Some Concepts on Constant Interval Valued Intuitionistic Fuzzy Graphs

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1 IOS Journal of Mathematcs (IOS-JM) e-issn: , p-issn: X. Volume, Issue 6 Ver. IV (Nov. - Dec. 05), PP Some Concepts on Constant Interval Valued Intutonstc Fuzzy Graphs S. Yahya Mohamed and. Jahr Hussan, P.G and esearch Department of Mathematcs, Govt. rts College, Truchrappall-60 0, Inda. P.G and esearch Department of Mathematcs, Jamal Mohamed College, Truchrappall-60 00, Inda. bstract: The fundamental characterstc of the IVIFS s that the values of ts membershp functon and nonmembershp functon are ntervals rather than exact numbers. In ths paper, we defne degree, order and sze of IVIFGs. lso constant Interval valued ntutonstc fuzzy graphs and totally constant IVIFGs are ntroduced and dscussed some propertes of constant IVIFG. Keywords: Intutonstc fuzzy graph, Interval valued ntutonstc fuzzy set, Interval valued ntutonstc fuzzy graph, Strong IVIFG, Constant IVIFG, totally constant IVIFG. MS Mathematcs Subect Classfcaton (00): 03E7, 05C69, 05C7. I. Introducton: Fuzzy set (FS) was ntroduced by Zadeh [6] n 965, as a generalzaton of crsp sets. The concept of ntutonstc fuzzy sets (IFSs), as a generalzaton of fuzzy set was ntroduced by K. tanassov [] and defned new operatons on ntutonstc fuzzy graphs. ater, K. ttannasov and G. Gargov [3] ntroduced the Interval valued ntutonstc fuzzy sets (IVIFSs) theory, as a generalzaton of both nterval valued fuzzy sets(ivfss) and ntutonstc fuzzy sets (IFSs). Hence, many scholars appled IFS and IVIFS to decson analyss and pattern recognton wdely. By ntroducng the degree of membershp M (x), the degree of non-membershp N (x) and the degree of hestancy H (x), the IFS theory and IVIFS theory establshed. ccordng to the IVIFS defnton, M (x), N (x) and H (x) are ntervals, where M (x) denotes the range of support party, N (x) denotes the range of opposton party and H (x) denotes the range of absent party. tannasov s IVIFS s based on pont estmaton, whch means that these ntervals can be regarded as estmaton result of an experment. K. tanassov [] n 994 ntroduced some operatons over nterval valued fuzzy sets (IVFSs). In 000, Some Operatons on ntutonstc fuzzy sets were proposed by De S. K., Bswas. and oy. [6]. In 006, some operators defned over Interval-valued ntutonstc fuzzy sets were proposed by N. K. Jana and M. Pal [8]. In 008, propertes of some IFS operators and operatons were proposed by Q. u, C. Ma and X. Zhou [9]. In 0, Intutonstc fuzzy sets: Some new results were proposed by. K. Verma and B. D. Sharma []. In 04, n Interval-valued Intutonstc Fuzzy Weghted Entropy (IVIFWE) Method for Selecton of Vendor was proposed by D. Ezhlmaran and S. Sudharsan [7].. Nagoor Gan and S. Satha Begum [] defned degree, Order and Sze n ntutonstc fuzzy graphs and extend the propertes. In 04, M. Ismayl[0] et.al. ntroduced the noton of strong IVIFG and dscussed some operatons. In ths paper, we defne the degree, order and sze of IVIFGs wth some results. lso, Constant IVIFGs and totally constant IVIFGs are ntroduced, and some propertes are analyzed. II. Prelmnary Defnton.: fuzzy graph G = (σ,µ) s a par of functons σ : V [ 0,] and µ : V x V [ 0,], where for all u, v ЄV, we have µ(u,v) σ(u) Λ σ(v). Defnton.: n ntutonstc fuzzy graph s of the form G = (V, E ), where () V={v,v,,v n } such that μ : V[0,]and γ : V [0,] denote the degree of membershp and nonmembershp of the element v V, respectvely, and 0 μ (v ) + γ (v ) for every v V, ( =,,, n), () E V x V where μ : V x V [0,] and γ : V x V [0,] are such that μ (v, v ) mn [μ (v ), μ (v )] and γ (v,v ) max [γ (v ), γ (v ) ] and 0 μ (v, v ) + γ (v, v ) for every (v, v ) E,, =,,, n. Defnton.3: n edge e = (x, y) of an IFG G = ( V, E ) s called an effectve edge f μ (x, y) = μ (x) Λ μ (y) and γ (x, y) = γ (x) V γ (y). Defnton.4: n Intutonstc fuzzy graph s complete f μ = mn ( μ, μ ) and γ = max (γ,γ ) for all ( v, v ) Є V. DOI: / Page

2 Some Concepts on Constant Interval Valued Intutonstc Fuzzy Graphs Defnton.5: n Intutonstc fuzzy graph G s sad to be strong IFG f μ (x, y) = μ (x) Λ μ (y) and γ (x, y) = γ (x) V γ (y) for all ( v, v ) Є E. That s every edge s effectve edge. Defnton.6: n Interval valued ntutonstc fuzzy set n the fnte unverse X s defned as = { x,[μ (x), γ (x)] x X }, where μ : X [0,] and γ :X [0,] wth the condton 0 sup(μ (x)) + sup(γ (x)), for any x X. The ntervals μ (x) and γ (x) denote the degree of membershp functon and the degree of non-membershp of the element x to the set. For every x X, μ (x) and γ (x) are closed ntervals and ther eft and ght end ponts are denoted by (x), (x) and (x), (x). et us denote = { [x, ( (x), (x)), ( (x), (x))] x X}where 0 (x)+ (x), (x) 0, (x) 0. We call the nterval [ (x) (x), (x) (x)], abbrevated by [ (x), (x)] and denoted by π (x), the nterval-valued ntutonstc ndex of x n, whch s a hestancy degree of x to. Especally f μ (x) = (x) = (x) and γ (x) = (x) = (x) then the gven IVIFS s reduced to an ordnary IFS III. Interval Valued ntutonstc fuzzy graph Defnton 3.: n nterval valued ntutonstc fuzzy graph (IVIFG) wth underlyng set V s defned to be a par G = (μ, γ), Where () μ : VD[0,]and γ : V D[0,] denote the degree of membershp and non-membershp of the element v V, respectvely, and 0 ( v) ( v) for every v V, ( =,,, n), () The functon µ : E V x V D[0,] and γ : E V x V D[0,] where ( v, ) mn[ ( ), v v ( v )] and ( v, ) mn[ ( ), v v ( v, v ) Max[ ( v ), ( v )] and ( v, ) [ ( ), v Max v such that 0 ( v, v ) ( v, v ) for every (v, v ) E,, =,,, n. Example 3.: Fg -: Interval Valued Intutonstc Fuzzy Graph (IVIFG) Defnton 3.3: n IVIFG s called Strong Interval valued ntutonstc fuzzy graph (SIVIFG) f ( v, ) mn[ ( ), v v v v v v ( v, v ) Max[ ( v ), ( v )] (, ) mn[ ( ), ( )], ( v, v ) Max[ ( v ), ( v )] Defnton 3.4: et G = (V, E ) be an IVIFG. Then the degree of a vertex v s defned by d(v) = (d μ (v), d γ ) and d γ (v) = ( v, u, v, u ) where d μ (v) = ( v, u, v, u uv uv Defnton 3.5: The total degree of a vertex v n IVIFG s defned as td μ (v) = uv uv and uv t(v) = [(td μ (v), td γ ], where and ( v, u (v), v, u uv. td γ (v) = ( (v, u) (v), v, u uv uv Defnton 3.6: et G be an IVIFG and all vertces are of same degree then G s sad to be regular IVIFG. DOI: / Page

