Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions Ratio Mathematica Volume 41, 2021, pp. 227-254 227 Some studies on products of fuzzy soft graphs Shashikala S* Anil P N† Abstract In this paper, alpha, beta and gamma product of two fuzzy soft graphs are defined. The degree of a vertex in these product fuzzy soft graphs are determined and its regular properties are studied. Keywords: alpha, beta and gamma product of fuzzy soft graphs, degree of a vertex, regular properties. 2010 AMS subject classification: 05C72, 05C99.‡ 1. Introduction Graph theory is used to model various types of relations that exist in different fields of physics, chemistry, medicine, electrical network, computer science such as networking, image processing etc. Whenever the information provided is imprecise, uncertainty exists. Molodtsov [1] initiated the concept of soft sets to deal with uncertainty. A. Rosenfeld [2] developed the theory of fuzzy graphs in 1975 based on fuzzy sets which were initiated by Zadeh [3] in 1965. Maji et al. [5,7] presented the definition of fuzzy soft sets and applied it in decision making problems. Later many researchers progressively worked on these concepts and developed it. Operations on fuzzy graphs were demonstrated by J. N. Mordeson and C. S. Peng [6]. Akram and Saira Nawaz [8,11] introduced fuzzy soft graphs, studied some of its properties and applied these concepts in social network and road network. Shashikala S and Anil P N [12,15] discussed connectivity in fuzzy soft graphs and studied hamiltonian fuzzy soft cycles. A. Pouhassani and H. Doostie [13] studied degree, total degree, regularity and total regularity of fuzzy soft graph and its properties. * Shashikala S (Global Academy of Technology, Bangalore, India); shashikala.s@gat.ac.in † Anil P N (Global Academy of Technology, Bangalore, India); anilpn@gat.ac.in ‡Received on August 19, 2021. Accepted on December 1, 2021. Published on December 31, 2021. doi: 10.23755/rm.v41i0.644. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published under the CC-BY licence agreement. Shashikala S and Anil P N 228 Regular fuzzy soft graphs and its related properties are studied by B Akhilandeswari [16]. Shovan Dogra [10] studied some types of fuzzy graph products such as modular product, homomorphic product and determined its degree of vertices. Union and intersection of fuzzy soft graphs and some of its properties are studied by Mohinta and Samanta [9]. Fuzzy soft theory provides a clear picture of the problems that allows parameterization finds applications in many areas. Recently, it is used to represent the oligopolistic competition among the wireless internet connection providers in Malaysia by Akram and Saba Nawaz [17]. In this paper, some products of fuzzy soft graphs namely alpha, beta and gamma products are defined and degree of a vertex in these products are determined and its regular properties are studied. 2. Preliminaries Definition 2.1: [11] A fuzzy soft graph G ~ over a graph ),(: * EVG is a triple ), ~ , ~ ( AKF where: a) A is a nonempty set of parameters b) ), ~ ( AF is a fuzzy soft set over V c) ), ~ ( AK is a fuzzy soft set over E d) ( ))(~),(~ ii eKeF is a fuzzy graph on AeG i  * i.e. )})(( ~ ),)(( ~ min{))(( ~ yeFxeFxyeK iii  for all Aei  and ., Vyx  Definition 2.2: [14] The underlying crisp graph of a fuzzy soft graph G ~ is denoted by ( )*** , KFG = where  AesomeforxeFVxF ii = 0))(( ~ : * ,  AesomeforyxeKVVyxK ii = 0),)(( ~ :),( * . Definition 2.3: [8] Let G ~ be a fuzzy soft graph on *G . The degree of a vertex x is defined as     = Ae xyVy iG i xyeKx , ~ ))(( ~ )(deg . Definition 2.4: [8] ] G ~ is said to be a regular fuzzy soft graph if ( ))(~),(~ ii eKeF is regular fuzzy graph for all Ae i  . If ( ))(~),(~ ii eKeF is a regular fuzzy graph of degree k for all Ae i  then G ~ is a k-regular fuzzy soft graph. Definition 2.5: [4] The degree )(* xd G of a vertex v in *G is the number of edges incident with x . Some studies on products of fuzzy soft graphs 229 In this paper, we assume that ),(: * EVG of any fuzzy soft graph G ~ is finite and simple. Notation: Let ), ~ , ~ (: ~ 1111 AKFG and ), ~ , ~ (: ~ 2222 AKFG be two fuzzy soft graphs. The relation )( ~ )( ~ 21 ji eKeF  for all 1 Ae i  , 2Ae j  means that ))(( ~ ))(( ~ 21 eeKxeF ji  21 , EeVx  where 1 ~ F is a fuzzy soft subset of 1 V and 2 ~ K is a fuzzy soft subset of 2 E . 