Regular product vague graphs and product vague line graphs

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1 APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Regular product vague graphs and product vague lne graphs Ganesh Ghora 1 * and Madhumangal Pal 1 Receved: 26 December 2015 Accepted: 08 July 2016 Frst Publshed: 21 July 2016 *Correspondng author: Ganesh Ghora Department of Appled Mathematcs wth Oceanology and Computer Programmng Vdyasagar Unversty Mdnapore West Bengal Inda E-mal: ghoraganesh@gmalcom Revewng edtor: Ca Heng L Unversty of Western Australa Australa Addtonal nformaton s avalable at the end of the artcle Abstract: Vague graph s a generalzed structure of fuzzy graph whch gves more precson flexblty and compatblty to a system when compared wth systems that are desgned usng fuzzy graphs In ths paper we ntroduced the noton of regular totally regular product vague graphs and product vague lne graph We proved that under some condtons regular and totally regular product vague graph becomes equvalent Some propertes of product vague lne graph are nvestgated We showed that a product vague graph s somorphc to ts correspondng product vague lne graph under some condtons Subjects: Advanced Mathematcs; Combnatorcs; Dscrete Mathematcs; Mathematcal Logc; Mathematcs & Statstcs; Scence Keywords: vague sets; product vague graphs; regular product vague graphs; totally regular product vague graphs; product vague lne graphs AMS subject classfcatons: 05C72; 05C76 1 Introducton Now a days most mathematcal models are developed usng fuzzy sets to handle varous types of systems contanng elements of uncertanty In 1993 Gau and Buehrer (1993) ntroduced the noton of vague set theory as a generalzaton of Zadeh fuzzy set theory (1965) Vague sets are hgher order ABOUT THE AUTHORS Ganesh Ghora s an assstant professor n the Department of Appled Mathematcs Vdyasagar Unversty Inda Hs research nterests nclude fuzzy sets fuzzy graphs and graph theory Madhumangal Pal s a professor of Appled Mathematcs Vdyasagar Unversty Inda He receved jontly wth G P Bhattacherjee the Computer Dvson Medal from the Insttute of Engneers (Inda) n 1996 for best research work He receved the Bharat Jyot Award from Internatonal Frendshp Socety New Delh n 2012 He has publshed more than 250 artcles n nternatonal and natonal journals and 31 artcles n edted books and n conference proceedngs Hs specalzatons nclude computatonal graph theory genetc algorthms and parallel algorthms fuzzy correlaton and regresson fuzzy game theory fuzzy matrces and fuzzy algebra He s the edtorn-chef of Journal of Physcal Scences and Annals of Pure and Appled Mathematcs and a member of the edtoral boards of several other journals PUBLIC INTEREST STATEMENT The theoretcal concepts of graphs are hghly utlzed by computer scence applcatons especally n research areas of computer scence such as data mnng mage segmentaton clusterng mage capturng and networkng The vague graphs are more flexble and compatble than fuzzy graphs due to the fact that they have many applcatons n networks In ths paper the concept of vague sets s appled to defne and study many mportant propertes of regular totally regular product vague graphs and product vague lne graphs 2016 The Author(s) Ths open access artcle s dstrbuted under a Creatve Commons Attrbuton (CC-BY) 40 lcense Page 1 of 13

2 fuzzy sets Applcaton of hgher order fuzzy sets makes the soluton-procedure more complex but f the complexty on computaton-tme computaton-volume or memory-space are not matters of concern then we can acheve better results In a fuzzy set each element s assocated wth a pontvalue selected from the unt nterval [0 1] whch s termed as the grade of membershp n the set Instead of usng pont-based membershp as n fuzzy sets nterval-based membershp s used n a vague set The nterval-based membershp n vague sets s more expressve n capturng vagueness of data There are some nterestng features for handlng vague data that are unque to vague sets For example vague sets allow for a more ntutve graphcal representaton of vague data whch facltates sgnfcantly better analyss n data relatonshps ncompleteness and smlarty measures Consderng the