On total regularity of the join of two interval valued fuzzy graphs

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1 International Jornal of Scientific and Research Pblications, Volme 6, Isse 12, December On total reglarity of the join of two interval valed fzzy graphs Soriar Sebastian 1 and Ann Mary Philip 2 1 Department of Mathematics, St. Albert s College, Kochi , Kerala, India. 2 Department of Mathematics, Assmption College, Changanasserry , Kerala, India. Abstract- In this paper, we define the total degree (TD) of a vertex in the join of two interval valed fzzy graphs (IVFGs) and investigate their totally reglar property (TRP). In general, the join of two totally reglar IVFGs need not be a totally reglar interval valed fzzy graph (TRIVFG). We obtain some necessary and sfficient conditions for the join of two TRIVFGs to be totally reglar. Index Terms- Interval valed fzzy graph, total reglarity, join F I. INTRODUCTION zzy graph theory is finding an increasing nmber of applications in modeling real time systems where the level of information inherent in the system varies with different levels of precision. In the applied field, the sccess of the se of fzzy set theory depends on the choice of the membership fnction that we make. However, there are applications in which experts do not have precise knowledge of the fnction that shold be taken. In these cases, it is appropriate to represent the membership degree of each element of the fzzy set by means of an interval. From these considerations arises the extension of fzzy sets called the theory of interval valed fzzy sets. That is, fzzy sets sch that the membership degree of each element of the fzzy set is given by a closed sbinterval of the interval [0,1]. Replacing the membership fnctions of vertices and edges in fzzy graphs by interval valed fzzy sets sch that they satisfy some particlar conditions, interval valed fzzy graphs (IVFGs) were defined. Ths IVFGs provide a better description of vageness and ncertainty within the specific interval than the traditional fzzy graph. The basic concepts of fzzy sets, interval valed fzzy sets and fzzy graph theory can be fond in [8], [19] and [18] to [23]. In 2009, Hongmei and Lianha[6] introdced the definition of IVFG. Mhammad Akram and Wieslaw A. Ddek[1], defined the operations of cartesian prodct, composition, nion and join on IVFGs and investigated some properties. They also introdced the notion of interval valed fzzy complete graphs and obtained some properties of self complementary and self weak complementary interval valed fzzy complete graphs. Mhammad Akram [2] also introdced interval valed fzzy line graphs. A. A. Talebi and H. Rashmanlo [20] stdied on isomorphism of IVFGs. H. Rashmanlo and Yong Bae Jn[13] defined the three new operations, direct prodct, semi strong prodct and strong prodct of IVFGs and discssed their properties on complete IVFGs. Pradip Debnath[11] introdced domination in IVFGs. H. Rashmanlo and Madhmangal Pal defined irreglar IVFG[9], balanced IVFG[14] and antipodal IVFG[15] and stdied their properties. They also stdied on the properties of highly irreglar IVFG[17] and defined isometry on IVFG[16]. Mhammad Akram, Nora Omair Alshehri and W.A Ddek [3] introdced certain types of IVFG sch as balanced IVFGs, neighborly irreglar IVFGs, neighborly total irreglar IVFGs, highly irreglar IVFGs, highly total irreglar IVFGs. Again Mhammad Akram, M. Mrtaza Yosaf and W A Ddek [4] stdied the properties of self centered IVFGs. Madhmangal Pal, Sovan Samanta and H. Rashmanlo [10] defined the degree and total degree of an edge in the cartesian prodct and composition of two IVFG and obtained some reslts. Basheer Ahammed Mohideen [5] stdied on strong and reglar IVFGs. S. Ravi Narayanan and N. R. Santhi Maheswari [18] introdced strongly edge irreglar and strongly edge totally irreglar IVFG and made a comparative stdy between the two. A. A. Talebi, H. Rashmanlo and Reza Ameri [21] discssed prodct IVFGs. K. Radha and M. Vijaya [12] discssed the totally reglar property of the join of two fzzy graphs. In this paper, we introdce and analyse the notion of TRIVFGs. Also we obtain some necessary and sfficient conditions for total reglarity of IVFGs. II. BASIC CONCEPTS Graph theoretic terms and reslts sed in this work are either standard or are explained as and when they first appear Definition (Fzzy graph) [8]. Let be a non empty set. A fzzy graph is a pair of fnctions where is a fzzy sbset of and is a symmetric fzzy relation on. That is, and sch that for all in