3. Alpha product )( product− , Beta product )( product− and Gamma product )( product− of Fuzzy soft graph Definition 3.1: Let ), ~ , ~ (: ~ 1111 AKFG and ), ~ , ~ (: ~ 2222 AKFG be two fuzzy soft graphs on * 1 G and * 2 G respectively. The product− ), ~~ , ~~ (: ~~ 21212121 AAKKFFGG   is defined as follows: )(:) ~~ ( 212121 VVFSAAFF →  by 21212121 ,,)()( ~ )()( ~ )(),() ~~ ( VVyxAeAeyeFxeFyxeeFF lkjiljkilkji =  and )(:) ~~ ( 212121 EEFSAAKK →  by               = = = 2 1211 2 1122 112 221 21 ,)()( ~ )()( ~ )()( ~ ,)()( ~ )()( ~ )()( ~ ,)()( ~ )()( ~ ,)()( ~ )()( ~ )()(),() ~~ ( Eyy ExxifyyeKxeFxeF Eyy ExxifxxeKyeFyeF ExxyyifxxeKyeF EyyxxifyyeKxeF yxyxeeKK nl mknljmiki nl mkmkinjlj mknlmkilj nlmknljki nmlkji  Definition 3.2: Let ), ~ , ~ (: ~ 1111 AKFG and ), ~ , ~ (: ~ 2222 AKFG be two fuzzy soft graphs on * 1 G and * 2 G respectively. The product− ), ~~ , ~~ (: ~~ 21212121 AAKKFFGG   is defined as follows: )(:) ~~ ( 212121 VVFSAAFF →  by 21212121 ,,)()( ~ )()( ~ )(),() ~~ ( VVyxAeAeyeFxeFyxeeFF lkjiljkilkji =  and )(:) ~~ ( 212121 EEFSAAKK →  by Shashikala S and Anil P N 230                = 2121 1 122 2 211 21 ,)()( ~ )()( ~ ,)()( ~ )()( ~ )()( ~ ,)()( ~ )()( ~ )()( ~ )()(),() ~~ ( EyyExxifyyeKxxeK Exx yyifxxeKyeFyeF Eyy xxifyyeKxeFxeF yxyxeeKK nlmknljmki mk nlmkinjlj nl mknljmiki nmlkji  Definition 3.3: Let ), ~ , ~ (: ~ 1111 AKFG and ), ~ , ~ (: ~ 2222 AKFG be two fuzzy soft graphs on * 1 G and * 2 G respectively. The product− ), ~~ , ~~ (: ~~ 21212121 AAKKFFGG   is defined as follows: )(:) ~~ ( 212121 VVFSAAFF →  by 21212121 ,,)()( ~ )()( ~ )(),() ~~ ( VVyxAeAeyeFxeFyxeeFF lkjiljkilkji =  and )(:) ~~ ( 212121 EEFSAAKK →  by                 = = = 2121 1 122 2 211 112 221 21 ,)()( ~ )()( ~ ,)()( ~ )()( ~ )()( ~ ,)()( ~ )()( ~ )()( ~ ,)()( ~ )()( ~ ,)()( ~ )()( ~ )()(),() ~~ ( EyyExxifyyeKxxeK Exx yyifxxeKyeFyeF Eyy xxifyyeKxeFxeF ExxyyifxxeKyeF EyyxxifyyeKxeF yxyxeeKK nlmknljmki mk nlmkinjlj nl mknljmiki mknlmkilj nlmknljki nmlkji  Example 3.4: Consider two fuzzy soft graphs ), ~ , ~ (: ~ 1111 AKFG and ), ~ , ~ (: ~ 2222 AKFG on ),(: 11 * 1 EVG and ),(: 22 * 2 EVG respectively such that },{ 211 xxV = , }{ 211 xxE = , },,{ 3212 yyyV = , },{ 32212 yyyyE = , }{ 1 i eA = where i=1,2 and }{2 jeA = where j=3,4. Let ), ~ ( 11 AF , ), ~ ( 22 AF , ), ~ ( 11 AK and ), ~ ( 22 AK be represented by the following Table 1. 1 ~ F 1 x 2 x 2 ~ F 1 y 2y 3 y 1 e 0.4 0.6 3e 0.3 0.5 0.8 2 e 0.7 0.5 4 e 0.5 0.6 0.7 1 ~ K 21 xx 2 ~ K 21 yy 32 yy 1 e 0.1 3e 0.3 0.4 2 e 0.3 4 e 0.4 0.5 Table 1 : Tabular representation of two fuzzy soft graphs Some studies on products of fuzzy soft graphs 231 1 ~ .1. GFig 2 ~ .2. GFig 21 ~~ .3. GGFig   21 ~~ .4. GGFig   Shashikala S and Anil P N 232 21 ~~ .5. GGFig   4. Degree of a vertex in Alpha product )( product− of two fuzzy soft graphs and its regular properties Theorem 4.1: Let ), ~ , ~ (: ~ 1111 AKFG and ), ~ , ~ (: ~ 2222 AKFG be two fuzzy soft graphs on ),(: 11 * 1 EVG and ),(: 22 * 2 EVG respectively. If )( ~ )( ~ 21 ji eKeF  and )( ~ )( ~ 12 ij eKeF  then )()](1[)()](1[),(deg 1 ~ 11 ~ 111 ~~ 1 * 2 2 * 1 21 xdeydydexdyx GjGG i GGG cc +++=  Proof:      = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ ))(( ~ 2111221211 Ae Ae Exx yy ij Ae Ae Eyy xx ji j i j i xxeKyeFyyeKxeF       + 2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2112212 Ae Ae Eyy Exx ijj j i xxeKyeFyeF        2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2122111 Ae Ae Eyy Exx jii j i yyeKxeFxeF ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ 211212 Ae Ae Exx yy i Ae Ae Eyy xx j j i j i xxeKyyeK       + 2 1 221 121 ))(( ~ 211 Ae Ae Eyy Exx i j i xxeK Some studies on products of fuzzy soft graphs 233       2 1 221 121 ))(( ~ 212 Ae Ae Eyy Exx j j i yyeK )()](1[)()](1[ )()()()()()( 1 ~ 11 ~ 1 1 ~ 11 ~ 11 ~ 1 ~ 1 * 2 2 * 1 2 * 1 1 * 2 12 xdeydydexd ydxdexdydexdeyde GjGG i G GG iGG jGjGi cc cc +++= +++= This is true for any vertex ),( 11 yx in 21 ~~ GG   with 21 ~~ KF  and 12 ~~ KF  . Theorem 4.2 : Let 1 ~ G and 2 ~ G be two fuzzy soft graphs on ),(: 11 * 1 EVG and ),(: 22 * 2 EVG respectively. If )( ~ )( ~ 21 ji eKeF  and )( ~ 1 i eF is a constant function with 11 ))(( ~ VxcxeF kki = then )](1)[()](1)[(),(deg 11 ~ 1111 ~~ * 2 1 * 1 * 221 ydxdexdydeecyx cc GG j GG jiGG +++=   Proof: Given )( ~ )( ~ 21 ji eKeF  then )( ~ )( ~ 12 ij eKeF  . 11 ))(( ~ VxcxeF kki = , for any 21 ),( VVyx lk       = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ ))(( ~ 2111221211 Ae Ae Exx yy ij Ae Ae Eyy xx ji j i j i xxeKyeFyyeKxeF       + 2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2112212 Ae Ae Eyy Exx ijj j i xxeKyeFyeF        2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2122111 Ae Ae Eyy Exx jii j i yyeKxeFxeF ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ 21111 Ae Ae Exx yy i Ae Ae Eyy xx i j i j i xxeKxeF       + 2 1 221 121 ))(( ~ 211 Ae Ae Eyy Exx i j i xxeK       2 1 221 121 ))(( ~ 11 Ae Ae Eyy Exx i j i xeF +++=   )()()())(( ~ )( 1 ~ 11 ~ 111 1 * 2 1 1 * 2 xdydexdexeFyde GG jGj Ae iGj c i )())(( ~ )( 1111 * 2 1 * 1 ydxeFxde G Ae i G j i c   Shashikala S and Anil P N 234 )()()()()()( 111 ~ 11 ~ 1 * 2 * 1 1 * 2 1 * 2 ydecxdexdydexdeecyde GiG jGG jGjiGj cc +++= )](1)[()](1)[( 11 ~ 11 * 2 1 * 1 * 2 ydxdexdydeec cc GG j GG ji +++= Theorem 4.3 : Let 1 ~ G and 2 ~ G be two fuzzy soft graphs on ),(: 11 * 1 EVG and ),(: 22 * 2 EVG respectively. If )( ~ )( ~ 12 ij eKeF  and )( ~ 2 j eF is a constant function with 22 ))(( ~ VymyeF lli = then )](1)[()](1)[(),(deg 11 ~ 1111 ~~ * 1 2 * 2 * 121 xdydeydxdeemyx cc GG i GG jiGG +++=   Proof: Given )( ~ )( ~ 12 ij eKeF  then )( ~ )( ~ 21 ji eKeF       = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ ))(( ~ 2111221211 Ae Ae Exx yy ij Ae Ae Eyy xx ji j i j i xxeKyeFyyeKxeF       + 2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2112212 Ae Ae Eyy Exx ijj j i xxeKyeFyeF        2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2122111 Ae Ae Eyy Exx jii j i yyeKxeFxeF ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ 12212 Ae Ae Exx yy j Ae Ae Eyy xx j j i j i yeFyyeK       + 2 1 221 121 ))(( ~ 12 Ae Ae Eyy Exx j j i yeF       2 1 221 121 ))(( ~ 212 Ae Ae Eyy Exx j j i yyeK +++=   )())(( ~ )())(( ~ )()( 11211211 ~ * 1 2 * 2 2 * 12 xdyeFydeyeFxdeyde G Ae j G i Ae jGiGi j c j )()( 1 ~ 1 2 * 1 ydxde GG i c )](1)[()](1)[( 11 ~ 11 * 1 2 * 2 * 1 xdydeydxdeem cc GG i GG ji +++= Theorem 4.4 : Let 1 ~ G and 2 ~ G be two fuzzy soft graphs on complete graphs ),(: 11 * 1 EVG and ),(: 22 * 2 EVG respectively. If )( ~ )( ~ 21 ji eKeF  and )( ~ )( ~ 12 ij eKeF  then )()(),(deg 1 ~ 1 ~ 11 ~~ 1221 xdeydeyx GjGiGG +=   Proof:      = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ ))(( ~ 2111221211 Ae Ae Exx yy ij Ae Ae Eyy xx ji j i j i xxeKyeFyyeKxeF Some studies on products of fuzzy soft graphs 235       + 2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2112212 Ae Ae Eyy Exx ijj j i xxeKyeFyeF        2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2122111 Ae Ae Eyy Exx jii j i yyeKxeFxeF ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ 211212 Ae Ae Exx yy i Ae Ae Eyy xx j j i j i xxeKyyeK       + 2 1 221 121 ))(( ~ 211 Ae Ae Eyy Exx i j i xxeK       2 1 221 121 ))(( ~ 212 Ae Ae Eyy Exx j j i yyeK )()](1[)()](1[ 1 ~ 11 ~ 1 1 * 2 2 * 1 xdeydydexd GjGG i G cc +++= )()( 1 ~ 1 ~ 12 xdeyde GjGi += (since * 1 G and * 2 G are complete graphs) Theorem 4.5 : Let ), ~ , ~ (: ~ 1111 AKFG and ), ~ , ~ (: ~ 2222 AKFG be two fuzzy soft graphs on regular graphs ),(: 11 * 1 EVG and ),(: 22 * 2 EVG respectively. If )( ~ )( ~ 21 ji eKeF  and )( ~ )( ~ 12 ij eKeF  then 21 ~~ GG   is a regular fuzzy soft graph if and only if 1 ~ G and 2 ~ G are regular fuzzy soft graphs. Proof: Let 1 ~ G and 2 ~ G be regular fuzzy soft graphs of degree 1k and 2 k respectively. For any vertex 2111 )( VVyx  , )()](1[)()](1[),(deg 1 ~ 11 ~ 111 ~~ 1 * 2 2 * 1 21 xdeydydexdyx GjGG i GGG cc +++=  (From Theorem 4.1) 122111 ~~ ]1[]1[),(deg 21 keEkeEyx j c i c GG +++=   This is true 2111 )( VVyx  Hence, 21 ~~ GG   is a regular fuzzy soft graph. Conversely, Let 21 ~~ GG   be a regular fuzzy soft graph. For any two vertices )( 11 yx and )( 22 yx in 21 VV  , =  ),(deg 11 ~~ 21 yx GG  ),(deg 22 ~~ 21 yx GG   )()](1[)()](1[ 1 ~ 11 ~ 1 1 * 2 2 * 1 xdeydydexd GjGG i G cc +++ )()](1[)()](1[ 2 ~ 22 ~ 2 1 * 2 2 * 1 xdeydydexd GjGG i G cc +++= (From Theorem 4.1) Fix 1 Vx , consider )( 11 yx and )( 22 yx in 21 VV  , Shashikala S and Anil P N 236 )()](1[)()](1[ 1 * 2 2 * 1 ~ 11 ~ xdeydydexd GjGG i G cc +++ )()](1[)()](1[ 1 * 2 2 * 1 ~ 22 ~ xdeydydexd GjGG i G cc +++= )()](1[)()](1[ 2 ~ 1 ~ 2 * 1 2 * 1 ydexdydexd GiGG i G cc +=+ )()( 2 ~ 1 ~ 22 ydyd GG = This is true for all vertices of 2 V . 2 ~ G is a regular fuzzy soft graph. Fix 2 Vy , consider )( 11 yx and )( 22 yx in 21 VV  , )()](1[)()](1[ 1 ~~ 1 1 * 2 2 * 1 xdeydydexd GjGG i G cc +++ )()](1[)()](1[ 2 ~~ 2 1 * 2 2 * 1 xdeydydexd GjGG i G cc +++= )()](1[)()](1[ 2 ~ 1 ~ 1 * 2 1 * 2 xdeydxdeyd GjGG j G cc +=+ )()( 2 ~ 1 ~ 11 xdxd GG = This is true for all vertices of 1 V . 1 ~ G is a regular fuzzy soft graph. Theorem 4.6 : Let 1 ~ G and 2 ~ G be two fuzzy soft graphs on complete graphs ),(: 11 * 1 EVG and ),(: 22 * 2 EVG respectively. If )( ~ )( ~ 21 ji eKeF  and )( ~ )( ~ 12 ij eKeF  then 21 ~~ GG   is regular if and only if 1 ~ G and 2 ~ G are regular fuzzy soft graphs. Proof: Let 1 ~ G and 2 ~ G be regular fuzzy soft graphs of degree 1k and 2 k respectively. Let * 1 G and * 2 G are complete graphs. )()(),(deg 1 ~ 1 ~ 11 ~~ 1221 xdeydeyx GjGiGG +=   (From Theorem 4.4) This is true 2111 )( VVyx  Hence, 21 ~~ GG   is a regular fuzzy soft graph. Conversely, Let 21 ~~ GG   be a regular fuzzy soft graph. For any two vertices )( 11 yx and )( 22 yx in 21 VV  , =  ),(deg 11 ~~ 21 yx GG  ),(deg 22 ~~ 21 yx GG   )()( 1 ~ 1 ~ 12 xdeyde GjGi + )()( 2~2~ 12 xdeyde GjGi += Fix 1 Vx , consider )( 11 yx and )( 22 yx in 21 VV  , )()( 12 ~ 1 ~ xdeyde GjGi + )()( 12 ~ 2 ~ xdeyde GjGi += )()( 2 ~ 1 ~ 22 ydyd GG = Some studies on products of fuzzy soft graphs 237 This is true for all vertices of 2 V . 2 ~ G is a regular fuzzy soft graph. Fix 2 Vy , consider )( 11 yx and )( 22 yx in 21 VV  , )()( 1 ~~ 12 xdeyde GjGi + )()( 2 ~~ 12 xdeyde GjGi += )()( 2 ~ 1 ~ 11 xdxd GG = This is true for all vertices of 1 V . 