Strong Regular L Fuzzy Graphs

1. International Journal of Trend in Scientific Research and Development (IJTSRD) UGC Approved International Open Access Journal ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume - 1 | Issue – 5 Strong Regular L-Fuzzy Graphs R. Seethalakshmi Saiva Bhanu Kshatriya College, Aruppukottai, Tamilnadu, India R.B. Gnanajothi V.V.Vanniaperumal College For Women, Virudhunagar, Tamilnadu, India ABSTRACT In this paper, the notion of regular and strong regular L-fuzzy graph is introduced. The conditions for strong regularity of L-fuzzy graphs on cycle and star graph are derived. Keywords:L-fuzzy graph, strong L-fuzzy graph, regular L-fuzzy graph, strong regular L-fuzzy graph AMS Classification: 05C72, 03E72 1.INTRODUCTION The first definition of fuzzy graph was proposed by Kaufmann [3] using fuzzy relations introduced by Zadeh[9]. The theory of fuzzy graphs was introduced by Azriel Rosenfeld [7] in 1975. Generalizing the notion of fuzzy sets, L. Goguen [2] introduced the notion of L-fuzzy sets. Pramada Ramachandran and Thomas [6] introduced the notion of L-fuzzy graphs and the degree of a vertex in a L-fuzzy graph. This motivated us to introduce the notion of regular L-fuzzy graphs and strong regular L-fuzzy graphs. Some preliminary, but primary work has been carried out. Comparison of set magic graphs and L-fuzzy structure introduces a new platform to work on fuzzy graph. 2.PRELIMINARIES Definition 2.1: [4] A fuzzy graph ? = ( ? ,? ) is a pair of functions ? ∶ ? → [0,1] and ?:? ? ? → [0,1] with ?(?,?) ≤ ?(?) ∧ ?(?),∀ ?,? ∈ ?, where V is a finite nonempty set and ∧ denote minimum. Definition 2.2: [4] The graph ?∗= ( ? ,? ) is called the underlying crisp graph of the fuzzy graph G, where ? = { ?/?(?) ≠ ? } and ? = {(?,?) ∈ ? ? ? /?(?,?) ≠ 0}. Definition 2.3: [5] A fuzzy graph ? = ( ? ,? ) is defined to be a strong fuzzy graph if ?(?,?) = ?(?) ∧ ?(?),∀ (?,?) ∈ ?. @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 1 | Issue – 5 | July-Aug 2017 Page: 503

2. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Definition 2.4: [5] A fuzzy graph ? = ( ? ,? ) is defined to be a complete fuzzy graph if ?(?,?) = ?(?) ∧ ?(?),∀ ?,? ∈ ?. Definition 2.5: [5] Let ? = ( ? ,? ) be a fuzzy graph on ?∗= ( ? ,? ). The fuzzy degree of a node ? ∈ ? is defined as (??)(?) = ∑ ?(?,?) ?≠?,?∈? . G is said to be regular fuzzy graph if each vertex has same fuzzy degree. If (??)(?) = ?,∀ ? ∈ ?, then G is said to be ? −regular fuzzy graph. Definition 2.6: [6] Let (?,∨,∧) be a complete lattice. A nonempty set V together with a pair of functions ? ∶ ? → ? and ?:? ? ? → ? with ?(?,?) ≤ ?(?) ∧ ?(?),∀ ?,? ∈ ?, is called an L-fuzzy graph. It is denoted by ??= (?,?,?). Definition 2.7: [6] Let ??= (?,?,?) be an L-fuzzy graph. The L-fuzzy degree ??(?) of a vertex ? in ?? is defined as ??(?) = ⋁ ?(?,?) ? ∈?,?≠? . Definition 2.8 : [1] A graph G=(V,E) is said to have a set-magic labeling if its edges can be assigned distinct subsets of a set X such that for every vertex u of G, the union of the subsets assigned to the edges incident at u is X. A graph is said to be a set-magic graph if it admits a set-magic labeling. 3.REGULAR L-FUZZY GRAPH The above definition of degree in an L-fuzzy graph does not coincide with usual definition of degree in a fuzzy graph. Any fuzzy graph can be treated as an L-fuzzy graph taking the lattice ( [0,1],≤ ). Fuzzy degree of a vertex ?, denoted as ??(?), defined as the sum of ?(?,?), where ? ∈ ?,? ≠ ?. For further discussion, L-fuzzy degree will be taken into account. Definition 3.1:Let ??= (?,?,?) be an L-fuzzy graph. ?? is said to be a regular L-fuzzy graph if each vertex has the same L-fuzzy degree. If ??(?) = ?,∀ ? ∈ ?, for some ? ∈ ?, then ??is called a ? − regular L-fuzzy graph. Example 3.2 : Let ? = {?,?,?} and L= ℘(?), the power set of ?. Then (?,⊆ ) is a complete lattice. Let ? = { ?1 ,?2 ,?3,?4,?5 } Define ?:? → ? and ?:? ? ? → ? by ?(?1 ) = {?},?(?2 ) = {?,?} = ?(?4 ), ?(?3 ) = ?(?5 ) = {?,?} and ?(?1, ?2) = ?(?1, ?3) = ?(?1, ?4) = ?(?2, ?3) = ?(?3, ?5) = ?(?3, ?4) = {?}. Then ??(?) = ∨? ∈?,?≠? ?(?,?) = {?},∀? ∈ ?. Then ??= (?,?,?) is a regular L-fuzzy graph. Example 3.3 : Consider the set S and the complete lattice L= ℘(?) given as in Example 3.2. Let ? = { ?1 ,?2 ,?3} @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 1 | Issue – 5 | July-Aug 2017 Page: 504

3. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Define ?:? → ? and ?:?? ? → ? by ?(?1 ) = {?,?,?},?(?2 ) = {?,?},?(?3 ) = {?,?} and ?(?1,?2) = {?}, ?(?2,?3) = {?,?}, ?(?3,?1) = {?}. Then ??= (?,?,?) is a L-fuzzy graph. Example 3.4 : Consider the set S and ℘(?) as given in example 3.2. Let ? = { ?1 ,?2 ,?3,?4,?5 } Define ?:? → ? and ?:?? ? → ? by ?(?1 ) = {?},?(?2 ) = {?,?},?(?3 ) = {?,?},?(?4 ) = {?,?,?},?(?5 ) = {?,?} and ?(?1, ?2) = ?(?2, ?3) = ?(?1, ?3) = ?(?1, ?4) = {?}, ?(?3, ?5) = {?},?(?3, ?4) = {?,?},?(?4, ?5) = {?,?} Then ??= (?,?,?) is a non-regular L-fuzzy graph. Remark : In general, an L-fuzzy graph, ??= (?,?,?), where ? is a constant function, is a regular L-fuzzy graph. Theorem 3.5 : Any set magic graph admits a regular L-fuzzy graph structure. Proof : Let G=(V,E) be a set magic graph. Then G admits a set magic labeling l. That is , the edges of G can be assigned distinct subsets of a set X such that for every vertex u of G, union of subsets assigned to the edges incident at u is X. Let L= ℘(?), the power set of ?. Then (?,⊆ ) is a complete lattice. Define ?:? → ? and ?:? → ? as follows: ?(?) = ?,∀ ? ∈ ? and ?(?) = ?(?), for all edges ? of G and ?(?) ∈ ?. Since, ?(?) ⊆ ?,?(?) ≤ ?(?) ∧ ?(?), for all ? = (?,?) ∈ ?. Then, ??= (?,?,?) is an L-fuzzy graph. For all u ∈ ?, ??(?) = ⋁ ?(?,?) ?∈?,?≠? = ⋁ ?(?) ?=(?,?),?≠? = ⋁ ?(?) ?=(?,?),?≠? =X, (since l is a set magic labeling) Hence, ??is a regular L-fuzzy graph. @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 1 | Issue – 5 | July-Aug 2017 Page: 505

4. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 4.STRONG REGULAR L-FUZZY GRAPH Definition 4.1 : An L-fuzzy graph ??= (V,σ,μ) is said to be a strong regular L-fuzzy graph if ?(?,?) = ?(?) ∧ ?(?), for all edges (u,v) of ?? and the L-fuzzy degree ??(?) is constant, for all ? ∈ ?. Example 4.2 : Let ? = {?,?,?} and L= ℘(?), the power set of ?. Then (?,⊆ ) is a complete lattice. Let ? = { ?1 ,?2 ,?3,?4 } Define ?:? → ? by ?(?1 ) = ?(?3 ) = {?,?,?},?(?2 ) = ?(?4 ) = {?,?} and ?(?1, ?2) = ?(?2, ?3) = ?(?3, ?4) = ?(?4, ?1) = {?,?,?}. Then ??= (?,?,?) is a strong regular L-fuzzy graph. Example 4.3 : If ? and ? are constants in a L-fuzzy graph??, then ?? is strong regular L-fuzzy graph. Remark : If ?? is a strong regular L-fuzzy graph, then ?? is a regular L-fuzzy graph. However, the converse need not be true, in general. This is seen from Example 3.3. Note In [8], it has been proved that no fuzzy graph on a star graph with at least two spokes is strong regular. But in case of L-fuzzy graph, there exist a strong regular L-fuzzy structure on star graph. Example 4.3 : Let L= {1,2,3}.(?,≤ ) is a complete lattice. Let ? = { ?,?1 ,?2 ,?3,?4 } Define ?:? → ? by ?(? ) = 1,?(?1 ) = 1,?(?2 ) = 2,?(?3 ) = 3,?(?4 ) = 4 and ?(?,??) = 1, ? = 1,2,3,4. Then, ??(?1 ) = ??(?2 ) = ??(?3 ) = ??(?4) = 1. Hence, ??= (?,?,?) is a strong regular L-fuzzy graph on a star graph. Example 4.4 :Let ? = {?,?,c} and L= ℘(?), the power set of ?.(?,⊆ ) is a lattice. 1.Let = { ?1 ,?2 ,?3,?4 }. Define ?:? → ? by ?(?1 ) = {?,?},?(?2 ) = {?},?(?3 ) = {?,?},?(?4 ) = {?} and ?(?1, ?2) = ?(?2, ?3) = ?(?3, ?4) = ?(?4, ?1) = {?}. Then, ??= (?,?,?) is strong regular L-fuzzy graph. 2.Let = { ?,?1 ,?2 ,?3,?4 }. Define ?:? → ? by ?(?1 ) = {?,?,?},?(?2 ) = {?},?(?3 ) = {?,?},?(?4 ) = {?,?},?(? ) = {?} and ?(?, ?1) = ?(?, ?2) = ?(?, ?3) = ?(?, ?4) = {?}. Then, ??= (?,?,?) is a strong regular L-fuzzy graph. Theorem 4.5 Let ??= (?,?,?) be the L-fuzzy graph. 1.If the underlying graph ?? all ? ∈ ?, then ??is a strong regular L-fuzzy graph. 2.If the underlying graph ?? if ?(?) ≤ ?(??),∀? = 1,2,3,…..? and ?(?,??) = ?(?), for all the edges (?,??), then ??is a strong regular L-fuzzy graph. ∗ is a cycle and if ? is a constant say k, for all edges (u,v) and ?(?) ≥ ?, for ∗ is a star graph with ? = { ?,?1,?2,…..??} and @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 1 | Issue – 5 | July-Aug 2017 Page: 506

5. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Proof 1.Let ? = constant = k(say). Since, ?(?) ≥ ?, for all ? ∈ ?,?(?) ∧ ?(?) ≥ ? = ?(?,?). From the definition of L-fuzzy graph, ?(?,?) ≤ ?(?) ∧ ?(?). Then, ?(?,?) = ?(?) ∧ ?(?). Also, ??(?) = ⋁? ∈?,?≠? ?(?,?) = ⋁? ∈?,?≠? ? = ?,∀? ∈ ?. Hence, ?? is strong regular L-fuzzy graph. 2.Let ?(?) ≤ ?(??),∀? = 1,2,3,…..?. Therefore, ?(??) ∧ ?(?) ≤ ?(?) = ?(?,??). From the definition of L-fuzzy graph, ?(?,??) ≤?(??) ∧ ?(?). Hence, ?(?,??) =?(?) ∧ ?(??), ∀? = 1,2,3,…..?. Now, ??(?? ) = ⋁? ∈? ?(?? ,?),∀? = 1,2,3,…..?. = ∨ ?(?) = ?(?) and ??(?) = ⋁?? ∈? ?(?,?? ),∀? = 1,2,3,…..?. = ∨ ?(?) = ?(?) Thus, ?? is strong regular L-fuzzy graph. Theorem 4.6 : For any strong regular L-fuzzy graph ??= (?,?,?) whose crisp graph ?? constant function. ∗ is a path, ? is a Proof Let ? = { ?0 ,?1 ,?2,…….,?? }.?(?? ∗ ) = {?0?1,?1?2,?2?3 ,………..,??−1?? } ?(?? ) = ??,??? ? = 0,1,2,……….? and ?(??−1, ??) = ?? , for ? = 1,2,……….?, where ?? ,??∈ (?,≤) Since, ?? is a strong regular L-fuzzy graph, we have ?1= ?.??∨ ??+1= ?,??? ? = 1,2,………? − 2,??= ?,?ℎ??? ? ∈ ? -----------------------(1) ??−1∧ ??= ??,??? ? = 1,2,………? ------------------------------------------------(2) (1) ⇒ ??≤ ?,? = 1,2,……….,? -----------------------------------------------------(3) Also (2) ⇒ ??≤ ??−1 ??? ??≤ ??,? = 1,2,……….,? ⇒ (?1≤ ?0,?1≤ ?1),(?2≤ ?1,?2≤ ?2),………,(??≤ ??−1,??≤ ??) ⇒ (?1≤ ?0),(?1,?2≤ ?1),(?2≤ ?2,?3≤ ?2),………,(??−1≤ ??,??≤ ??),(??≤ ??) @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 1 | Issue – 5 | July-Aug 2017 Page: 507

6. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 ⇒ ? ≤ ?0,??∨ ??+1≤ ??,? = 1,2,………,? − 1,? ≤ ??. ⇒ ? ≤ ?0,? ≤ ??,? = 1,2,………,? − 1,? ≤ ??. ⇒ ? ≤ ?0,??−1∧ ??,? = 1,2,………,?. ⇒ ? ≤ ??,??? ? = 1,2,………,? ---------------------------------------------(4) Equations (3) and (4) ⇒, ??= ?,??? ? = 1,2,………,?. Hence, ? is a constant function. Theorem 4.7 : For any strong regular L-fuzzy graph ??= (?,?,?) whose crisp graph ?? constant function. ∗ is a cycle, ? is a Proof Let ? = { ?1 ,?2,…….,?? }.?(?? ∗ ) = { ?1?2,?2?3 ,………..,??−1??,???1 } ?(?? ) = ??,??? ? = 1,2,……….? and (??, ??+1) = ?? , for ? = 1,2,……….?, where ??+1= ?1 ??? ?? ,??∈ (?,≤) Since, ?? is a strong regular L-fuzzy graph, we have ?1= ?.??∨ ??+1= ?,??? ? = 1,2,………? − 1,?ℎ??? ? ∈ ? -----------------------(1) ??∧ ??+1= ??,??? ? = 1,2,………? ,????? ??+1= ?1,??+1= ?1-------------------(2) (1)⇒ ??≤ ?,? = 1,2,……….,? -----------------------------------------------------(3) Also (2) ⇒ ??≤ ?? ??? ??≤ ??+1,? = 1,2,……….,? ⇒ (?1≤ ?1,?1≤ ?2),(?2≤ ?2,?2≤ ?3),………,(??≤ ??,??≤ ??+1= ?1) ⇒ (?1≤ ?1),(?1,?2≤ ?2),(?2,?3≤ ?3),………,(??−1,??≤ ??) ⇒ ? ≤ ?1,??∨ ??+1≤ ??+1,? = 1,2,………,? − 1 ⇒ ? ≤ ?1,? ≤ ??+1,? = 1,2,………,? − 1 ⇒ ? ≤ ??∧ ??+1,? = 1,2,………,?. ⇒ ? ≤ ??,??? ? = 1,2,………,? ---------------------------------------------(4) @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 1 | Issue – 5 | July-Aug 2017 Page: 508

7. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Equations (3) and (4) ⇒, ??= ?,??? ? = 1,2,………,?. Hence, ? is a constant function. REFERENCES 1)Acharya B D, Set valuations of a graph and their applications, MRI Lecture Notes in Applied Mathematics, No 2, Mehta Research Intitute, Allahabad, 1983. 2)Goguen. J. A, L-Fuzzy sets, J. Math. Anal.Appl. 18, (1967), 145 - 174. 3)Kaufmann, Introduction a la theorie des sous - ensembles flous, Elements theoriques de base, Parisi Masson et cie 1976. 4)Morderson, J.N, Nair, P.S, Fuzzy Graphs and Fuzzy Hyper Graphs, Physics - verlag, Heidelberg (2000) 5)Nagoorgani, A and Chandrasekaran V.T, A First Look at Fuzzy Graph Theory(2010) 6)Pramada Ramachandran, Thomas K V, On Isomorphism of L-Fuzzy graphs, Annals of Fuzzy Mathematics and Informatics, Vol 11, No 2(February 2016), pp 301-313 7)Rosenfeld A, Fuzzygraphs In : Zadeh, L.A. Fu, K.S.Shimura, M (Eds), Fuzzy sets and their Applications (Academic Press, Newyark) pp.77-95(1975). 8)Seethalakshmi R and Gnanajothi R.B, On Strong regular fuzzy graph, International Journal of Mathematical Sciences and Engineering Applications, ISSN 0973-9424, Vol 6 No.1( January 2012),pp 161-173. 9)Zadeh L.A. Similarity relations and fuzzy ordering Information sciences 971; 3(2): 177-200. @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 1 | Issue – 5 | July-Aug 2017 Page: 509