Ratio Mathematica Volume 44, 2022 Various Product on Multi Fuzzy Graphs R. Muthuraj 1 K. Krithika 2 S. Revathi 3 Abstract In this paper, the definition of complement of multi fuzzy graph, direct sum of two multi fuzzy graphs are given and derived some theorems related to them. Also, we examine the different product on multi fuzzy graphs such as Direct product, Cartesian product, Strong product, Composition, Corona product and some properties are analyzed. Key Words: Multi fuzzy graph, complement of multi fuzzy graph, direct sum, direct product, cartesian product, strong product, composition, corona product, AMS Subject Classification: 03E72, 05C72, 05C07 4 1 PG and Research Department of Mathematics, H.H. The Rajah’s College, Pudukkottai -622001, Affiliated to Bharathidasan University, Tiruchirappalli, Tamilnadu, India. E-mail: rmr1973@yahoo.co.in;rmr1973@gmail.com 2 Part time Research Scholar, PG and Research Department of Mathematics, H.H. The Rajah’s College, Pudukkottai, 622001, (Affiliated to Bharathidasan University, Tiruchirappalli), Tamilnadu, India. Department of Mathematics, Dhaanish Ahmed College of Engineering, Chennai – 601301, Tamilnadu, India E-mail: krithika.cv1982@gmail.com 3 Department of Mathematics, Saranathan College of Engineering, Trichy – 620012, Tamilnadu, India. E-mail: revathi.soundar@gmail.com 4 Received on June 26 th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.911. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published under the CC-BY license agreement. 231 mailto:rmr1973@gmail.com mailto:krithika.cv1982@gmail.com mailto:revathi.soundar@gmail.com R. Muthuraj, K. Krithika and S. Revathi 1. Introduction The notion of fuzzy set and fuzzy relations were proposed by L.A Zadeh [18] in 1965 for representing uncertainty. The concept of fuzzy graph was first introduced by Kauffman [2] from the concept fuzzy relation introduced by L.A Zadeh in 1973. In 1975, Rosenfeld [14] developed the theory of fuzzy graph and several fuzzy analogs of graph theoretic concepts such as paths, cycles and connectedness. Thereafter in 1987, Bhattacharya [1] defined some remarks on fuzzy graphs. The operations of union, join, cartesian product and composition of two fuzzy graphs were defined by Mordeson. J.N, and Prem Chand S. Nair, [3] in 2000. After that M.S. Sunitha and A. Vijayakumar [17] extended the concept of operations on fuzzy graph in 2002.Sebu Sebastian, T.V. Ramakrishnan [15] defined Multi fuzzy set in 2010. Radha. K and Arumugam. S [11, 12] defined the direct sum of two fuzzy graphs in 2013 and strong product of two fuzzy graphs in 2014.OzgeColakogluHavare and Hamza Menken [10] defined the Corona Product of Two Fuzzy Graphs in 2016.In 2020R.Muthuraj and S. Revathi [5] introduced the concept of multi fuzzy graph which is the extension of a fuzzy graph with single phenomenon into a multi-phenomenon which suits to describe the real-life problems in a better manner than fuzzy graph. Later on, and Multi anti fuzzy graph defined by Muthuraj. Ret.al [6]. In