1 ~ G is a regular fuzzy soft graph. Theorem 4.7 : Let 1 ~ G and 2 ~ G be two fuzzy soft graphs on complete graphs * 1 G and * 2 G respectively then 21 ~~ GG   becomes a Cartesian product of fuzzy soft graphs. Proof: By the definition of alpha product of fuzzy soft graphs, for any two vertices )( 11 yx and )( 22 yx in 21 VV  ,               = = = 221 1212122111 221 1212112212 1212121112 2212121211 221121 ,)()( ~ )()( ~ )()( ~ ,)()( ~ )()( ~ )()( ~ ,)()( ~ )()( ~ ,)()( ~ )()( ~ )()(),() ~~ ( Eyy ExxifyyeKxeFxeF Eyy ExxifxxeKyeFyeF ExxyyifxxeKyeF EyyxxifyyeKxeF yxyxeeKK jii ijj ij ji ji  Since * 1 G and * 2 G are complete graphs,     = = = 1212121112 2212121211 221121 ,)()( ~ )()( ~ ,)()( ~ )()( ~ )()(),() ~~ ( ExxyyifxxeKyeF EyyxxifyyeKxeF yxyxeeKK ij ji ji  )()(),() ~~ ()()(),() ~~ ( 221121221121 yxyxeeKKyxyxeeKK jiji =  is a Cartesian product of 1 ~ G and 2 ~ G . Theorem 4.8 : Let 1 ~ G and 2 ~ G be two fuzzy soft graphs such that )( ~ )( ~ 21 ji eKeF  . Let )( ~ 1 i eF be a constant and * 1 G is a complete graph then 21 ~~ GG   is regular fuzzy soft graph if and only if 1 ~ G is a regular fuzzy soft graph and * 2 G is a regular graph. Proof: Let 11 ))(( ~ VxcxeF kki = Given )( ~ )( ~ 21 ji eKeF  then )( ~ )( ~ 12 ij eKeF  . Let * 1 G be a complete graph. From Theorem 4.2, )](1)[()](1)[(),(deg 11 ~ 1111 ~~ * 2 1 * 1 * 221 ydxdexdydeecyx cc GG j GG jiGG +++=   )()()()( 1 ~ 11 ~ 1 1 * 2 1 * 2 xdydexdeydeec GG jGjGji c++= Shashikala S and Anil P N 238 Let 1 ~ G be a regular fuzzy soft graph with degree m and * 2 G is a regular graph of degree n. mydemeneecyx c G jjjiGG )(),(deg 111 ~~ * 2 21 ++=   ]1[ 2 c jji Emecnee ++= Hence, 21 ~~ GG   is regular fuzzy soft graph. Conversely, assume that 21 ~~ GG   is regular fuzzy soft graph. ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx GGGG   = =+++ )()()()()()( 111 ~ 11 ~ 1 * 2 * 1 1 * 2 1 * 2 ydecxdexdydexdeydeec GiG jGG jGjGji cc )()()()()()( 222 ~ 22 ~ 2 * 2 * 1 1 * 2 1 * 2 ydecxdexdydexdeydeec GiG jGG jGjGji cc +++ Fix 1 Vx , consider )( 1 xy and )( 2 xy in 21 VV  , )](1)[()()](1)[()( 2 ~ 21 ~ 1 * 2 1 * 2 * 2 1 * 2 ydxdeydeecydxdeydeec cc GG jGjiGG jGji ++=++ This is true when degree of all vertices in * 2 G as well as in its complement are equal. Hence, * 2 G is regular. Fix 2 Vy , )](1)[()()](1)[()( * 2 1 * 2 * 2 1 * 2 2 ~ 1 ~ ydxdeydeecydxdeydeec cc GG jGjiGG jGji ++=++ )()( 2 ~ 1 ~ 11 xdxd GG = Hence, 1 ~ G is a regular fuzzy soft graph. Theorem 4.9 : Let 1 ~ G and 2 ~ G be two fuzzy soft graphs such that )( ~ )( ~ 21 ji eKeF  . If 21 ~~ GG   , 1 ~ G are regular fuzzy soft graphs, * 1 G and * 2 G are regular graphs then   1 ))(( ~ 1 Ae ki i xeF is same for all k=1,2,3..... Proof: Let 1 ~ G be a regular fuzzy soft graph of degree m, * 1 G and * 2 G are regular graphs of degree s and n respectively. Given )( ~ )( ~ 21 ji eKeF  then )( ~ )( ~ 12 ij eKeF  . Using the definition of 21 ~~ GG   ,      = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   Some studies on products of fuzzy soft graphs 239 ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ 21111 Ae Ae Exx yy i Ae Ae Eyy xx i j i j i xxeKxeF       + 2 1 221 121 ))(( ~ 211 Ae Ae Eyy Exx i j i xxeK        2 1 221 121 ))(( ~ ))(( ~ 2111 Ae Ae Eyy Exx ii j i xeFxeF   +++= 1 * 2 * 1 1 * 2 1 1 * 2 ))(( ~ )()()()()())(( ~ )( 1111 ~ 11 ~ 111 Ae kiGG jGG jGj Ae iGj i cc i xeFydxdexdydexdexeFyde )](1)[(])(1[))(( ~ )( 11 ~ 1111 * 2 1 * 1 1 * 2 ydxdexdxeFyde cc i GG j G Ae iGj +++=   Since 21 ~~ GG   is a regular fuzzy soft graph, ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx GGGG   = =+++  )](1)[(])(1[))(( ~ )( 11 ~ 1111 * 2 1 * 1 1 * 2 ydxdexdxeFyde cc i GG j G Ae iGj )](1)[(])(1[))(( ~ )( 22 ~ 2212 * 2 1 * 1 1 * 2 ydxdexdxeFyde cc i GG j G Ae iGj +++  =+++  )](1[])(1[))(( ~ 1111 * 2 * 1 1 ydmexdxeFne cc i G j G Ae ij )](1[])(1[))(( ~ 2221 * 2 * 1 1 ydmexdxeFne cc i G j G Ae ij +++  ])(1[))(( ~ ])(1[))(( ~ 221111 * 1 1 * 1 1 xdxeFnexdxeFne c i c i G Ae ij G Ae ij +=+   (Since c G * 1 is regular)   = 11 ))(( ~ ))(( ~ 2111 Ae i Ae i ii xeFxeF Theorem 4.10 : Let 1 ~ G and 2 ~ G be two fuzzy soft graphs such that )( ~ )( ~ 12 ij eKeF  . If 21 ~~ GG   , 2 ~ G are regular fuzzy soft graphs, * 1 G and * 2 G are regular graphs then   21 ))(( ~ 1 Ae lj j yeF is same for all ...3,2,1=l Proof: Proof is similar to the proof of theorem 4.9. Theorem 4.11 : Let 1 ~ G and 2 ~ G be two fuzzy soft graphs with )( ~ )( ~ 12 ij eKeF  and ceF j =)( ~ 2 then 21 ~~ GG   is a regular FSG if and only if * 1 G and * 2 G is a regular graph and 2 ~ G is a regular FSG. Proof: Given ceF j =)( ~ 2 , Since )( ~ )( ~ 12 ij eKeF  , )( ~ )( ~ 21 ji eKeF  . Let 2 ~ G be a regular FSG of degree m, * 1 G and * 2 G are regular graphs of degree n and p respectively. Shashikala S and Anil P N 240 From Theorem 4.3, )](1)[()](1)[(),(deg 11 ~ 1111 ~~ * 1 2 * 2 * 121 xdydeydxdeecyx cc GG i GG jiGG +++=   )](1[)](1[ 11 * 1 * 2 xdemydeecn cc G i G ji +++= 21 ~~ GG   is a regular FSG. Conversely, Let 21 ~~ GG   be a regular FSG. ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx GGGG   = =+++ )](1)[()](1)[( 11 ~ 11 * 1 2 * 2 * 1 xdydeydxdeec cc GG i GG ji )](1)[()](1)[( 22 ~ 22 * 1 2 * 2 * 1 xdydeydxdeec cc GG i GG ji +++ Fix 1 Vx , =+++ )](1)[()](1)[( * 1 2 * 2 * 1 1 ~ 1 xdydeydxdeec cc GG i GG ji )](1)[()](1)[( * 1 2 * 2 * 1 2 ~ 2 xdydeydxdeec cc GG i GG ji +++ )()( 2 ~ 1 ~ 22 ydyd GG = and )()( 21 * 2 * 2 ydyd cc GG = 2 ~ G is a regular FSG and * 2 G is regular. Fix 2 Vy , =+++ )](1)[()](1)[( 1 ~ 1 * 1 2 * 2 * 1 xdydeydxdeec cc GG i GG ji )](1)[()](1)[( 2 ~ 2 * 1 2 * 2 * 1 xdydeydxdeec cc GG i GG ji +++ )()( 21 * 1 * 1 xdxd GG = and )()( 21 * 1 * 1 xdxd cc GG = This is true when * 1 G is regular. Theorem 4.12 : Let 1 ~ G and 2 ~ G be two fuzzy soft graphs where * 1 G is a complete graph. If )( ~ 1 i eF and )( ~ 1 i eK are constant and )( ~ )( ~ 21 ji eKeF  then 21 ~~ GG   is a regular FSG if and only if 1 ~ G is regular and * 2 G is a regular graph. Proof: Let ceKeF ii == )( ~ )( ~ 11 and 21 ~~ GG   be regular. Given )( ~ )( ~ 21 ji eKeF  , )( ~ )( ~ 12 ij eKeF  . ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx GGGG   = Using the result of theorem 4.9, =+++  )](1)[(])(1)[)(( ~ )( 11 ~ 1111 * 2 1 * 1 1 * 2 ydxdexdxeFyde cc i GG j G Ae iGj Some studies on products of fuzzy soft graphs 241 )](1)[(])(1)[)(( ~ )( 22 ~ 2212 * 2 1 * 1 1 * 2 ydxdexdxeFyde cc i GG j G Ae iGj +++  =++  )](1)[())(( ~ )( 11 ~ 111 * 2 1 1 * 2 ydxdexeFyde c i GG j Ae iGj )](1)[())(( ~ )( 22 ~ 212 * 2 1 1 * 2 ydxdexeFyde c i GG j Ae iGj ++  (Since * 1 G is a complete graph) Given ceF i =)( ~ 1 )](1)[()()](1)[()( 22 ~ 211 ~ 1 * 2 1 * 2 * 2 1 * 2 ydxdydecydxdydec cc GGG i GGG i ++=++ Fix 1 Vx , )](1)[()()](1)[()( 2 ~ 21 ~ 1 * 2 1 * 2 * 2 1 * 2 ydxdydecydxdydec cc GGG i GGG i ++=++ )()( 21 * 2 * 2 ydyd GG = and )()( 21 * 2 * 2 ydyd cc GG = This is true only when * 2 G is a regular graph. Similarly, fix 2 Vy , )()( 2 ~ 1 ~ 11 xdxd GG = 1 ~ G is regular FSG. Conversely, Let 1 ~ G and * 2 G be regular with degree m and n respectively. )](1)[(])(1)[)(( ~ )(),(deg 11 ~ 111111 ~~ * 2 1 * 1 1 * 221 ydxdexdxeFydeyx cc i GG j G Ae iGjGG +++=     )1( pmecene jij ++= Therefore, 21 ~~ GG   is a regular FSG. 5. Degree of a vertex in Beta product )( product− of two fuzzy soft graphs and its regular properties Theorem 5.1: Let 1 ~ G and 2 ~ G be two FSGs on complete graphs * 1 G and * 2 G respectively. i) If )( ~ )( ~ 21 ji eKeK  then )()(),(deg 11 ~ 11 ~~ * 2121 ydxdeyx GGjGG =   ii) If )( ~ )( ~ 12 ij eKeK  then )()(),(deg 11 ~ 11 ~~ * 1221 xdydeyx GGiGG =   Proof:      = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   +=       2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 2122111 Ae Ae Eyy xx jii j i yyeKxeFxeF Shashikala S and Anil P N 242 +      2 1 121 21 ))(( ~ ))(( ~ ))(( ~ 2112212 Ae Ae Exx yy ijj j i xxeKyeFyeF        2 1 221 121 ))(( ~ ))(( ~ 212211 Ae Ae Eyy Exx ji j i yyeKxxeK i) For any vertex )( 11 yx 21 VV  Since * 1 G and * 2 G are complete graphs, we have        = 2 1 221 121 21 ))(( ~ ))(( ~ ),(deg 21221111 ~~ Ae Ae Eyy Exx jiGG j i yyeKxxeKyx  Given )( ~ )( ~ 21 ji eKeK         = 2 1 221 121 21 ))(( ~ ),(deg 21111 ~~ Ae Ae Eyy Exx iGG j i xxeKyx  )()( 11 ~ * 21 ydxde GGj = ii)        = 2 1 221 121 21 ))(( ~ ))(( ~ ),(deg 21221111 ~~ Ae Ae Eyy Exx jiGG j i yyeKxxeKyx  Since )( ~ )( ~ 12 ij eKeK         = 2 1 221 121 21 ))(( ~ ),(deg 21211 ~~ Ae Ae Eyy Exx jGG j i yyeKyx  )()( 11 ~ * 12 xdyde GGi = Theorem 5.2: Let ), ~ , ~ (: ~ 1111 AKFG and ), ~ , ~ (: ~ 2222 AKFG be two fuzzy soft graphs on ),(: 11 * 1 EVG and ),(: 22 * 2 EVG respectively. If )( ~ )( ~ 21 ji eKeF  and )( ~ )( ~ 12 ij eKeF  then i) If )( ~ )( ~ 21 ji eKeK  then )()()]()()[(),(deg 1 ~ 1111 ~ 11 ~~ 1 * 1 * 2 * 2 121 ydxdeydydxdeyx GG iGGG jGG cc ++=   ii) If )( ~ )( ~ 12 ij eKeK  then )()()]()()[(),(deg 1 ~ 1111 ~ 11 ~~ 1 * 2 * 1 * 1 221 xdydexdxdydeyx GG jGGG iGG cc ++=   Proof:      = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   +=       2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 2122111 Ae Ae Eyy xx jii j i yyeKxeFxeF Some studies on products of fuzzy soft graphs 243 +      2 1 121 21 ))(( ~ ))(( ~ ))(( ~ 2112212 Ae Ae Exx yy ijj j i xxeKyeFyeF        2 1 221 121 ))(( ~ ))(( ~ 212211 Ae Ae Eyy Exx ji j i yyeKxxeK i) Given )( ~ )( ~ 21 ji eKeF  and )( ~ )( ~ 12 ij eKeF  . Also )( ~ )( ~ 21 ji eKeK              += 2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ),(deg 21121211 ~~ Ae Ae Exx yy i Ae Ae Eyy xx jGG j i j i xxeKyyeKyx        + 2 1 221 121 ))(( ~ 211 Ae Ae Eyy Exx i j i xxeK )()()()([)()( 11 ~ 1 ~ 11 ~ 1 * 211 * 2 2 * 1 ydxdexdydeydxde GGjGG jGG i cc ++= )()()]()()[( 1 ~ 1111 ~ 1 * 1 * 2 * 2 1 ydxdeydydxde GG iGGG j cc ++= ii) Given )( ~ )( ~ 21 ji eKeF  , )( ~ )( ~ 12 ij eKeF  & )( ~ )( ~ 12 ij eKeK              += 2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ),(deg 21121211 ~~ Ae Ae Exx yy i Ae Ae Eyy xx jGG j i j i xxeKyyeKyx        + 2 1 221 121 ))(( ~ 212 Ae Ae Eyy Exx j j i yyeK )()()()([)()( 11 ~ 1 ~ 11 ~ 1 * 121 * 2 2 * 1 xdydexdydeydxde GGiGG jGG i cc ++= )()()]()()[( 1 ~ 1111 ~ 1 * 2 * 1 * 1 2 xdydexdxdyde GG jGGG i cc ++= Theorem 5.3: Let ), ~ , ~ (: ~ 1111 AKFG and ), ~ , ~ (: ~ 2222 AKFG be two fuzzy soft graphs on ),(: 11 * 1 EVG and ),(: 