2 International Jornal of Scientific and Research Pblications, Volme 6, Isse 12, December Definition (Interval nmber) [1]. An interval nmber is an interval with 2.3. Remark. (i) The interval nmber is identified with the nmber (ii) denotes the set of all interval nmbers Definition (Some operations on D[0,1]) [1]. For interval nmbers and Then is a complete lattice with as the least element and as the greatest Definition (Interval valed fzzy set (IVFS)) [1]. The interval valed fzzy set in is defined by where and are fzzy sbsets of sch that for all. We shall sometimes denote the IVFS A by. For any two IVFSs and in, we define 2.6. Definition (Interval valed fzzy relation (IVFR)) [1]. If is a graph, then by an interval valed fzzy relation on the set we mean an IVFS sch that 2.7. Definition (Interval valed fzzy graph (IVFG)) [1]. By an interval valed fzzy graph of a graph, we mean a pair, where is an IVFS on and is an IVFR on In what follows, G = (A, B) denotes sch an IVFG of a graph G * = (V, E) where V is a non-empty finite set and E 2.8. Definition (Order of an IVFG) [9]. The order of denoted by is defined by where and Definition (Degree of a vertex) [9]. The negative degree of a vertex is defined by Similarly, positive degree of a vertex is defined by Then the degree of the vertex is defined as Definition (Total degree (TD) of a vertex) [9]. The total degree of the vertex is defined as where, and.

3 International Jornal of Scientific and Research Pblications, Volme 6, Isse 12, December Definition (Totally reglar IVFG (TRIVFG)[9]. If each vertex of has the same total degree then the graph is called a TRIVFG of degree or -TRIVFG Definition (Join of two IVFGs) [1]. The join of two IVFG and of and is defined as follows: i. ii. iii. where is the set of all edges joining the nodes of and It is assmed that both are finite and. if III. TOTALLY REGULAR PROPERTY OF THE JOIN 3.1. Remark. Since, we have. Hence by definition 2.10, For any Similarly, Also, if, In what follows, the nmber of vertices in and are denoted as respectively. The proof of the following lemma is straight forward and so is omitted Lemma. Let and be two IVFGs. 1) If

4 International Jornal of Scientific and Research Pblications, Volme 6, Isse 12, December ) If 3.3. Remark. If and are TRIVFGs, then need not be a TRIVFG which is clear from the following examples Example. The following diagram presents two TRIVFGs and and their join which is not a TRIVFG. 1 G 1 G 2 G 1 + G 2 Fig 3.1: An example to show that TRP is not carried over to joins Example. We give below another example to show that join of two TRIVFGs need not be a TRIVFG. [0.4,0.7] 1 [0.4,0.7] [0.2,0.4] [0.2,0.4] [0.4,0.7] [0.4,0.7] 3 G 1 G 2 G 1 + G 2 Fig 3.2: Another example to show that TRP is not carried over to joins Example. In the following example, and are totally reglar and is also totally reglar.

5 International Jornal of Scientific and Research Pblications, Volme 6, Isse 12, December G 1 G 2 G 1 + G 2 Fig 3.3: An example to show that TRP is sometimes carried over to joins. Now, we proceed to obtain some necessary and sfficient conditions for the join of two IVFGs to be totally reglar. Theorem 3.7. Let and be two TRIVFGs of the same degree sch that is a constant fnction. Then the join is a TRIVFG if and only if. Proof: Let and be two -TRIVFGs. Also let for all where are constants. Then so that and For any, Similarly, Hence, For any, Ths is a TRIVFG Remark 3.8. Theorem 3.7 says that the join of two TRIVFGs of the same total degree is a TRIVFG if and only if the nmber of vertices of the two graphs are same. In Fig 3.2, both are TRIVFGs of the same total degree, bt is not a TRIVFG. This is becase the nmber of vertices of are distinct. In Fig 3.3, both are TRIVFGs with the same total degree and same nmber of vertices and we can see that is a TRIVFG. Corollary 3.9. Let and be two -TRIVFGs sch that and is a constant fnction. Then the join is a TRIVFG if and only if.

