论文研究 区间值强模糊图的运算性质.pdf

利用经典图和模糊图定义和性质，给出了区间值模糊关系、模糊变换以及区间值强模糊图的定义，相应地定义了区间值强模糊图弱直积、半直积运算，并且证明了其弱直积、半直积运算封闭的性质。42014,50(17)Computer Engineering and Applications计算机工程与应用以上两个条件满足定义2.1的条件:区间值强模糊图(n#)(,w1)(4,w2)=min(A(.p(w1w2)证明设G1=(A1,B)是G1=VE1的区间值强模(B81)(u,w)(a,2)=min(P(n)pE2(w2)糊图和G2-(42,B2)是G2=(2E2)的区间值强模糊图以上三个条件满足定义2.2的条件,再给出以下条件且V1∩V2=中。又(t,v1)(u,v2)∈E,有:(n)(1,v)(2,)=min(A、(D),2(a22)4E(142)=min{44(t1,(2)}(B#)(u1,v)(n2,)=min(u1(①)(u1n2)2(x42)=mm(),x2(x)因此:3区间值模糊图的运算性质(1gmB)(u,v)(,v2)=mn{()2(v1v2)}=命题3.设G1=(4,B)是C=V,E1)的区间值强min{4(u)A(v1),(v2)}=模糊图和G2=(42,B2)是G2=(V2,E2)的区间值强模糊min(g③p1)(n,)(pA②Pp)(x,v2)图,且V∩V2=。则G米G2=(42*A2B*B2)也是区(urdu)(u, v)(u, v,)=min(u,(u),/R(v,v)3=值强模糊图。min(u4(u),u(v,u(v,))证明设G1=(41,B)是G1=(1,E)的区间值强模糊图和G2=(42,B2)是G2=(2E2)的区间值强模糊图,min(r③1)(a,)(②)2)n,v2)由命题31知,G1⑧G2=(A1CA2,B1②B2也是区间值且V1∩V2=p。又a1a2=(u11)(u2v2)∈E,有强模糊图。例如图2所示。从(x22)-min{(x1)4x4(a2)0.1,1.3AB,u2 2)=minquA, (u),u, (u2);(02,0.5)(0,0.5(0.1,0.5[O.1,0.5因此0.2,0.41[0.1,0.3[0.1,0.3][0.1,0.3](n1)(2)=(a+2)(n,2y"2)(0.2,0.4])(01.03(01.03ymin{B(u12.g2(v12)}「0.1,0.3[0.1,0.3min{(1,(v)pA(2)(v2)}=图2区间值强模糊图的半直积*.)(41,1)(PA*A)(u2v2)推论3.2若G⑧G2=(A1A2,B1QB,是区间值(1)(x1以2)=(2*)(1v1)(2,v2)强模糊图,则G1=(41B或G2=(A2,B2)是区间值强模min{p(12.p,(v12)糊图命题33设G1=(1B是G1=(1,F)的区间值强min{(a),;4(1),p4(ax2,4(v2)模猢图G2=(A2,B3)是G2=(2,E,)的区间值强模糊min(4*4)(1,v1)(*)(n2,v2)}图,且V∩V2=。则G1#G2=(41+A2,B1#B2)也是区间则G*G,=(A12B*B)也是区间值强模糊图。如图1值强模糊图所示证明设G1=(A13B是G1=(1,E的区间值强模0.2.0.50.1,0.5」5)糊图和G2=(4,6)是G,=(2E2)的区间值强模糊图,且V∩V2=φ。有:[0.2,0.4][0.1,0.3][0.1,0.3[0.1,0.342(42)-mim()(g2[0.2,0.4(.031)(01,0311.042(x43=mim2)又u,w1)(u,w2)∈E,图1区间值强模糊图的弱直积(uB, #.(u, wD(u, w2)=mintu (u),u, (ww2)3推论3.1若G*G2=(A*A2B*B2)是区间值强模糊图,则G1=(A1,B)或G,=(A2,B)是区间值强模糊图。min{2(a)(w1).pA、(w2)}=命題3.2设G1-(4、B)是G=(,E1)的区间值强min{(4形4.)(2w1,(#)(,w2)模糊图和G2(42B2)是G2=(2E2)的区间值强模糊({H2)(u,w1)(,w2)=min{A(m),a(ww2)图,且∩V2=,则C1G2=(1A2,B⑧B2)也是min{4(u,(w1),(w2)索南仁欠,李生刚:区间值强模糊图的运算性质2014,50(17ISmin{(#4),w1)(4#1)(,w2)参考文献:又(1v)(u2,yv)∈E,LI Bhattacharya P.Some remarks on fuzzy graphs[J]. Pattern(FB#B,)(H12)(2)=min{(412),()Recognition Letters, 987,6: 297-302[2] Mordeson J N, Peng C.S. Operations on fuzzy graphs[]min{4(),(u2),(u2)}=Information Sciences. 