this paper complement of multi fuzzy graph, direct sum of two multi fuzzy graphs and various product on multi fuzzy graphs are defined and proved some theorems related to them. 2. Preliminaries Definition 2.1 [2] A fuzzy graph ),( G defined on the underlying crisp graph ),( EVG   where VVE  is a pair of functions ]1,0[: V and ]1,0[: VV ,  is a symmetric fuzzy relation on  such that  )(),(min)( vuuv   for Vvu , Definition 2.2 [15] Let X be a non-empty set. A Multi Fuzzy Set A in X is defined as a set of ordered sequences:  XxxxxxA i  :)...)(),......(),(,( 21  where ]1,0[: X i  for all i. Definition 2.3 [5] A Multi fuzzy Graph (MFG) of dimension m defined on the underlying crisp graph ),( EVG  where VVE  , is denoted as  ),...,(),,...,( 2121 mm G  and ]1,0[: V i  and ]1,0[: VV i  , i is a symmetric fuzzy relation on i  such that  )(),(min)( vuuv iii   for all mi ....3,2,1 where Vvu , and Euv 232 Various Product on Multi Fuzzy Graphs 3. Complement of Multi Fuzzy Graph Definition 3.1 The complement of a multi-fuzzy graph )),....,(),,....,(( 2121 mmG  of dimension m is a multi-fuzzy graph )),...,(),,...,(( 2121 mm G  of dimension m where ii   and ),())()((),( vuvuvu iiii   for all Vvu , and for all mi ....3,2,1 Example 3.2 Theorem 3.3 If G is a strong multi fuzzy graph then G is also strong multi fuzzy graph. Proof: Let Evu , . Then ),())()((),( vuvuvu iiii   ))()(())()(( vuvu iiii   0 since G is strong. Let Evu , . Then ),())()((),( vuvuvu iiii   0))()((  vu ii  )()(( vu ii   Theorem 3.4 The complement of complete multi fuzzy graph is a null graph. Proof: Let ),( EVG  be a multi-fuzzy graph with the underlying crisp graph ),( EVG  is complete. ie., EuvVvuvuvu iii  &,))()((),(  Let Evu , ),())()((),( vuvuvu iiii   ))()(())()(( vuvu iiii   0 since G is complete. So, we have the edge set of G is empty when G is a complete multi fuzzy graph. Hence the complement of complete multi fuzzy graph is a null graph. 4. Various Product On Multi Fuzzy Graphs In this section  ),...,(),,...,( 21211 mm G  denotes the multi fuzzy graph with dimension m with the underlying crisp graph ),( 111 EVG   and  ),...,(),,...,( 21212 nn G  denotes the multi fuzzy graph with dimension n with the underlying crisp graph ),( 222 EVG   233 R. Muthuraj, K. Krithika and S. Revathi Definition 4.1 The operation Direct sum between two MFG 1 G and 2 G is defined as follows, )),......,(),,......,(( 2211221121 kkkk GG   with the underlying crisp graph ),( 212121 EEVVGG   ,          21 12 21 )(),(max )( )( ))(( VVuifuu VVuifu VVuifu u ii i i ii     for all ki ....3,2,1 and       2 1 ),(),( ),(),( ),)(( Evuifvu Evuifvu vu i i ii    for all ki ....3,2,1 If , let k = max (m, n). Suppose , then let us introduce n – m membership values of multi fuzzy graph G1 into 0 so as to convert the multi fuzzy graphs G1 and G2 have the same dimension as k. Theorem 4.2 The direct sum of two multi fuzzy graph is also a multi-fuzzy graph, Proof: Let  ),...,(),,...,( 21211 mm G  and  ),...,(),,...,( 21212 nn G  be the multi fuzzy graph with