fuzzy relatons between fuzzy sets Rosenfeld (1975) frst ntroduced the concept of fuzzy graphs n 1975 and developed the structure of fuzzy graphs obtanng analogous of several graph concepts Ramakrshna (2009) ntroduced the concept of vague graphs Akram et al studed some propertes of vague graphs (Akram Chen & Shum 2013) vague hypergraphs (Akram Gan & Saed 2014) regularty n vague ntersecton graphs (Akram Dudek & Yousaf 2014) rregular and hghly rregular vague graphs (Akram Feng Sarwar & Jun 2014) and vague cycles and vague trees (Akram Feng Sarwar & Jun 2015) Taleb Rashmanlou and Mehdpoor (2013) Taleb Mehdpoor and Rashmanlou (2014) studed somorphsm and operatons on vague graphs Borzooe and Rashmanlou (2015a 2015b 2015c 2016) Rashmanlou and Borzooe ( ) ntroduced many new concepts of vague graphs Samanta et al ntroduced fuzzy competton graphs (Samanta Akram & Pal 2013; Samanta & Pal 2015) fuzzy planar graphs (Samanta & Pal 2015) bpolar fuzzy ntersecton graphs (Samanta & Pal 2014) and strength of vague graphs (Samanta Pal Rashmanlou & Borzooe 2016) Later on Ghora and Pal studed some propertes of generalzed m-polar fuzzy graphs (2016a) defned operatons and densty of m-polar fuzzy graphs (2015) ntroduced m-polar fuzzy planar graphs (2016b) and defned faces and dual of m-polar fuzzy planar graphs (2016c) In ths paper the concept of regular totally regular product vague graphs and product vague lne graphs are ntroduced Necessary and suffcent condton s establshed under whch regular and totally regular product vague graph becomes equvalent and a product vague graph s somorphc to ts correspondng product vague lne graph 2 Prelmnares In ths secton we pont out some basc defntons of graphs The readers are encouraged to see these references (Balakrshnan 1997; Harary 1972; Mordeson & Nar 2000) for further study Defnton 21 Harary (1972) A graph s an ordered par G =(V E) where V s the set of vertces of G and E s the set of all edges of G Two vertces x and y n an undrected graph G are sad to be adjacent n G f xy s an edge of G A smple graph s an undrected graph that has no loops and not more than one edge between any two dfferent vertces A subgraph of a graph G =(V E) s a graph H =(W F) where W V and F E We wrte xy E to mean (x y) E and f e = xy E we say x and y are adjacent Formally gven a graph G =(V E) two vertces x y V are sad to be neghbors or adjacent nodes f xy E The neghborhood of a vertex v n a graph G s the nduced subgraph of G consstng of all vertces adjacent to v and all edges connectng two such vertces The neghborhood of v s often denoted by N(v) The degree deg(v) of vertex v s the number of edges ncdent on v The open neghborhood for a vertex v n a graph G conssts of all vertces adjacent to v but not ncludng v e N(v) ={u V:uv E} If v s ncluded n N(v) then t s called closed neghborhood for v and s denoted by N[v] e N[v] =N(v) {v} A regular graph s a graph where each vertex has the same open neghborhood degree A complete graph s a smple graph n whch every par of dstnct vertces has an edge Page 2 of 13

3 An somorphsm φ of the graphs G 1 and G 2 s a bjecton between the vertex sets of G 1 and G 2 such that any two vertces v 1 and v 2 of G 1 are adjacent n G 1 f and only f φ(v ) 1 and φ(v 2 ) are adjacent n G 2 If G 1 and G 2 are somorphc then we denote t by G 1 G 2 The lne graph L(G ) of a smple graph G s a graph whch represents the adjacentness between edges of G For a graph G ts lne graph L(G ) s a graph such that: () Each vertex of L(G ) represents an edge of G and () Two vertces of L(G ) are adjacent f and only f ther correspondng edges have a common end pont n G Let G =(V E) be a graph where V ={v 1 v n } Let S ={v x 1 x q } where x j E and x j has v as a vertex j = 1 2 q ; = 1 2 n Let S ={S 1 S 2 S n } Let T ={S :S S S j} Then P(S) =(S T) s an ntersecton graph and P(S) =G The lne graph L(G ) of G s by defnton the ntersecton graph P(E) That s L(G )=(Z W) where Z = {{x} {u x v x }:x E u x v x V x = u x v x } and W ={S x :S x x y E x y} and S x ={x} {u x v x } x E Defnton 22 Gau and Buehrer (1993) A vague set on a non-empty set X s a par (t A f A ) where t A :X [0 1] f A :X [0 1] are true and false membershp functons