3 Defnton 3.7: et G be an IVIFG, then the order of G s defned to be [( v, v ),( v v )] O(G) = vv vv vv vv Some Concepts on Constant Interval Valued Intutonstc Fuzzy Graphs Defnton 3.8: et G be the IVIFG, then the sze of G s defned to be [( v, u, v, u ),( v, u, v, u )] S(G) = uv uv uv uv Defnton 3.9: In any IVIFG G, d ( v) [( v, u, v, u ),( v, u, v, u )] G = S(G). vv uv uv uv uv IV. Constant Interval Valued Intutonstc Fuzzy Graph Defnton 4. : et G be an IVIFG. If d ( v ) ( K, K ) and d ( v ) ( K, K ) for all v V the graph G s constant IVIFG wth degree [( K, K ),( K, K )]. Defnton 4.: et G be an IVIFG. If td ( v ) = (c, c ) and td ( v ) v V s called totally constant IVIFG wth degree [(c, c ), (c, c )] = (c, c ) for all, then, then the IVIFG Theorem 4.3: et G be the IVIFG and f = and = for all vertces f and only f the followng are equvalent () G s constant IVIFG () G s totally constant IVIFG et = = a, and = = b for all vertces, where a and b are constants. ssume that G s constant IVIFG wth constant degree [(c, c ), (c, c )].Then d ( v ) =(c, c ), d ( v ) =(c, c ) for all v V. So td ( v ) = (c + a, c +b), td ( v ) = (c + a, c +b) for all v V. Hence G s totally constant IVIFG. Thus () () s proved. Now, suppose that G s totally constant IVIFG, then td ( v ) = (k, k ), td ( v ) = (k, k ) for all v V. We get d ( v ) +(a, b) = (k, k ) d ( v ) = (k - a, k - b) and smlarly, d ( v ) = (k - a, k - b). So G s constant IVIFG. Thus () () s proved. Hence () and () are equvalent. Conversely, ssume G s both constant IVIFG and Totally constant IVIFG. Suppose and for at least one par of vertces say v, v V. et G be constant IVIFG, then d ( v ) = d ( v) = (c,c ). So td ( v ) = (c, c ) + ( ( v ), ( v )) = (c ( v ), c ( v ) ). Smlarly td ( v ) = (c ( v ), c ( v ) ) lso, td ( v ) = (c, c ) + ( ( v ), ( v )) = (c ( v ), c ( v ) and td ( v) = (c ( v ), c + ( v) ). Snce and, then td ( v ) td ( v), td ( v ) td ( v ). So G s not totally constant IVIFG, whch s a contradcton to our assumpton. Now, we take G s totally constant IVIFG, we get d ( v ) d ( v), d ( v ) d ( v). So, G s not constant IVIFG, whch s a contradcton to our assumpton. Hence, we get, = and =. Theorem 4.4: The sze of a constant IVIFG wth degree [( k, k),(( k, k )] s where p s order of IVIFG. pk pk pk pk [(, ),(, )] The sze of IVIFG s S(G) = [( v, u, v, u ),( v, u, v, u )] uv uv uv uv Snce G s [( k, k),(( k, k )] - regular IVIFG. Then, d ( v) ( k, k), d ( v) ( k, k) for all v ϵ V. DOI: / Page

4 We have d d d d (( v, v ),( v, v )) vv vv vv vv Some Concepts on Constant Interval Valued Intutonstc Fuzzy Graphs = S(G) = [( v, u, v, u ),( v, u, v, u )] uv uv uv uv = pk pk pk pk That s, S(G) = [( k, k),( k, k)] vv vv vv vv [(, ),((, )] pk pk pk pk Therefore, S(G) = [(, ),(, )] Theorem 4.5: et G be an IVIFG and G s totally constant IVIFG wth [(c,c )(c,c )] as total degree for all vertces, then S(G) + O(G) = [(pc,pc )(pc,pc )]. Where p = O(G). Gven G s [(c,c )(c,c )]- totally constant IVIFG,. then (c,c ) = ( v, u (v), v, u uv uv lso (c,c ) = ( (v, u) (v), v, u uv Therefore, [( ( c, c ), ( c, c ))] = vv vv uv [( v, u (v), v, u,( v, u (v), v, u ] vv uv uv uv uv [( v, u, v, u ) ( (v),, vv uv uv vv vv = ( (v, u), v, u ) ( (v), ]. vv uv uv vv vv That s [(pc, pc ),(pc,pc )] = S(G) + O(G) where p = O(G) Theorem 4.6: et G be Strong IVIFG, and are constant functons for all vertces then and are also wth same constant functons for all edges, but the converse need not true. Suppose, = c and = c for all the vertces. Snce gven s strong IVIFG, we get ( v, ) mn[ ( ), v v, ( v, v ) mn[ ( v ), ( v )] and ( v, ) [ ( ), v Max v, ( v, v ) Max[ ( v ), ( v )] So, we get = c and = c for all the edges. Conversely, for the edges, the maxmum s taken from ether two vertces and then the maxmum value may be dffer n the one of the vertces and = c, but may not equal to c for all vertces. lso + may not equal to c for all vertces. Therefore, the converse need not true. Example 4.7: Fg -: Strong IVIFG wth and are constant functons DOI: / Page