dimension m and n respectively To prove: 21 GGG  is also multi fuzzy graph with dimension k where k=max (m, n)          21 12 21 )(),(max )( )( ))(( VVuifuu VVuifu VVuifu u ii i i ii     Case (i): Let 1 ),( Evu  ),(),)(( vuvu iii   ))(),(min( vu ii  )))((),)(min(( vu iiii   )))((),)(min((),)(( vuvu iiiiii   Case (ii): Let 2 ),( Evu  ),(),)(( vuvu iii   ))(),(min( vu ii  )))((),)(min(( vu iiii   )))((),)(min((),)(( vuvu iiiiii   Definition4.3 The operation Direct Product between two MFG 1 G and 2 G is defined as follows, )),......,(),,......,(( 2211221121 kkkk GG   with the underlying crisp graph ),( 21 EVGG   where 21 VVV  and   2211212211 ),(&),/(),(),,( EvvEuuvuvuE  with   211121111111 ),(&,)(),(min),)(( VVvuVvVuallforvuvu iiii     22112121212211 ),(&),(),(),,(min)),(),,)((( EvvEuuallforvvuuvuvu iiii   for all I = 1, 2, 3, … k. If nm  ,let k = min (m, n). Suppose nm  , then we take first m dimensions for 2 G so as to convert the MFG 1 G and 2 G have the same dimension k. 234 Various Product on Multi Fuzzy Graphs Example 4.4 Figure1 Figure2 Theorem 4.5 Direct product of two multi fuzzy graph is also a multi-fuzzy graph. Proof: Let  ),...,(),,...,( 21211 mm G  and  ),...,(),,...,( 21212 nn G  be the multi fuzzy graph with dimension m and n respectively To Prove: 21 GGG  is a multi-fuzzy graph of dimension k where k=min (m, n)  )(),(min),)(( 1111 vuvu iiii    ),(),,(min)),(),,)((( 21212211 vvuuvuvu iiii    )(),(min()),(),(min(min 2121 vvuu iiii  ))()((())()(( 2121 vvuu iiii   ))()((())()(( 2211 vuvu iiii   )),)((()),)((( 2211 vuvu iiii   )),)((,),)(min(()),(),,)((( 22112211 vuvuvuvu iiiiii   . Theorem 4.6 If 1 G and 2 G are strong multi fuzzy graphs then 21 GG  is also a strong multi fuzzy graph. Proof:  )(),(min),)(( 1111 vuvu iiii    ),(),,(min)),(),,)((( 21212211 vvuuvuvu iiii    )(),(min()),(),(min(min 2121 vvuu iiii  ))()((())()(( 2121 vvuu iiii   ))()((())()(( 2211 vuvu iiii   )),)((()),)((( 2211 vuvu iiii   235 R. Muthuraj, K. Krithika and S. Revathi )),)((,),)(min(()),(),,)((( 22112211 vuvuvuvu iiiiii   Remark: If 1 G and 2 G are complete multi fuzzy graphs then 21 GG  is not a complete multi fuzzy graph. Definition 4.7 The operation Cartesian Product between two MFG 1 G and 2 G as follows, )),...,(),,...,(( 2211221121 kkkk GG   with the underlying crisp graph ),( 21 EVGG   where 21 VVV  and   12121221212211 ),(,),(,/),)(,( EuuvvorEvvuuvuvuE  with   211121111111 ),(&)(),(min),)(( VVvuVvandVuallforvuvu iiii            12122121 22112121 2211 ),(&,)(),,(min ),(&,),(),(min )),(),,)((( EuuVvallforvvvvuu EvvVuallforuuuvvu vuvu ii ii ii    for all i= 1, 2, 3, ... k If nm  let k = min (m, n). Suppose nm  then we take first m dimensions for 2G so as to convert the MFG 1 G and 2 G have the same dimension k. Example 4.8 Figure 3 Figure 4 Theorem 4.9 Cartesian product of two multi fuzzy graph is also a multi-fuzzy graph. Proof: Let  ),...,(),,...,( 21211 mm G  and  ),...,(),,...,( 21212 nn G  be the multi fuzzy graph with dimension m and n respectively To Prove: 21 GGG  is a multi-fuzzy