respectvely such that t A (x)+f A (x) 1 for all x X In the above defnton t A (x) s consdered as the lower bound for degree of membershp of x n A (based on evdence for) and f A (x) s the lower bound for negaton of membershp of x n A (based on evdence aganst) Therefore the degree of membershp of x n the vague set A s characterzed by the nterval [t A (x)1 f A (x)] So a vague set s a specal case of nterval-valued sets studed by many mathematcans and appled n many branches of mathematcs Vague sets also have many applcatons The nterval [t A (x)1 f A (x)] s called the vague value of x n A and s denoted by V A (x) We denote zero vague and unt vague value by 0 =[0 0] and 1 =[1 1] respectvely It s worth to menton here that nterval-valued fuzzy sets are not vague sets In nterval-valued fuzzy sets an nterval-valued membershp value s assgned to each element of the unverse consderng the evdence for x only wthout consderng evdence aganst x In vague sets both are ndependently proposed by the decson-maker Ths makes a major dfference n the judgment about the grade of membershp A vague relaton s a further generalzaton of a fuzzy relaton Defnton 23 Ramakrshna (2009) Let X and Y be ordnary fnte non-empty sets We call a vague relaton a vague subset of X Y that s an expresson R defned by R ={< (x y) t R (x y) f R (x y) > :x X y Y} where t R :X Y [0 1] and f R :X Y [0 1] whch satsfes the condton 0 t R (x y)+f R (x y) 1 for all (x y) X Y Defnton 24 Ramakrshna (2009) A vague relaton B on a set V s a vague relaton from V to V If A s a vague set on a set V then a vague relaton B on A s a vague relaton whch satsfes t B (x y) mn{t A (x) t A (y)} and f B (x y) max{t B (x) t B (y)} for all x y V Defnton 25 Ramakrshna (2009) Let G =(V E) be a crsp graph A par G =(V A B) s called a vague graph of G where A =(t A f A ) s a vague set on V and B =(t B f B ) s a vague set on E V V such that t B (x y) mn{t A (x) t A (y)} and f B (x y) max{t B (x) t B (y)} for each (x y) E We call A the vague vertex set of G and B as the vague edge set of G respectvely Page 3 of 13

4 A vague graph G s sad to be strong f t B (u v) = mn{t A (u) t A (v)} and f B (u v) = max{f A (u) f A (v)} for all (u v) E A vague graph G s sad to be complete f t B (u v) = mn{t A (u) t A (v)} and f B (u v) = max{f A (u) f A (v)} for all u v V 3 Regular and totally regular product vague graphs Throughout the paper G represents a crsp graph and G s a product vague graph of G Rashmanlou and Borzooe (2015) defned the product vague graphs as follows Here after we use xy E to denote (x y) E throughout the paper Defnton 31 A product vague graph of a graph G =(V E) s a par G =(V A B) where A =(t A f A ) s an vague set n V and B =(t B f B ) s a vague set on V 2 such that t B (xy) t A (x) t A (y) and f B (xy) f A (x) f A (y) for all x y V Note that every product vague graph s also a vague graph Example 32 Let us consder the graph G =(V E) where V ={v 1 v 3 v 4 } and E ={v 1 v 4 v 3 } A product vague graph G of G s shown n Fgure 1 Defnton 33 A product vague graph G =(V A B) of G =(V E) s sad to be strong f t B (xy) =t A (x) t A (y) and f B (xy) =f A (x) f A (y) for all xy E The product vague graph G of Fgure 1 s not strong Defnton 34 Let G =(V A B) be a product vague graph of G =(V E) The open neghborhood degree of a vertex v n G s defned by deg(v) =(deg t (v) deg f (v)) where deg t (v) = t B (uv) and u v deg f (v) = f B (uv) If all the vertces of G have the same open neghborhood degree (r 1 ) then G s u v uv E called (r 1 )-regular product vague graph uv E Defnton 35 Let G =(V A B) be a regular product vague graph of G =(V E) The order of G s defned as O(G) =(O t (G) O f (G)) where O t (G) = t A (v) and O f (G) = f A (v) The sze of G s defned as v V v V S(G) =(S t (G) S f (G)) where S t (G) = t B (uv) and S f (G) = f B (uv) uv E Defnton 36 Let G =(V A B) be a product vague graph of G =(V E) The closed neghborhood degree of a vertex v s defned by deg[v] =(deg t [v] deg f [v]) where deg t [v] = deg t (v)+t A (v) and deg f [v] = deg f (v)+f A (v) If each vertex of G has the same closed neghborhood degree (g 1 ) then G s called (g 1 )-totally regular product vague graph uv E Now we gve some examples whch show that product vague graphs may be both regular and totally regular nether totally regular nor regular and totally regular but not regular In other words there s no relaton between regular and totally regular product vague graphs Fgure 1 G s a product vague graph of G Page 4 of 13