5 Some Concepts on Constant Interval Valued Intutonstc Fuzzy Graphs Here, = 0.5 and = 0.9 for all vertces, Then we get for all edges, = 0.9. Example 4.8: = 0.5 and Fg -: Strong IVIFG wth and are constant functons Here, = 0.5 and = 0.9 for all edges, but 0.5 and 0.9 for all vertces. Proposton 4.9: For an Interval valued ntutonstc fuzzy graph G, It s mpossble that and are equal for any vertces, lso and are not also equal for any edges. In any IVIFG, membershp and non-membershp values are ntervals and n the form < and < respectvely. So, and are not equal for any ntervals. Smlarly, for the edges also we get and are not equal. V. Concluson Here we defned order and sze of IVIFG and verfy the property that sum of degrees of vertces s equal to twce the sze of the graph. lso, constant IVIFGs and totally constant IVIFGs are defned and some results are proved whch s very rch both n theoretcal developments and applcatons. In future, more theorems and results of IVIFGs wll be proposed. eferences []. tanassov. K, Intutonstc fuzzy sets. Fuzzy Sets & Systems, 0 (986), []. tanassov. K, Operatons over nterval valued fuzzy set, Fuzzy Sets & Systems, 64 (994), [3]. tanassov. K and Gargov. G, Interval-valued ntutonstc fuzzy sets, Fuzzy Sets & Systems. 3 (989), [4]. tanassov. K. More on ntutonstc fuzzy sets. Fuzzy Sets & Systems, 33 (989), Some new denttes 739. [5]. tanassov. K. New operatons defned over the ntutonstc fuzzy sets. Fuzzy Sets & Systems, 6 (994), [6]. De. S. K., Bswas. and oy., Some Operatons on ntutonstc fuzzy sets, Fuzzy Sets & Systems, 4 (000), [7]. Ezhlmaran. D and Sudharsan. S, n Interval-valued Intutonstc Fuzzy Weghted Entropy (IVIFWE) Method for Selecton of Vendor, Internatonal Journal of ppled Engneerng esearch, 9 () (04), [8]. Jana. N.K and Pal. M. Some operators defned over Interval-valued ntutonstc fuzzy sets. Fuzzy ogc & Optmzaton. (006), 3-6. [9]. u. Q, Ma. C and Zhou. X, On propertes of some IFS operators and operatons, Notes on Intutonstc Fuzzy Sets, 4 (3) (008), 7-4. [0]. Mohamed Ismayl. and Mohamed l., On Strong Interval-valued Intutonstc Fuzzy Graph, Int.Journal of Fuzzy Mathematcs and Systems, 4(04), []. Nagoor Gan. and Shatha Begum.S, Degree, Order and Sze n Intutonstc Fuzzy Graphs, Internatonal Journal of lgorthms, Computng and Mathematcs,(3)3 (00). []. Verma..K and Sharma. B.D, Intutonstc fuzzy sets: Some new results, Notes on Intutonstc Fuzzy Sets, 7 (3), (0), - 0. [3]. Vaslev. T, Four equaltes connected wth ntutonstc fuzzy sets, Notes on Intutonstc Fuzzy Sets, 4 (3), (008), - 4. [4]. Yahya Mohamed. S et. al., Irregular Intutonstc fuzzy graphs, IOS-JM, Vol. 9, Issue 6, (Jan. 04), pp [5]. Yahya Mohamed. S., and Jahr Hussan.., Sem global domnatng set of Intutonstc fuzzy graphs, IOS-JM, Vol. 0, Issue 4, Ver. II (Jul-ug04), pp 3-7. [6]. Zadeh..., Fuzzy Sets, Informaton & Control, 8 (965), DOI: / Page