graph of dimension k where k=min (m, n)  )(),(min),)(( 1111 vuvu iiii    ),(),(min)),(),,)((( 2121 vvuvuvu iiii   236 Various Product on Multi Fuzzy Graphs   )(),(min),(min 21 vvu iii      )(),(min,)(),(minmin 21 vuvu iiii   ),)((),,)((min 21 vuvu iiii    )(),,(min)),(),,)((( 2121 vuuvuvu iiii     )(,)(),(minmin 21 vuu iii      )(),(min,)(),(minmin 21 vuvu iiii   ),)((),,)((min 21 vuvu iiii   Theorem 4.10 Cartesian product of two strong multi fuzzy graph is also a strong multi fuzzy graph. Proof: Let  ),...,(),,...,( 21211 mm G  and  ),...,(),,...,( 21212 nn G  be the multi fuzzy graph with dimension m and n respectively To Prove: 21 GGG  is a multi-fuzzy graph of dimension k where k=min (m,n)  )(),(min),)(( 1111 vuvu iiii    ),(),(min)),(),,)((( 2121 vvuvuvu iiii     )(),(min),(min 21 vvu iii      )(),(min,)(),(minmin 21 vuvu iiii   ),)((),,)((min 21 vuvu iiii    )(),,(min)),(),,)((( 2121 vuuvuvu iiii     )(,)(),(minmin 21 vuu iii      )(),(min,)(),(minmin 21 vuvu iiii   ),)((),,)((min 21 vuvu iiii   Remark: If 1 G and 2 G are complete multi fuzzy graphs then 21 GG  is not a complete multi fuzzy graph. Theorem 4.11 If 21 GG  is a strong multi fuzzy graph then at least one 1G or 2G is a strong multi fuzzy graph. Proof: Suppose assume that the contrary that 1 G and 2 G are not strong fuzzy graphs. )()(),( 1111 vuvu iii   and )()(),( 2222 vuvu iii   (1) Without loss of generality, we assume that )()()(),(),( 1111122 uvuvuvu iiiii   Let   12121221212211 ),(,),(,/),)(,( EuuvvorEvvuuvuvuE  Consider Evuvu ),)(,( 2211 , by definition of 21 GG  & inequality (1) )()()(),()()),(),,)((( 2112112111 vvuvvuvuvu iiiiiii   (2) )()(),)(( 1111 vuvu iiii   & )()(),)(( 2121 vuvu iiii   )()()()(),)((),)(( 21112111 vuvuvuvu iiiiiiii   )()()( 211 vvu iii   (3) 237 R. Muthuraj, K. Krithika and S. Revathi From (2) and (3), ),)((),)(()()()()),(),,)((( 21112112111 vuvuvvuvuvu iiiiiiiii   ),)((),)(()),(),,)((( 21112111 vuvuvuvu iiiiii   This implies that 21 GG  is not a strong multi fuzzy graph. This gives a contradiction. So, if 21 GG  is a strong multi fuzzy graph then atleast one 1 G or 2 G is a strong multi fuzzy graph. Definition 4.12 The operation Strong Product between two MFG 1 G and 2 G is defined as follows, )),...,(),,...,(( 2211221121 kkkk GG   with the underlying crisp graph ),( 21 EVGG   where 21 VVV  and   22112112121221212211 ),(&),(),(,),(,/),)(,( EvvEuuorEuuvvorEvvuuvuvuE  with   211121111111 ),(&)(),(min),)(( VVvuVvandVuallforvuvu iiii                2211212121 12122121 22112121 2211 ),(&),(),(),,(min ),(&,)(),,(min ),(&,),(),(min )),(),,)((( EvvEuuallforvvuu EuuVvallforvvvvuu EvvVuallforuuuvvu vuvu ii ii ii ii     for all i= 1, 2, 3, ... k If nm  , let k = min (m, n). Suppose nm  , then we take first m dimensions for 2 G so as to convert the MFG 1 G and 2 G have the same dimension k. Example 4.13 Figure 5 Figure 6 Theorem 4.14 Strong product of two multi fuzzy graph is also a multi-fuzzy graph. Proof: Let  ),...,(),,...,( 21211 mm G  and  ),...,(),,...,( 21212 nn G  be the multi fuzzy