5 Fgure 2 G s (0304)-regular and (0808)-totally regular product vague graph Frst we gve an example of a product vague graph whch s both regular and totally regular (see Fgure 2) Example 37 Let us consder the graph G =(V E) where V ={v 1 v 3 v 4 } and E ={v 1 v 2 v 4 v 3 v 4 v 1 v 3 } and consder the product vague graph G =(V A B) of G (see Fgure 2) Here deg(v 1 )= deg(v 2 )= deg(v 3 )= deg(v 4 )=(03 04) and deg[v 1 ]= deg[v 2 ]= deg[v 3 ]= deg[v 4 ]=(08 08) Hence G s both (0304)-regular and (0808)-totally regular product vague graph Now we gve an example of a product vague graph whch s nether regular nor totally regular (see Fgure 3) Example 38 Let us consder a product vague graph G =(V A B) of G =(V E) where V ={v 1 v 3 } and E ={v 1 v 2 v 1 v 3 }(see Fgure 3) We have deg(v 1 )=(04 045) deg(v 2 )=(02 02) deg(v 3 )=(02 025) and deg[v 1 ]=(08 095) deg[v 2 ]=(07 06) deg[v 3 ]=(08 065) Hence G s nether regular nor totally regular product vague graph The followng example shows that a product vague graph may be totally regular but not regular (see Fgure 4) Example 39 Consder the product vague graph G gven n Fgure 4 Snce deg(v 1 )= deg(v 2 )=( ) deg(v 3 )=( ) and deg[v 1 ]= deg[v 2 ]= deg[v 3 ]=( ) therefore G s ( )-totally regular and but not regular product vague graph Smlarly we can gve example of a product vague graph whch s regular but not totally regular (see Fgure 5) We now state the followng propostons wthout proof Proposton 310 Let G =(V A B) be a (r 1 )-regular product vague graph of G =(V E) Then S(G) = n (r 1 2 ) where V = n Fgure 3 G s nether regular nor totally regular product vague graph Page 5 of 13

6 Fgure 4 G s (067028)-totally regular but not regular product vague graph Fgure 5 G s (024008)-regular but not totally regular product vague graph Proposton 311 Let G =(V A B) be a (g 1 )-totally regular product vague graph of G =(V E) Then 2S(G)+O(G) =n(g 1 ) where V = n Proposton 312 Let G =(V A B) be a (r 1 )-regular and (g 1 )-totally regular product vague graph of G =(V E) Then O(G) =n(g 1 r 1 r 2 ) where V = n Theorem 313 Let G =(V A B) be a product vague graph of G =(V E) Then A =(t A f A ) s constant functon f and only f the followng are equvalent: () G s (r 1 )-regular vague graph () G s (g 1 )-totally regular vague graph Proof Let us assume that A =(t A f A ) s constant functon Therefore let t A (v) =a 1 and f A (v) =a 2 for all v V where a 1 a 2 [0 1] We wll now show that the statements () and () are equvalent () (): Let G be a (r 1 )-regular product vague graph Therefore deg(v) =(deg t (v) deg f (v)) = (r 1 ) for all v V Now deg[v] =(deg t [v] deg f [v]) = ( deg t (v)+t A (v) deg f (v)+f A (v)) = (r 1 + a 1 + a 2 ) for all v V Hence G s (r 1 + a 1 + a 2 )-totally regular product vague graph () (): Let G be a (g 1 )- totally regular product vague graph Then deg[v] =(deg t [v] deg f [v]) = (g 1 ) for all v V e deg t (v)+t A (v) =g 1 and deg f (v)+f A (v) =g 2 for all v V or deg t (v) =g 1 a 1 and deg f (v) =g 2 a 2 for all v V Hence G s (g 1 a 1 a 2 )-regular product vague graph Conversely let () and () are equvalent Suppose A s not constant functon Ths means there exst at least two vertces u v V such that t A (u) t A (v) and f A (u) f A (v) Let G be a (r 1 )-regular product vague graph Then Page 6 of 13