22 * 2 EVG respectively. If )( ~ )( ~ 21 ji eKeF  and )( ~ 1 i eF is a constant function with cxeF ki =))(( ~ 1 1Vxk  then )()()]()()[(),(deg 11111 ~ 11 ~~ * 2 * 1 * 2 * 2 121 ydxdeecydydxdeyx GG jiGGG jGG cc ++=   Proof:      = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   Using the given conditions,                 ++= 2 1 221 121 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 21121111 Ae Ae Eyy Exx i Ae Ae Exx yy i Ae Ae Eyy xx i j i j i j i xxeKxxeKxeF Shashikala S and Anil P N 244 )()()()()()(),(deg 11 ~ 11 ~ 1111 ~~ * 21 * 2 1 * 2 * 1 21 ydxdeydxdeeydxdceyx GGjGG jjGG iGG cc ++=   )()()]()()[( 11111 ~ * 2 * 1 * 2 * 2 1 ydxdeecydydxde GG jiGGG j cc ++= Theorem 5.4: Let 1 ~ G and 2 ~ G be two FSGs on ),(: 11 * 1 EVG and ),(: 22 * 2 EVG respectively. If )( ~ )( ~ 12 ij eKeF  and )( ~ 2 j eF is a constant function with myeF lj =))(( ~ 2 2Vyl  then )()()]()()[(),(deg 11111 ~ 11 ~~ * 1 * 2 * 1 * 1 221 xdymdeexdxdydeyx GG jiGGG iGG cc ++=   Proof:      = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   Using the given conditions,                 ++= 2 1 221 121 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 21212212 Ae Ae Eyy Exx j Ae Ae Exx yy j Ae Ae Eyy xx j j i j i j i yyeKyeFyyeK )()()()()()( 11 ~ 1111 ~ * 12 * 1 * 2 * 1 2 xdydeexdymdexdyde GGiiGG j GG i cc ++= )()()]()()[( 11111 ~ * 1 * 2 * 1 * 1 2 xdymdeexdxdyde GG jiGGG i cc ++= Theorem 5.5: Let 1 ~ G and 2 ~ G be two FSGs on complete graphs * 1 G and * 2 G respectively. 21 ~~ GG   is a regular FSG if and only if 1 ~ G and 2 ~ G are regular. Proof: Let 1 ~ G and 2 ~ G be regular FSGs with degrees m and n respectively. Let * 1 G and * 2 G be complete graphs with degrees p and q respectively.      = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg         = 2 1 221 121 ))(( ~ ))(( ~ 212211 Ae Ae Eyy Exx ji j i yyeKxxeK Case (i): If )( ~ )( ~ 21 ji eKeK  then        = 2 1 221 121 21 ))(( ~ ),(deg 21111 ~~ Ae Ae Eyy Exx iGG j i xxeKyx  )()( 11~ * 21 ydxde GGj = qme j = 21 ~~ GG   is a regular FSG. Some studies on products of fuzzy soft graphs 245 Case (i): If )( ~ )( ~ 12 ij eKeK  then =  ),(deg 11 ~~ 21 yx GG        2 1 221 121 ))(( ~ 212 Ae Ae Eyy Exx j j i yyeK )()( 11 ~ * 12 xdyde GGi = pnei= 21 ~~ GG   is a regular FSG. Conversely, Let 21 ~~ GG   be a regular FSG. ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx GGGG   = Case (i): If )( ~ )( ~ 21 ji eKeK  then )()( 11 ~ * 21 ydxde GGj )()( 22 ~ * 21 ydxde GGj = Fix 1 Vx , )()()()( 2 ~ 1 ~ * 21 * 21 ydxdydxd GGGG = )()( 21 * 2 * 2 ydyd GG = * 2 G is a regular graph. Fix 2 Vy , )()()()( * 21 * 21 2 ~ 1 ~ ydxdydxd GGGG = )()( 2 ~ 1 ~ 11 xdxd GG = 1 ~ G is regular. Case (ii): Similarly for )( ~ )( ~ 12 ij eKeK  , we get * 1 G and 2 ~ G as regular. Theorem 5.6: Let 1 ~ G and 2 ~ G be two FSGs and * 1 G is a complete graph and * 2 G is a regular graph. If )( ~ )( ~ 21 ji eKeF  , )( ~ )( ~ 12 ij eKeF  and )( ~ )( ~ 21 ji eKeK = then 21 ~~ GG   is a regular FSG if and only if 1 ~ G is a regular FSG. Proof: Let * 2 G be a regular graph of degree p and * 1 G is a complete graph. Let ceKeK ji == )( ~ )( ~ 21 , )( ~ )( ~ 21 ji eKeF  , )( ~ )( ~ 12 ij eKeF  . Let us assume that 1 ~ G is a regular FSG of degree m. Using the definition of degree of a vertex in beta product of FSGs and the above conditions, we get Shashikala S and Anil P N 246                  ++= 2 1 221 121 2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ))(( ~ ),(deg 21121121211 ~~ Ae Ae Eyy Exx i Ae Ae Exx yy ij Ae Ae Eyy xx GG j i j i j i xxeKxxeKyyeKyx             += 2 1 221 121 2 1 121 21 ))(( ~ ))(( ~ 211211 Ae Ae Eyy Exx i Ae Ae Exx yy i j i j i xxeKxxeK (Since * 1 G is a complete graph) )()()()( 11 ~ 1 ~ 1 * 211 * 2 ydxdexdyde GGjGG j c += )]()([)( 111 ~ * 2 * 21 ydydxde c GGG j += ][ spme j += 21 ~~ GG   is a regular FSG. Conversely, Let 21 ~~ GG   be a regular FSG, * 2 G is a regular graph of degree p and * 1 G is a complete graph. ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx GGGG   = )]()([)()]()([)( 222 ~ 111 ~ * 2 * 21 * 2 * 21 ydydxdeydydxde cc GGG j GGG j +=+ ][)(][)( 2 ~ 1 ~ 11 tpxdetpxde GjGj +=+ where tyd c G =)( * 2 )()( 2 ~ 1 ~ 11 xdxd GG = 1 ~ G is regular. Theorem 5.7: Let 1 ~ G and 2 ~ G be two FSGs and * 2 G is a complete graph and * 1 G is a regular graph. If )( ~ )( ~ 21 ji eKeF  , )( ~ )( ~ 12 ij eKeF  and )( ~ )( ~ 21 ji eKeK = then 21 ~~ GG   is a regular FSG if and only if 1 ~ G is a regular FSG. Proof: Proof is similar to the proof of Theorem 5.6. Theorem 5.8: If 1 ~ G and 2 ~ G are two regular FSGs with )( ~ )( ~ 21 ji eKeF  , )( ~ )( ~ 12 ij eKeF  , * 1 G and * 2 G are regular but not complete graphs, then beta product of two FSGs 1 ~ G and 2 ~ G is a regular FSG. Proof: Let 1 ~ G and 2 ~ G be regular FSGs with degrees m and n respectively. Let * 1 G and * 2 G be regular graphs of degrees p and q respectively and suppose that * 1 G and * 2 G are not complete graphs. Let )( ~ )( ~ 21 ji eKeF  , )( ~ )( ~ 12 ij eKeF  . Some studies on