graph with dimension m and n respectively 238 Various Product on Multi Fuzzy Graphs To Prove: 21 GGG  is a multi-fuzzy graph of dimension k where k=min (m, n)   211121111111 ),(&)(),(min),)(( VVvuVvandVuallforvuvu iiii    ),(),(min)),(),,)((( 2121 vvuvuvu iiii     )(),(min),(min 21 vvu iii      )(),(min,)(),(minmin 21 vuvu iiii  ))()(())()(( 21 vuvu iiii    ),)((),,)((min 21 vuvu iiii    )(),,(min)),(),,)((( 2121 vuuvuvu iiii     )(,)(),(minmin 21 vuu iii      )(),(min,)(),(minmin 21 vuvu iiii  ))()(())()(( 21 vuvu iiii    ),)((),,)((min 21 vuvu iiii    ),(),,(min)),(),,)((( 21212211 vvuuvuvu iiii       )(),(min,)(),(minmin 2121 vvuu iiii  ))()(())()(( 2121 vvuu iiii   ))()(())()(( 2211 vuvu iiii    ),)((),,)((min 2211 vuvu iiii   Theorem 4.15 If 1 G and 2 G are strong multi fuzzy graphs then 21 GG  is also a strong multi fuzzy graph. Proof: Let  ),...,(),,...,( 21211 mm G  and  ),...,(),,...,( 21212 nnG  be the multi fuzzy graph with dimension m and n respectively To Prove: 21 GG  is a strong multi fuzzy graph of dimension k where k=min (m, n)   211121111111 ),(&)(),(min),)(( VVvuVvandVuallforvuvu iiii    ),(),(min)),(),,)((( 2121 vvuvuvu iiii     )(),(min),(min 21 vvu iii      )(),(min,)(),(minmin 21 vuvu iiii  ))()(())()(( 21 vuvu iiii    ),)((),,)((min 21 vuvu iiii    )(),,(min)),(),,)((( 2121 vuuvuvu iiii     )(,)(),(minmin 21 vuu iii      )(),(min,)(),(minmin 21 vuvu iiii  ))()(())()(( 21 vuvu iiii    ),)((),,)((min 21 vuvu iiii    ),(),,(min)),(),,)((( 21212211 vvuuvuvu iiii       )(),(min,)(),(minmin 2121 vvuu iiii  ))()(())()(( 2121 vvuu iiii   ))()(())()(( 2211 vuvu iiii    ),)((),,)((min 2211 vuvu iiii   Theorem 4.16 If 1 G and 2 G are complete multi fuzzy graphs then 21 GG  is also a complete multi fuzzy graph. Proof: Let  ),...,(),,...,( 21211 mm G  and  ),...,(),,...,( 21212 nnG  be the two complete multi fuzzy graphs with dimension m and n respectively. Then 1 G and 2 G 239 R. Muthuraj, K. Krithika and S. Revathi are strong multi fuzzy graphs where  21 GandG are complete graphs. Therefore, 21 GG  is a strong multi fuzzy graph by the theorem (4.15) with  21 GandG are complete graphs. Hence 21 GG  is a complete multi fuzzy graph. Theorem 4.17 The strong product of two multi fuzzy graphs 1 G and 2 G is the direct sum of the cartesian product of 1 G and 2 G and the direct product of 1 G and 2 G . Proof: Let  ),...,(),,...,( 21211 mm G  and  ),...,(),,...,( 21212 nn G  be the multi fuzzy graph with dimension m and n respectively. Let 21 GG  and 21 GG  be the cartesian product and direct product of 1 G and 2 G with dimension k where k = min(m,n) To Prove: )()( 212121 GGGGGG    2111111111 ),()(),(min),)((),)(( VVvuvuvuvu iiiiii   So,   2111111111 ),()(),(min),)((),)(( VVvuvuvuvu iiiiii    )(),(min),)(( 1111 vuvu iiii   ),)((),)((),)(( 111111 vuvuvu iiiiii         12121211 22121211 2211 ),(),()( ),(),()( )),(),,)((( Euuandvvifuuv Evvanduufvvu vuvu ii ii ii      22112121212211 ),(&),(,),(),,(min)),(),,)(( EvvEuuifvvuuvuvu iiii            2211212121 12121211 22121211 2211 ),(&),(),(),,(min ),(),()( ),(),()( )),(),,))(()(( EvvEuuifvvuu Euuandvvifuuv Evvanduuifvvu vuvu ii ii ii iiii     )),(),,)((( 2211 vuvu ii   Result: Let  ),...,(),,...,( 21211 mm G  and  ),...,(),,...,( 21212 nn G  be two strong multi fuzzy graph with dimension m and n respectively and 21 GG  & 21 GG  be the cartesian product and direct product of 1 G and 2 G with dimension k where k = min(m,n) and 21 GG  and 21 GG  be the complement of two multi fuzzy graphs then 21212121 GGGGGGGG  . Definition 4.18 The operation Composition between two MFG 1 G and 2 G as follows )),...,(),,...,((][ 2211221121 kkkk GG   with the underlying crisp graph ),(][ 21 EVGG   where 21 VVV  and   1212112121221212211 ),(,),(,),(,/),)(,( EuuvvorEuuvvorEvvuuvuvuE  with   211121111111 ),(&)(),(min),)(( VVvuVvandVuallforvuvu iiii                 1212121 12122121 22112121 2211 ),(),(),(),(min ),(&,)(),,(min ),(&,),(),(min )),(),,)((( Euuallforuuvv EuuVvallforvvvvuu EvvVuallforuuuvvu vuvu iii ii ii ii      for all i= 1, 2, 3, ... k. 240 Various Product on Multi Fuzzy Graphs If nm  , let k = min (m, n). Suppose nm  , then we take first m dimensions for 2 G so as to convert the MFG 1 G and 2 G have the same dimension k Example 4.19 Figure 7 Figure 8 Theorem 4.20 Composition of two multi fuzzy graph is also a multi-fuzzy graph. Proof: Let  ),...,(),,...,( 21211 mm G  and  ),...,(),,...,( 21212 nn G  be the multi fuzzy graph with dimension m and n respectively To Prove: 21 GGG  is a multi-fuzzy graph of dimension k where k = min (m, n)   211121111111 ),(&)(),(min),)(( VVvuVvandVuallforvuvu iiii     ),(),(min)),(),,)((( 2121 vvuvuvu iiii     )(),(min),(min 21 vvu iii      )(),(min,)(),(minmin 21 vuvu iiii  ))()(())()(( 21 vuvu iiii    ),)((),,)((min 21 vuvu iiii    )(),,(min)),(),,)((( 2121 vuuvuvu iiii     )(,)(),(minmin 21 vuu iii      )(),(min,)(),(minmin 21 vuvu iiii  ))()(())()(( 21 vuvu iiii    ),)((),,)((min 21 vuvu iiii    ),(),(),(min)),(),,)((( 21212211 uuvvvuvu iiiii     )(),(min),(),(min 2121 uuvv iiii   ))()((),(),(min 2121 uuvv iiii   ))()(())()(( 2211 vuvu iiii   241 R. Muthuraj, K. Krithika and S. Revathi  ),)((),,)((min 21 vuvu iiii   Theorem 4.21 If  ),...,(),,...,( 21211 mm G  and  ),...,(),,...,( 21212 nn G  are two strong multi fuzzy graphs with dimension m and n respectively and 21 GG  is a strong multi fuzzy graph of dimension k where k = min (m, n). Prove that 2121 GGGG   Proof: Let )),...,(),,...,(( 2211221121 kkkk GGG    )),...,(),,...,(( 2211221121 kkkk GG     ),...,(),,...,( 21211 mm G   ),...,(),,...,( 21212 nn G  )),...,(),,...,(( 2211221121 kkkk GG    To prove 2121 GGGG   It is enough to prove iiii    for all i= 1, 2, 3, ... k. To prove the above result, there are different cases may arise depending upon the edges joining the vertices Case(i): Consider the edge 22121 ),()),,(),,(( Evvvuvue  Then Ee and G is a strong multi fuzzy graph, so 0)( e ii   Also 0)()( 11 e  since 221 ),( Evv  If )),(),,(( 21 vuvue  , 22121 ),( Evvandvv  then Ee 0)),(),,)((( 21 vuvu ii   Now )),)((()),)((()( 