7 deg[u] =(deg t (u)+t A (u) deg f (u)+f A (u))=(r 1 + t A (u) + f A (u)) and deg[v] =(deg t (v)+t A (v) deg f (v)+f A (v)) = (r 1 + t A (v) + f A (v)) Ths shows that deg[u] deg[v] snce t A (u) t A (v) and f A (u) f A (v) Hence G s not totally regular whch s a contradcton to the assumpton that () and () are equvalent Therefore A must be constant In a smlar way we can show that f A s not constant functon then G totally regular does not mply G s regular Proposton 314 Let G =(V A B) be a product vague graph whch s both regular and totally regular Then A =(t A f A ) s constant Proof Let G be a (r 1 )-regular and (g 1 )-totally regular product vague graph Now deg t [v] =deg t (v)+t A (v) =r 1 + t A (v) =g 1 and deg f [v] = deg f (v)+f A (v) =r 2 + f A (v) =g 2 for all v V Hence t A (v) =g 1 r 1 and f A (v) =g 2 r 2 for all v V Ths shows that A(v) =(t A (v) f A (v)) = (g 1 r 1 r 2 ) for all v V e A s constant Remark 315 The converse of the Proposton 314 s not true always For example consder the product vague graph G =(V A B) of G =(V E) where V ={v 1 v 3 } and E ={v 1 v 2 v 3 v 3 v 1 } (see Fgure 6) Here A(v) =(t A (v) f A (v)) = (04 02) for all v V e A s constant but G s nether regular nor totally regular Theorem 316 Let G =(V A B) be a product vague graph of G =(V E) where G s an odd cycle Then G s regular product vague graph of G f and only f B =(t B f B ) s constant Proof Let G be a (r 1 )-regular product vague graph Let e 1 e 2 e 2n+1 be the edges of G such that e = v 1 v E v 0 v V = 1 2 2n + 1 and v 0 = v 2n+1 Let t B )=k 1 and f B )=k 2 where k 1 k 2 [0 1] G s (r 1 )-regular mples that deg t (v 1 and deg f (v 1 Ths means deg t (v 1 )+t B and deg f (v 1 )+f B e k 1 + t B and k 2 + f B e t B k 1 and f B k 2 Agan deg t (v 2 )+t B (e 3 and deg f (v 2 )+f B (e 3 Ths mples t B (e 3 (r 1 k 1 )=k 1 and f B (e 3 (r 2 k 2 )=k 2 and so on { { k1 f s odd Therefore t B (e )= (r 1 k 1 ) f s even and f (e )= k2 f s odd B (r 2 k 2 ) f s even Fgure 6 A s constant but G s nether regular nor totally regular Page 7 of 13

8 Therefore t B n+1 )=k 1 and f B n+1 )=k 2 Snce e 1 and e 2n+1 are ncdent on the vertex v 0 and deg(v 0 )=(r 1 ) we have t B )+t B n+1 and f B )+f B n+1 e 2k 1 = r 1 and 2k 2 = r 2 e k 1 = r 1 2 and k 2 = r 2 2 Therefore t B (e )= r 1 and f (e )= r 2 2 B for all = 1 2 2n + 1 Hence B s constant 2 Conversely let B be a constant functon Let B(uv) =(t B (uv) f B (uv)) = (k 1 k 2 ) for all uv E where k 1 k 2 [0 1] Then deg(v) =(deg t (v) deg f (v)) = ( t B (uv) t B (uv))=(2k 1 2k 2 ) for all v V u v u v uv E uv E Consequently G s (2k 1 2k 2 )-regular product vague graph 4 Product vague lne graphs In ths secton we frst defne a product vague ntersecton graph of a product vague graph Fnally we defne the product vague lne graphs Defnton 41 Let P(S) =(S T) be an ntersecton graph of a smple graph G =(V E) Let G =(V A B) be a product vague graph of G We defne a product vague ntersecton graph P(G) =(A 1 ) of P(S) as follows: () A 1 and B 1 are vague subsets of S and T respectvely () t A1 )=t A ) f A1 )=f A ) () t B1 ) f B1 ) for all S S S T In other words any product vague graph of P(S) s called a product vague ntersecton graph The followng proposton s mmedate Proposton 42 Let G =(V A B) be a product vague graph of G =(V E) and P(G) =(A 1 ) be a product vague ntersecton graph of P(S) Then the followng holds: (a) P(G) s a product vague graph of P(S) (b) G P(G) Proof (a) Snce G =(V A B) s a product vague graph we have by Defnton 41 t B1 ) t A ) t A ( )=t A1 ) t A1 ( ) and f B1 ) f A ) f A ( )=f A1 ) f A1 ( ) Hence P(G) s a product vague graph (b) Let us defne a mappng φ:v S by φ )=S for = 1 2 n Then clearly φ s one to one mappng of V onto S Now v E f and only f S T and T ={φ )φ( ):v E} Also t A )=t A1 )=t A1 (φ )) and f A )=f A1 )=f A1 (φ )) for all v V t B 1 (φ )φ( )) and f B 1 (φ )φ( )) for all v E Hence φ s an somorphsm of G onto P(G) e G P(G) Ths proposton shows that any product vague graph s somorphc to a product vague ntersecton graph The product vague lne graph of a product vague graph s defned as below Page 8 of 13

9 Defnton 43 Let L(G )=(Z W) be a lne graph of a smple graph G =(V E) Let G =(V A B) be a product vague graph of G Then a product vague lne graph L(G) =(A 1 ) of G s defned as follows: () A 1 and B 1 are vague subsets of Z and W respectvely () t A1 (x) =t B (u x v x ) () f A1 (x) =f B (u x v x ) (v) t B1 (x) t B (y) =t B (u x v x ) t B (u y v y ) (v) f B1 (x) f B (y) =f B (u x v x ) f B (u y v y ) for all S x Z and S x W Example 44 Consder now a graph G =(V E) where V ={v 1 v 3 v 4 } and E ={x 1 = v 1 v 2 x 2 = v 2 v 3 x 3 = v 3 v 4 x 4 = v 4 v 1 } Let G =(V A B) be a product vague graph of G (see Fgure 7) Now consder a lne graph L(G )=(Z W) such that Z ={S x1 S x2 S x3 S x4 } and