6 International Jornal of Scientific and Research Pblications, Volme 6, Isse 12, December Proof: Since, and. Again since is a constant fnction, and are constant fnctions. Hence, is a constant. Similar is the case when Hence the reslt follows from theorem 3.7 Theorem Let and be two IVFGs sch that and is a constant fnction. Then the join is a TRIVFG if and only if both are both TRIVFGs of the same degree. Proof: Let and let for all where are constants. For any, and. Hence,. Similarly, for any, Ths is a TRIVFG where and are arbitrary. Hence the theorem Remark Theorems 3.7 and 3.10 fail to hold if example. is not a constant fnction which is clear from the following Example Here, both are TRIVFGs with the same total degree and same nmber of vertices. Bt is not a constant fnction. We can observe from the figre that is not a TRIVFG

7 International Jornal of Scientific and Research Pblications, Volme 6, Isse 12, December v 3 G 1 G v 3 G 1 + G 2 Fig 3.4 : An example to show that the condition rmin = constant is essential for TRP of G 1 + G 2 Remark Theorem 3.10 says that the join of two TRIVFGs with the same nmber of vertices is a TRIVFG if and only if they have the same total degree. In Fig 3.1, both are TRIVFGs and have the same nmber of vertices, bt is not a TRIVFG. This is becase they have distinct total degrees. Corollary Let and be two IVFGs sch that and is a constant fnction. Then the join is a TRIVFG if and only if both are TRIVFG of the same degree. Proof: Proof is similar to Corollary 3.9 Theorem Let and be two TRIVFGs sch that If is a TRIVFG, then is a constant fnction.

8 International Jornal of Scientific and Research Pblications, Volme 6, Isse 12, December Proof. Let and be two TRIVFGs of total degrees and respectively. Also let For any, Similarly, Hence, For any, Similarly, Hence, Ths is a TRIVFG For any, and is a constant fnction. is a constant fnction The following example shows that the converse of the above theorem is not tre. Example Here are TRIVFGs sch that and is a constant fnction. Bt, is not a TRIVFG.

9 International Jornal of Scientific and Research Pblications, Volme 6, Isse 12, December [0.3,0.6] 1 [0.1,0.3] [0.3,0.6] [0.1,0.3] [0.1,0.3] [0.3,06] [0.1,0.3] [0.3,0.6] 3 G 1 G 2 G 1 + G 2 Fig 3.5 : Example to show that converse of theorem 3.15 is not tre In the next theorem, we give a necessary and sfficient condition for the join of two TRIVFGs with distinct total degrees and distinct nmber of vertices to be a TRIVFG. Theorem Let and be two TRIVFGs of total degrees and respectively and let be a constant fnction. Then the join is a TRIVFG if and only if where is the constant vale of Proof: For any, and Ths is a TRIVFG. Hence,. Similarly, for any,. Corollary Let and be two TRIVFGs sch that and is a constant fnction. Then the join is a TRIVFG if and only if where is the constant vale of Example 3.19

