets based on normal forms Fuzzy Sets Syst (986) 9- [7] J Hongme W Lanhua Interal-alued fuzzy subsemgroups and subgroups assocated by nteralalued suzzy graphs 9 WRI lobal Congress on Intellgent Systems [8] JM Mendel ncertan rule-based fuzzy logc systems: Introducton and new drectons Prentce-Hall pper Saddle Rer New Jersey [9] JM Mendel X ang Fast computaton of centrods for constant-wdth nteral-alued fuzzy sets Fuzzy Informaton Processng Socety NAFIPS (6)6-66 [] JN Mordeson CS Peng Operatons on fuzzy graphs Informaton Sc 79 (994) 59-7 [] JN Mordeson Fuzzy lne graphs Pattern Recognton Letter 4 (993) [] JN Mordeson PS Nar Fuzzy graphs and fuzzy hypergraphs Physca erlag Hedel berg 998; Second dton [3] KP Huber MR Berthold Applcaton of fuzzy graphs for metamodelng Proceedngs of the I Conference [4] KR Bhutan A Battou On M-strong fuzzy graphs Informaton Sc 55 (3) 3-9 [5] KR Bhutan A Rosenfeld Strong arcs n fuzzy graphs Informaton Sc 5 (3) 39-3 [6] KR Bhutan On automorphsm of fuzzy graphs Pattern Recognton Letter 9 (989) 59-6 [7] KT Atanasso Intutonstc fuzzy sets: Theory and applcatons Studes n fuzzness and soft computng Hedelberg New York Physca-erl 999 [8] LA Zadeh Fuzzy sets Informaton Control 8 (965) [9] LA Zadeh Smlarty relatons and fuzzy orderngs Informaton Sc 3 (97) 77- [] LA Zadeh The concept of a lngustc and applcaton to approxmate reasonng I Informaton Sc 8 (975) [] M Akram KH ar eneralzed fuzzy K-algebras M erlag pp88 ISBN [] MAkram Fuzzy Le deals of Le algebras wth nteral-alued membershp functons Quasgroups Related Systems 6 (8) - [3] MB orzalczany A method of nference n approxmate reasonng based on nteral-alued fuzzy sets Fuzzy Sets Syst (987) -7 [4] MB orzalczany An nteral-alued fuzzy nference method some basc propertes Fuzzy Sets Syst 3 (989) 43-5 [5] MK Roy R Bswas I- fuzzy relatons and Sanchez s approach for medcal dagnoss Fuzzy Sets Syst 47 (99) [6] MS Suntha A ayakumar Complement of a fuzzy graph Indan J Pure Appl Math 33 () [7] P Bhattacharya Some remarks on fuzzy graphs Pattern Recognton Letter 6 (987) 97-3 [8] S Mathew MS Suntha Node connectty and arc connectty of a fuzzy graph Informaton Scences 8(4)() 59-53

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