W ={S x1 S x2 S x2 S x3 S x3 S x4 S x4 S x1 } Let A 1 and B 1 be vague subsets of Z and W respectvely Then we have t A1 1 (x 1 )=015 t A1 2 (x 2 )=014 t A1 3 (x 3 )=025 t A1 4 (x 4 )=025 f A1 1 (x 1 )=007 f A1 2 (x 2 )=015 f A1 3 (x 3 )=017 f A1 4 (x 4 )=009 t B1 1 S x2 (x 1 ) t B (x 2 )= = 0021 t B1 2 S x3 (x 2 ) t B (x 3 )= = 0035 t B1 3 S x4 (x 3 ) t B (x 4 )= = t B1 4 S x1 (x 4 ) t B (x 1 )= = f B1 1 S x2 (x 1 ) f B (x 2 )= = f B1 2 S x3 (x 2 ) f B (x 3 )= = f B1 3 S x4 (x 3 ) f B (x 4 )= = f B1 4 S x1 (x 4 ) f B (x 1 )= = Fgure 7 G s a product vague graph Fgure 8 The product vague lne graph L(G) of G Page 9 of 13

10 Hence L(G) =(A 1 ) s the product vague lne graph of G We see that L(G) s nether regular nor totally regular product vague lne graph Proposton 45 A product vague lne graph s a strong product vague graph Proof Follows from the defnton of product vague lne graph Proposton 46 lne graph of G If L(G) s a product vague lne graph of the product vague graph G then L(G ) s the Proof Snce G =(V A B) s a product vague graph and L(G) =(A 1 ) s a product vague lne graph therefore t A1 (x) and f A1 (x) for all x E and so S x Z x E Also t B1 (x) t B (y) and f B1 (x) f B (y) for all S x Z and so W ={S x :S x x y E x y} Ths completes the proof Proposton 47 L(G) =(A 1 ) s a product vague lne graph of some product vague graph G =(V A B) f and only f t B1 )=t A1 ) t A1 ( )) and f B1 )=f A1 ) f A1 ( ) for all S x W Proof Suppose that t B1 )=t A1 ) t A1 ( ) and f B1 )=f A1 ) f A1 ( ) for all S x W Let us now defne t A (x) =t A1 ) and f A (x) =f A1 ) for all x E Then t B1 )=t A1 ) t A1 ( )) = t A (x) t A (y)) and f B1 )=f A1 ) f A1 ( )=f A (x) f A (y) A vague set A =(t A f A ) that yelds that the property t B (xy) t A (x) t A (y) and f B (xy) f A (x) f A (y) wll suffce The converse part follows from the Defnton 43 Proposton 48 L(G) =(A 1 ) s a product vague lne graph of some product vague graph f and only f L(G )=(Z W) s a lne graph satsfyng t B1 (uv) =t A1 (u) t A1 (v) and f B1 (uv) =f A1 (u) f A1 (v) for all uv W Proof Follows from the Propostons 46 and 47 Defnton 49 Let G 1 =(V 1 A 1 ) and G 2 =(V 2 A 2 B 2 ) be two product vague graphs of the graphs G =(V E ) and G =(V E ) respectvely A homomorphsm between G and G s a mappng φ:v 1 V 2 such that () t A1 (x) t A2 (φ(x)) and f A1 (x) f A2 (φ(x)) for all x V 1 () t B1 (xy) t B2 (φ(x)φ(y)) and f B1 (xy) f B2 (φ(x)φ(y)) for all xy V 2 1 A bjectve homomorphsm wth the property that t A1 (x) =t A2 (φ(x)) and f A1 (x) =f A2 (φ(x)) for all x V 1 s called a (weak) vertex-somorphsm A bjectve homomorphsm wth the property that t B1 (xy) =t B2 (φ(x)φ(y)) and f B1 (xy) =f B2 (φ(x)φ(y)) for all xy V 2 1 s called a (weak) lne-somorphsm If φ s both (weak) vertex somorphsm and (weak) lne-somorphsm then φ s called a (weak) somorphsm of G 1 onto G 2 If G 1 s somorphc to G 2 then we wrte G 1 G 2 Proposton 410 Let G 1 =(A 1 ) and G 2 =(A 2 B 2 ) be two product vague graphs of the graphs G =(V E ) and G =(V E ) respectvely If φ s a weak somorphsm of G onto G then φ s an somorphsm of G onto 1 G 2 Proof Obvous Page 10 of 13

11 Proposton 411 Let L(G) =(A 1 ) be the product vague lne graph correspondng to the product vague graph G =(V A B) of G =(V E) Suppose that G s connected Then the followng hold: () There exsts a weak somorphsm of G onto L(G) f and only f G s a cycle and for all v V x E t A (v) =t B (x) f A (v) =f B (x) e A =(t A f A ) and B =(t B f B ) are constant functons on V and E respectvely takng on the same value () If φ s a weak somorphsm of G onto L(G) then φ s an somorphsm Proof Suppose that φ s a weak somorphsm of G onto L(G) By Proposton 410 φ s an somorphsm of G onto L(G ) By Proposton 46 G s a cycle (by Harary 1972) Theorem 82) Let V ={v 1 v n } and E ={x 1 = v 1 v 2 x 2 = v 2 v 3 x n = v n v 1 } where v 1 v 2 v n v 1 s a cycle Let us defne the vague sets t A )=s f A )= s and t B )=t f B )= t = 1 2 n v n+1 = v 1 s t s t [0 1] Then for s n+1 = s 1 ś n+1 = s 1 { t s s +1 t s ś +1 = 1 2 n (1) Now Z ={S x1 S x2 S xn } and W ={S x1 S x2 S x2 S x3 S xn S x1 } Also for t n+1 = t 1 and t n+1 = t 1 t A1 (x )=t f A1 (x )= t and t B1 S x+1 (x ) t B (x +1 )=t t +1 v n+1 = v 1 