1994.79. 159-170min(, HA)(u, v)uA, #uA, )(u2, D))[3] Sunitha M S, Vijaya Kumar A Complement of a fuzzy(pB#B)B1B2)=(B#)(u1,w)a2,v2)graph[j].Indian J Pure Appl Math. 2002, 33(9): 1452-1464min(FB(u,u)).uB(w,w,))4」彭祖赠,孙韫玉.模糊(上uzxy)数学及其应用[M」.2版.武汉:武汉大学出版社,2004:115-146min{(w1),4(w2)x(a1)x(u2)}[51 Bhutani K R, rosenfeld A Strong arcs in fuzzy graphs[JImin(u4A4,)(a,W1)(H1H,)(a2W2)Information Sciences. 2003. 152: 319-322又因为[6 Mathew S, Sunitha M STypes of arcs in fuzzy graph[].(B#B)B1B2)=(#)(u1、wa2,w2)Information Sciences. 2009, 179: 1760-1768[7] Bhutani K R, Ballou A On M-strong fuzzy graphs[JIn(4(4240w2)}=Information Sciences, 2003,155: 103-109A,( W1),A(w2),u, (u1),u, (u2))= [81 Al-Hawary T Complete fuzzy graphs J International J Mathmnin{(4#1)(u4,w1)(p4#x1)(u2,甲2)Combin,2011,4:26-34山命题3.2知G形(2=(4#A12B+B2)也是区间值强模91 Nagoor Gani A, Kacha K. On regular fuzzy graphs糊图。如图3所示j Physical sciences. 2008.12. 33-4010] Zadeh L A. The concept of a linguistic and application(02,0.5(01,.050.0301.03(0.5to approximate reasoning I[].Information Sciences, 1975o.1,0.31)×(00.1,0.318:199-249「0.2,0.4]0.1,0.310.1,0.3TIl Akram M, Alshehri N O, Dudek w A Certain types of[0.2,0.4」(.1,0.3)(0..0301,031(0.03interval-valued fuzzy graphs [J] Journal of Applied Math-G#Gematics,2013,7:1-11图3区间值强模糊图的直积[12] Akram M. Interal-valued fuzzy line graphs[J]. Neural Com推论33若G#G2=(A1#A2B#B2)是区间值强模糊uting Applicalions, 2012, 21: 145-150图,则G1=(41,B)或G2=(A2,B2)是区间值强模糊图13 Talebi AA, Rashmanlou H Isomorphism on interval-valuedfuzzy graphs[J. Annals of Fuzzy Mathematics and Infor4结语[14 Rashmanlou H, Jun Y B Complete interval-valued fuzzy在模糊图论中,有直观模糊图的相关性质,对于区graphs[J]. Annals of Fuzzy Mathematics and Informatics间值模糊图可否找到直观的区间值模糊图,能否建立模2010,23:31-42型,使已研究的区间值模糊图的理论应用到相关的领口15]杨文华,李生刚区间值模糊图的运算性质门模糊系纨与域,这些都是下步有待解决的问题。数学,2013(2):127-135从目前的发展趋势看来,模糊图论巳在聚类分析、[16]杨文华,李生刚区间值模糊图的分解性质口计算机工程数据理论、 Network分析以及信息理论等方面体现出重与应用,2012,48(31):25-29要的应用价值,关于模糊图论的研究也受到了众多学者[17] Akram M, Dudek w A Interval-valued fuzzy graphs[的关注。模糊图论必然会像经典图论一样,发展成为更Computers and Mathematics with Applications, 2011, 61系统、结构性更紧密的理论研究基础。(2):289299