21 vuvue iiiiii    ))()((())()(( 21 vuvu iiii   )()()( 21 vvu iii   ),()()()( 2111 vvue ii   )()()( 21 vvu iii   iiii    for all i= 1, 2, 3, ... k. Case(ii): Consider the edge 12121 ),()),,(),,(( Euuvuvue  Then Ee and G is a strong multi fuzzy graph, so 0)( e ii   Also 0)()( 11 e  since 121 ),( Euu  If )),,(),,(( 21 vuvue  121 ),( Euu  then Ee 0)),(),,)((( 21 vuvu ii   Now )),)((()),)((()( 21 vuvue iiiiii    ))()((())()(( 21 vuvu iiii   )()()( 21 vuu iii   since 121 ),( Euu  )(),()()( 2111 vuue ii   )()()( 21 vuu iii   iiii    for all i= 1, 2, 3, ... k. Case(iii): Consider the edge 211212211 &),()),,(),,(( vvEuuvuvue  Then Ee and G is a strong multi fuzzy graph, So 0)( e ii   since 121 ),( Euu  , 0)()( 11 e  If 211212211 &),()),,(),,(( vvEuuvuvue  Then Ee 0))(( e ii   )),)((()),)((()( 2211 vuvue iiiiii    ))()((())()(( 2211 vuvu iiii   242 Various Product on Multi Fuzzy Graphs Since 121 ),( Euu  we have )()(),()()( 212111 vvuue iii   )()()()( 2121 vvuu iiii   )(e ii   iiii    for all i= 1, 2, 3, ... k. Case(iv): Consider the edge 2211212211 ),(&),()),,(),,(( EvvEuuvuvue  Since Ee , 0))(( eii   )),)((()),)((()( 2211 vuvue iiiiii    ))()((())()(( 2211 vuvu iiii   If 121 ),( Euu  and if 21 vv  then we have case (ii) If 121 ),( Euu  and if 21 vv  then we have case (iii) In all the cases we have, iiii    for all i= 1, 2, 3, ... k. Definition 4.22 The operation Corona Product between two MFG 1 G and 2 G is defined as follows, )),...,(),,...,(( 2211221121 kkkk GG   with the underlying crisp graph ),(),( 212121 EEVVEVGG         2 1 ),( ),( ))(( Vuu Vuu u i i ii    and          ',)(),(min ),(),,( ),(),,( ),)(( 2 1 Euvvu Evuvu Evuvu vu ii i i ii     where E' is the set of all edges joining by an edge the i th vertex of 1 G to every vertex in the i th copy of 2 G If nm  , let k = max (m, n). Suppose nm  then let us introduce n – m membership values of multi fuzzy graph 1 G into 0 so as to convert the multi fuzzy graphs 1G and 2G have the same dimension as k. Example 4.23 Figure 9 Figure 10 243 R. Muthuraj, K. Krithika and S. Revathi Theorem 4.24 Corona product of two multi fuzzy graph is also a multi-fuzzy graph. Proof: Let  ),...,(),,...,( 21211 mm G  and  ),...,(),,...,( 21212 nn G  be the multi fuzzy graph with dimension m and n respectively To Prove: 21 GGG  is a multi-fuzzy graph of dimension k where k= max (m, n)       2 1 ),( ),( ))(( Vuu Vuu u i i ii    Case(i): If 1 ),( Evu  ),(),)(( vuvu iii    )(),(min vu ii   ))((),)((min vu iiii   Case(ii): If 2 ),( Evu  ),(),)(( vuvu iii    )(),(min vu ii   ))((),)((min vu iiii   Case(iii): If '),( Evu   )(),(min),)(( vuvu iiii    ))((),)((min vu iiii   5. Conclusion In this paper, the complement of multi fuzzy graph and direct sum of two multi fuzzy graphs are defined and proved some results connected to them. Also defined various product on multi fuzzy graphs such as direct product, strong product, cartesian product, composition, corona product and proved some properties related to them. 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