v n+2 = v 2 f B1 S x+1 (x ) f B (x +1 )= t t +1 = 1 2 n where Snce φ s an somorphsm of G onto L(G ) φ maps V one-to-one onto Z Also φ preserves adjacency Hence φ nduces a permutaton π of {1 2 n} such that φ )=S xπ() = S vπ() v π(+1) and x = v φ )φ+1 )=S vπ() v π(+1) S vπ(+1) v π(+2) = 1 2 (n 1) Now s = t A ) t A1 (φ )) = t A1 (S vπ() v π(+1) )=t π() s = f A ) f A1 (φ )) = f A1 (S vπ() v π(+1) )= t π() t = t B ) t B1 (φ )φ+1 )) = t B1 (S vπ() v π(+1) S vπ(+1) v π(+2) ) = t B1 (S vπ() v π(+1) ) t B1 (S vπ(+1) v π(+2) )=t π() t π(+1) t = f B ) f B1 (φ )φ+1 )) = f B1 (S vπ() v π(+1) S vπ(+1) v π(+2) ) = f B1 (S vπ() v π(+1) ) f B1 (S vπ(+1) v π(+2) )= t π() t π(+1) for = 1 2 n That s s t π() s t π() and { t t t π() π(+1) t t π() t π(+1) = 1 2 n By (2) we have t t π() t t π() for = 1 2 n and so t π() t π(π()) t π() t π(π()) for = 1 2 n Contnung we have (2) Page 11 of 13

12 t t π() t π j () t t t π() t π j () t and so t = t π() t = t π() = 1 2 n where π j+1 s the dentty map Agan by (2) we have t t π(+1) = t +1 t t π(+1) = t (+1) = 1 2 n where t n+1 = t n t n+1 = t n Hence by (1) and (2) we have t 1 = =t n = s 1 = s n t 1 = = t n = s 1 = = s n Thus we have not only proved the concluson about A and B beng constant functons but also we have shown that () holds Conversely suppose that G s a cycle and for all v V x E t A (v) =t B (x) f A (v) =f B (x) By Proposton 46 L(G ) s the lne graph of G Snce G s a cycle G L(G ) by (Harary (Harary (1972)) Theorem 82) Ths somorphsm nduces an somorphsm of G onto L(G) snce t A (v) =t B (x) f A (v) =f B (x) for all v V x E and so A = B = A 1 = B 1 on ther respectve domans Proposton 412 Let G 1 =(V 1 A 1 ) and G 2 =(V 2 A 2 B 2 ) be two product vague graphs of the graphs G =(V E ) and G =(V E ) respectvely such that G and 1 G s connected Let L(G )=(A B ) and L(G 2 )=(A 4 B 4 ) be the product vague lne graphs correspondng to G 1 and G 2 respectvely Suppose that t s not the case that one of G and 1 G s complete graph K 2 3 and other s bpartte complete graph K 13 If L(G 1 ) L(G 2 ) then G 1 and G 2 are lne somorphc Proof Snce L(G 1 ) L(G 2 ) therefore by Proposton 410 L(G 1 ) L(G ) Snce 2 L(G ) and 1 L(G 2 ) are the lne graphs of G and 1 G respectvely by Proposton 46 we have that 2 G 1 G 2 by (Harary (1972) Theorem 83) Let ψ be the somorphsm of L(G 1 ) onto L(G 2 ) and φ be the somorphsm of G onto 1 G Then 2 t A3 )=t A4 (ψ )) = t A4 (S φ(x) ) f A3 )=f A4 (ψ )) = f A4 (S φ(x) ) where the latter equaltes holds by the proof of (Harary (1972) Theorem 83) and so t B1 (x) =t B2 (φ(x)) f B1 (x) =f B2 (φ(x)) Hence G 1 and G 2 are lne somorphc 5 Conclusons Graph theory has several nterestng applcatons n system analyss operatons research computer applcatons and economcs Snce most of the tme the aspects of graph problems are uncertan t s nce to deal wth these aspects va the methods of fuzzy systems It s known that fuzzy graph theory has numerous applcatons n modern scences and technology especally n the felds of neural networks artfcal ntellgence and decson-makng In ths paper we defned the notons of regular totally regular product vague graphs and product vague lne graphs We nvestgated some propertes of them In our future work we wll focus on categorcal propertes on product vague graphs edge regular and rregular product vague graph product vague competton graph Fundng The authors receved no drect fundng for ths research Author detals Ganesh Ghora 1 E-mal: ghoraganesh@gmalcom Madhumangal Pal 1 E-mal: mmpalvu@gmalcom 1 Department of Appled Mathematcs wth Oceanology and Computer Programmng Vdyasagar Unversty Mdnapore West Bengal Inda Ctaton nformaton Cte ths artcle as: Regular product vague graphs and product vague lne graphs Ganesh Ghora & Madhumangal Pal Cogent Mathematcs (2016) 3: References Akram M Chen W & Shum K P (2013) Some propertes of vague graphs Southeast Asan Bulletn of Mathematcs Akram M Dudek W A & Yousaf M M (2014) Regularty n vague ntersecton graphs and vague lne graphs Abstract and Appled Analyss Hndwa Publshng Corporaton 10 Artcle ID do:101155/2014/ Akram M Farooq A Saed A B & Shum K P (2015) Certan types vague cycles and vague trees Journal of Intellgent and Fuzzy Systems Akram M Feng F Sarwar S & Jun Y B (2014) Certan types of vague graphs Unversty Poltehnca of Bucharest Scentfc Bulletn Seres A Akram M Gan A N & Saed A B (2014) Vague hypergraphs Journal of Intellgent and Fuzzy Systems Page 12 of 13