products of fuzzy soft graphs 247      = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   Case (i) : Assume that * 1 G and * 2 G are isomorphic graphs. Let ceKeK ji == )( ~ )( ~ 21 +=        2 1 221 21 21 ))(( ~ ))(( ~ ))(( ~ ),(deg 212211111 ~~ Ae Ae Eyy xx jiiGG j i yyeKxeFxeFyx  +      2 1 121 21 ))(( ~ ))(( ~ ))(( ~ 2112212 Ae Ae Exx yy ijj j i xxeKyeFyeF        2 1 221 121 ))(( ~ ))(( ~ 212211 Ae Ae Eyy Exx ji j i yyeKxxeK                 ++= 2 1 221 121 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 211211212 Ae Ae Eyy Exx i Ae Ae Exx yy i Ae Ae Eyy xx j j i j i j i xxeKxxeKyyeK )()()()()()( 11 ~ 1 ~ 11 ~ 1 * 211 * 2 2 * 1 ydxdexdydeydxde GGjGG jGG i cc ++= nsesqme ij ++= )( (Since * 1 G and * 2 G are regular graphs of degree p and q and are isomorphic, sydxd cc GG == )()( 11 * 2 * 1 ) 21 ~~ GG   is regular. Case (ii) : Assume that * 1 G and * 2 G are not isomorphic but are regular graphs of degrees p and q . By the definition, ++=             2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ),(deg 21121211 ~~ Ae Ae Exx yy i Ae Ae Eyy xx jGG j i j i xxeKyyeKyx         2 1 221 121 ))(( ~ ))(( ~ 212211 Ae Ae Eyy Exx ji j i yyeKxxeK If )( ~ )( ~ 21 ji eKeK  , ++=             2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ),(deg 21121211 ~~ Ae Ae Exx yy i Ae Ae Eyy xx jGG j i j i xxeKyyeKyx        2 1 221 121 ))(( ~ 211 Ae Ae Eyy Exx i j i xxeK )()()]()()[( 11 ~ 111 ~ * 1 2 * 2 * 21 xdydeydydxde cc GG i GGG j ++= snetqme ij ++= ][ Shashikala S and Anil P N 248 21 ~~ GG   is regular. If )( ~ )( ~ 12 ij eKeK  , ++=             2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ),(deg 21121211 ~~ Ae Ae Exx yy i Ae Ae Eyy xx jGG j i j i xxeKyyeKyx        2 1 221 121 ))(( ~ 212 Ae Ae Eyy Exx j j i yyeK )()()]()()[( 11 ~ 111 ~ * 2 1 * 1 * 12 ydxdexdxdyde cc GG j GGG i ++= tmespne ji ++= ][ 21 ~~ GG   is regular. 6. Degree of a vertex in Gamma product )( product− of two fuzzy soft graphs and its regular properties Theorem 6.1: Let 1 ~ G and 2 ~ G be two FSGs on complete graphs * 1 G and * 2 G respectively and )( ~ )( ~ 21 ji eKeF  , )( ~ )( ~ 12 ij eKeF  i) If )( ~ )( ~ 21 ji eKeK  then )()](1)[(),(deg 1 ~ 11 ~ 11 ~~ 2 * 2121 ydeydxdeyx GiGGjGG ++=   ii) If )( ~ )( ~ 12 ij eKeK  then )()](1)[(),(deg 1 ~ 11 ~ 11 ~~ 1 * 1221 xdexdydeyx GjGGiGG ++=   Proof: Given )( ~ )( ~ 21 ji eKeF  , )( ~ )( ~ 12 ij eKeF  .For any vertex 2111 )( VVyx  ,      = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ ))(( ~ 2111221211 Ae Ae Exx yy ij Ae Ae Eyy xx ji j i j i xxeKyeFyyeKxeF +      2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 2122111 Ae Ae Eyy xx jij j i yyeKxeFxeF        2 1 121 21 ))(( ~ ))(( ~ ))(( ~ 2112212 Ae Ae Exx yy ijj j i xxeKyeFyeF Some studies on products of fuzzy soft graphs 249 ))(( ~ ))(( ~ 212211 2 1 221 121 yyeKxxeK j Ae Ae Eyy Exx i j i        i) Given )( ~ )( ~ 21 ji eKeK  , ++=       =    =  2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ),(deg 21121211 ~~ Ae Ae Exx yy i Ae Ae Eyy xx jGG j i j i xxeKyyeKyx                  ++ 2 1 221 121 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 211211212 Ae Ae Eyy Exx i Ae Ae Exx yy i Ae Ae Eyy xx j j i j i j i xxeKxxeKyyeK Since * 1 G and * 2 G are complete graphs,            =    = ++= 2 1 221 121 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 211211212 Ae Ae Eyy Exx i Ae Ae Exx yy i Ae Ae Eyy xx j j i j i j i xxeKxxeKyyeK )()()()(),(deg 11 ~ 1 ~ 1 ~ 11 ~~ * 212121 ydxdeydexdeyx GGjGiGjGG ++=   )()](1)[( 1~11~ 2 * 21 ydeydxde GiGGj ++= ii) Given )( ~ )( ~ 12 ij eKeK       = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   ))(( ~ ))(( ~ ))(( ~ 212212211 2 1 221 121 2 1 221 21 2 1 121 21 yyeKyyeKxxeK j Ae Ae Eyy Exx Ae Ae Eyy xx j Ae Ae Exx yy i j i j i j i            =    = ++= )()()()(),(deg 11 ~ 1 ~ 1 ~ 11 ~~ * 122121 xdydeydexdeyx GGiGiGjGG ++=   )()](1)[( 1~11~ 1 * 12 xdexdyde GjGGi ++= Theorem 6.2: Let 1 ~ G and 2 ~ G be two FSGs on crisp graphs * 1 G and * 2 G respectively and )( ~ )( ~ 21 ji eKeF  , )( ~ )( ~ 12 ij eKeF  i) If )( ~ )( ~ 21 ji eKeK  then )](1)[()]()(1)[(),(deg 11 ~ 111 ~ 11 ~~ * 1 2 * 2 * 2 121 xdydeydydxdeyx cc GG iGGG jGG ++++=   ii) If )( ~ )( ~ 12 ij eKeK  then )](1)[()]()(1)[(),(deg 11 ~ 111 ~ 11 ~~ * 2 1 * 1 * 1 221 ydxdexdxdydeyx cc GG jGGG iGG ++++=   Proof: Given )( ~ )( ~ 21 ji eKeF  , )( ~ )( ~ 12 ij eKeF  i) If )( ~ )( ~ 21 ji eKeK  then Shashikala S and Anil P N 250 Using the definition of degree of 21 ~~ GG        = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   +++=            =    = 2 1 221 21 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 212211212 Ae Ae Eyy xx j Ae Ae Exx yy i Ae Ae Eyy xx j j i j i j i yyeKxxeKyyeK            + 2 1 221 121 2 1 121 21 ))(( ~ ))(( ~ 211211 Ae Ae Eyy Exx i Ae Ae Exx yy i j i j i xxeKxxeK )](1)[()]()(1)[(),(deg 11 ~ 111 ~ 11 ~~ * 1 2 * 2 * 2 121 xdydeydydxdeyx cc GG iGGG jGG ++++=   ii) If )( ~ )( ~ 12 ij eKeK  then      = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   +++=            =    = 2 1 221 21 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 212211212 Ae Ae Eyy xx j Ae Ae Exx yy i Ae Ae Eyy xx j j i j i j i yyeKxxeKyyeK             + 2 1 121 21 2 1 221 121 ))(( ~ ))(( ~ 212211 Ae Ae Exx yy j Ae Ae Eyy Exx i j i j i yyeKxxeK )](1)[()]()(1)[(),(deg 11 ~ 111 ~ 11 ~~ * 2 1 * 1 * 1 221 ydxdexdxdydeyx cc GG jGGG iGG ++++=   Theorem 6.3: Let ), ~ , ~ (: ~ 1111 AKFG and ), ~ , ~ (: ~ 2222 AKFG be two fuzzy soft graphs on * 1 G and * 2 G respectively. If )( ~ )( ~ 21 ji eKeF  and )( ~ 1 i eF is a constant function with cxeF ki =))(( ~ 1 1Vxk  , then )](1)[()]()(1)[(),(deg 11111 ~ 11 ~~ * 1 * 2 * 2 * 2 121 xdydeecydydxdeyx cc GG jiGGG jGG ++++=   Proof: Given )( ~ )( ~ 21 ji eKeF  then )( ~ )( ~ 12 ij eKeF  , )( ~ )( ~ 21 ji eKeK  and cxeF i =))(( ~ 11 By the definition, ++=       =    =  2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ),(deg 2111111 ~~ Ae Ae Exx yy i Ae Ae Eyy xx iGG j i j i xxeKxeFyx                  ++ 2 1 221 121 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ ))(( ~ 2112112111 Ae Ae Eyy Exx i Ae Ae Exx yy i Ae Ae Eyy xx ij j i j i j i xxeKxxeKxeFxeF Some studies on products of fuzzy soft graphs 251 +++=    jGG iGj Ae iGjGG eydxcdexdexeFydeyx c i )()()())(( ~ )(),(deg 111 ~ 11111 ~~ * 2 * 1 1 1 * 221  )()()()( 11 ~ 11 ~ * 21 * 2 1 ydxdeydxde GGjGG j c + )](1)[()]()(1)[( 11111 ~ * 1 * 2 * 2 * 2 1 xdydeecydydxde cc GG jiGGG j ++++= Theorem 6.4: Let ), ~ , ~ (: ~ 1111 AKFG and ), ~ , ~ (: ~ 2222 AKFG be two fuzzy soft graphs on * 1 G and * 2 G respectively. If )( ~ )( ~ 12 ij eKeF  and )( ~ 2 j eF is a constant function with cyeF lj =))(( ~ 2 2Vyl  , then )](1)[()]()(1)[(),(deg 11111 ~ 11 ~~ * 2 * 1 * 1 * 1 221 ydxdeecxdxdydeyx cc GG jiGGG iGG ++++=   Proof: Proof is analogues to the proof of Theorem 6.3. Theorem 6.5 : Let 1 ~ G and 2 ~ G be two fuzzy soft graphs with )( ~ )( ~ 21 ji eKeF  and 11 ))(( ~ VxcxeF kki = then 21 ~~ GG   is a regular FSG if and only if 1 ~ G is regular, * 1 G and * 2 G are regular graphs. Proof: Given cxeF ki =))(( ~ 1 , )( ~ )( ~ 21 ji eKeF  , )( ~ )( ~ 12 ij eKeF  then )( ~ )( ~ 21 ji eKeK  . Let 1 ~ G be a regular FSG of degree m, * 1 G and * 2 G are regular graphs of degree p and q respectively.      = Aee Eyxyx jiGG ji yxyxeeKKyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   Using Theorem 6.3, )](1)[()]()(1)[( 11111 ~ * 1 * 2 * 2 * 2 1 xdydeecydydxde cc GG jiGGG j ++++= ]1[]1[ 122 c ji c j EpeecEEme ++++= 21 ~~ GG   is a regular FSG. Conversely, Let 21 ~~ GG   be a regular FSG. ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx GGGG   = =++++ )](1)[()]()(1)[( 11111 ~ * 1 * 2 * 2 * 2 1 xdydeecydydxde cc GG jiGGG j )](1)[()]()(1)[( 22222 ~ * 1 * 2 * 2 * 2 1 xdydeecydydxde cc GG jiGGG j ++++= Fix 2 Vy , Shashikala S and Anil P N 252 =++++ )](1)[()]()(1)[( 11 ~ * 1 * 2 * 2 * 2 1 xdydeecydydxde cc GG jiGGG j )](1)[()]()(1)[( 22 ~ * 1 * 2 * 2 * 2 1 xdydeecydydxde cc GG jiGGG j ++++= )()( 2 ~ 1 ~ 11 xdxd GG = and )()( 21 * 1 * 1 xdxd cc GG = i.e. 1 ~ G and * 1 G are regular. Fix 1 Vx , =++++ )](1)[()]()(1)[( * 1 * 2 * 2 * 2 1 111 ~ xdydeecydydxde cc GG jiGGG j )](1)[()]()(1)[( * 1 * 2 * 2 * 2 1 222 ~ xdydeecydydxde cc GG jiGGG j ++++= This holds good when )()( 21 * 2 * 2 ydyd GG = and )()( 21 * 2 * 2 ydyd cc GG = * 2 G is regular. Theorem 6.6: Let 1 ~ G and 2 ~ G be two FSGs on complete graphs * 1 G and * 2 G respectively. If )( ~ )( ~ 21 ji eKeF  and )( ~ )( ~ 12 ij eKeF  then 21 ~~ GG   is a regular FSG if and only if 1 ~ G and 1 ~ G are regular FSGs. Proof: Let 1 ~ G and 2 ~ G be regular FSGs of degree m and n respectively. Given )( ~ )( ~ 21 ji eKeF  and )( ~ )( ~ 12 ij eKeF  , * 1 G and * 2 G are complete graphs of degree p and q respectively. From Theorem 6.1, i) If )( ~ )( ~ 21 ji eKeK  then )()](1)[(),(deg 1 ~ 11 ~ 11 ~~ 2 * 2121 ydeydxdeyx GiGGjGG ++=   neqme ij ++= ]1[ 21 ~~ GG   is a regular FSG. ii) If )( ~ )( ~ 12 ij eKeK  then )()](1)[(),(deg 1 ~ 11 ~ 11 ~~ 1 * 1221 xdexdydeyx GjGGiGG ++=   mepne ji ++= ]1[ 21 ~~ GG   is a regular FSG. Conversely, Let 21 ~~ GG   be a regular FSG. Some studies on products of fuzzy soft graphs 253 ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx GGGG   = For )(~)(~ 21 ji eKeK  , )()](1)[()()](1)[( 2 ~ 22 ~ 1 ~ 11 ~ 2 * 212 * 21 ydeydxdeydeydxde GiGGjGiGGj ++=++ Fix 1 Vx , )()](1)[()()](1)[( 2 ~ 2 ~ 1 ~ 1 ~ 2 * 212 * 21 ydeydxdeydeydxde GiGGjGiGGj ++=++ Since * 1 G and * 2 G are complete graphs, we get )()( 2~1~ 22 ydyd GG = 2 ~ G is a regular FSG. Fix 2 Vy , )()](1)[()()](1)[( 2 * 212 * 21 ~ 2 ~~ 1 ~ ydeydxdeydeydxde GiGGjGiGGj ++=++ )()( 2 ~ 1 ~ 11 xdxd GG = 1 ~ G is a regular FSG. ii) For )( ~ )( ~ 12 ij eKeK  , similar process is followed and we get 1 ~ G and 2 ~ G as regular FSGs. Theorem 6.7 : Let 1 ~ G and 2 ~ G be two FSGs with )( ~ )( ~ 12 ij eKeF  and 22 ))(( ~ VycyeF llj = then 21 ~~ GG   is a regular FSG if and only if 2 ~ G is regular, * 1 G and * 2 G are regular graphs. 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