13 Balakrshnan V K (1997) Graph theory McGraw-Hll Borzooe R A & Rashmanlou H (2015a) Domnaton n vague graphs and ts applcatons Journal of Intellgent and Fuzzy Systems Borzooe R A & Rashmanlou H (2015b) New concepts of vague graphs Internatonal Journal of Machne Learnng and Cybernetcs do:101007/s x Borzooe R A & Rashmanlou H (2015c) Degree of vertces n vague graphs Journal of appled mathematcs and nformatcs Borzooe R A & Rashmanlou H (2016) Sem global domnaton sets n vague graphs wth applcaton Journal of Intellgent and Fuzzy Systems Gau W L & Buehrer D L (1993) Vague sets IEEE Transacton on Systems Man and Cybernetcs Ghora G & Pal M (2015) On some operatons and densty of m-polar fuzzy graphs Pacfc Scence Revew A: Natural Scence and Engneerng Ghora G & Pal M (2016) Some propertes of m-polar fuzzy graphs Pacfc Scence Revew A: Natural Scence and Engneerng do:101016/jpsra Ghora G & Pal M (2016b) A study on m-polar fuzzy planar graphs Internatonal Journal of Computatonal Scence and Engneerng Ghora G & Pal M (2016c) Faces and dual of m-polar fuzzy planar graphs Journal of Intellgent and Fuzzy Systems do:103233/jifs Harary F (1972) Graph theory (3rd ed) Readng MA: Addson-Wesely Mordeson J N & Nar P S (2000) Fuzzy graphs and hypergraphs Physca Verlag Ramakrshna N (2009) Vague graphs Internatonal Journal of Computatonal Cognton Rashmanlou H & Borzooe R A (2015) A Note on vague graphs Journal of Algebrac Structures and Ther Applcatons Rosenfeld A (1975) Fuzzy graphs In L A Zadeh K S Fu & M Shmura (Eds) Fuzzy sets and ther applcatons (pp 77 95) New York: Academc Press Rashmanlou H & Borzooe R A (2015) Product vague graphs and ts applcatons Journal of Intellgent and Fuzzy Systems Rashmanlou H & Borzooe R A (2016) More results on vague graphs Unversty Poltehnca of Bucharest Scentfc Bulletn-Seres A Samanta S Akram M & Pal M (2015) M-step fuzzy competton graphs Journal of Appled Mathematcs and Computng Samanta S & Pal M (2013) Fuzzy k-competton graphs and p-compettons fuzzy graphs Fuzzy Informaton and Engneerng Samanta S & Pal M (2014) Some more results on bpolar fuzzy sets and bpolar fuzzy ntersecton graphs The Journal of Fuzzy Mathematcs 22(2) 1 10 Samanta S & Pal M (2015) Fuzzy planar graphs IEEE Transactons on Fuzzy Systems Samanta S Pal M Rashmanlou H & Borzooe R A (2016) Vague graphs and strengths Journal of Intellgent and Fuzzy Systems Taleb A A Mehdpoor N & Rashmanlou H (2014) Some operatons on vague graphs Journal of Advanced Research n Pure Mathematcs Taleb A A Rashmanlou H & Mehdpoor N (2013) Isomorphsm on vague graphs Annals of Fuzzy Mathematcs and Informatcs Zadeh L A (1965) Fuzzy sets Informaton and Control The Author(s) Ths open access artcle s dstrbuted under a Creatve Commons Attrbuton (CC-BY) 40 lcense You are free to: Share copy and redstrbute the materal n any medum or format Adapt remx transform and buld upon the materal for any purpose even commercally The lcensor cannot revoke these freedoms as long as you follow the lcense terms Under the followng terms: Attrbuton You must gve approprate credt provde a lnk to the lcense and ndcate f changes were made You may do so n any reasonable manner but not n any way that suggests the lcensor endorses you or your use No addtonal restrctons You may not apply legal terms or technologcal measures that legally restrct others from dong anythng the lcense permts Cogent Mathematcs (ISSN: ) s publshed by Cogent OA part of Taylor & Francs Group Publshng wth Cogent OA ensures: Immedate unversal access to your artcle on publcaton Hgh vsblty and dscoverablty va the Cogent OA webste as well as Taylor & Francs Onlne Download and ctaton statstcs for your artcle Rapd onlne publcaton Input from and dalog wth expert edtors and edtoral boards Retenton of full copyrght of your artcle Guaranteed legacy preservaton of your artcle Dscounts and wavers for authors n developng regons Submt your manuscrpt to a Cogent OA journal at wwwcogentoacom Page 13 of 13