10 International Jornal of Scientific and Research Pblications, Volme 6, Isse 12, December G 1 G 2 G 1 + G 2 Fig 3.6 : An example to verify theorem 3.17 Here and. given in theorem 3.17 and so. Hence satisfy the condition is a TRIVFG which is also clear from the figre. IV. CONCLUSION In this paper, we have introdced the notion of TRIVFGs. We have observed that the join of two TRIVFGs need not be a TRIVFG. Also, we have derived some necessary and sfficient conditions for the join of two IVFGs to be a TRIVFG. REFERENCES [1] M. Akram and W. A. Ddek; Interval-valed fzzy graphs; Compters and Mathematics with Applications; 61(2011) [2] M. Akram; Interval-valed fzzy line graphs; Neral Compting Applications; 21(2012) [3] M. Akram, N. O. Alshehri and W.A. Ddek; Certain types of interval-valed fzzy graphs; Jornal of Applied Mathematics (2013). [4] M. Akram, M.Mrtaza Yosaf and W.A.Ddek; Self centered interval-valed fzzy graphs; Africa Mathematika; 26(2015) [5] Basheer Ahamed Mohideen; Strong and reglar interval-valed fzzy graphs; Jornal of Fzzy Set Valed Analysis; 3(2015), pp [6] J. Hongmei and W. Lianha; Interval-valed fzzy sbsemigrops and sbgrops associated by interval - valed fzzy graphs; 2009 WRI Global Congress on Intelligent Systems; (2009) [7] Kafman. A; Introdction a la Theorie des Sos-emsembles Flos, Masson et Cie 1, (1973). [8] J.N.Mordeson and P.S.Nair, Fzzy Graphs and Fzzy Hypergraphs, Physica-verlag, Heidelberg, 1998, Second Edition [9] M. Pal and H. Rashmanlo; Irreglar interval-valed fzzy graphs; Annals of pre and Applied Mathematics 3(1) (2013), [10] M. Pal, S. Samanta and H. Rashmanlo; Some reslts on interval-valed fzzy graphs; International Jornal of Compter Science and Electronics Engineering, 3(3) (2015) [11] Pradip Debnath; Domination in interval-valed fzzy graphs; Annals of Fzzy Mathematics and Informatics (2013). [12] K.Radha and M.Vijaya, Totally reglar property of the join of two fzzy graphs, International Jornal of Fzzy Mathematical Archive, 8(1) (2015) [13] H. Rashmanlo, and Y. B. Jn; Complete interval-valed fzzy graphs; Annals of Fzzy Mathematics and Informatics 6(3) (2013) [14] H. Rashmanlo and M. Pal; Balanced interval-valed fzzy graphs; Jornal of Physical Sciences; 17 (2013) [15] H. Rashmanlo and M. Pal; Antipodal interval-valed fzzy graphs; International Jornal of Applications of Fzzy Sets and Artificial Intelligence; 3(2013)

11 International Jornal of Scientific and Research Pblications, Volme 6, Isse 12, December [16] H. Rashmanlo and M. Pal; Isometry on interval-valed fzzy graphs; International Jornal on Fzzy Mathematical Archive; 3(2013) [17] H. Rashmanlo and M. Pal; Some properties of Highly Irreglar interval-valed fzzy graphs; World Applied Sciences Jornal, 27(12) (2013) [18] S. Ravi Narayanan and N.R. Santhi Maheswari; Strongly Edge Irreglar interval-valed fzzy graphs; International Jornal of Mathematical Archive, 7(1) (2016) [19] A. Rosenfeld; Fzzy Graphs, Fzzy Sets and their Applications. In: Zadeh, L.A., F, K.S., Shimra, M.(eds.), Academic Press, New York, (1975) [20] A. A. Talebi and H. Rashmanlo; Isomorphism on interval-valed fzzy graphs; Annals of Fzzy Mathematics and Informatics; 6(1) (2013) [21] A. A. Talebi, H. Rashmanlo and Reza Ameri; New Concepts of Prodct interval-valed fzzy graphs; Jornal of Applied Mathematics and Informatics; 34 (2016) [22] L.A.Zadeh; Fzzy Sets; Information Control; 8(1965) [23] L.A.Zadeh; The concept of a Lingistic variables and its application to approximate reasoning; Information Science; 8(1975) AUTHORS First Athor Soriar Sebastian, Department of Mathematics, St. Albert s College, Kochi , Kerala, India., soriarsebastian@gmail.com Second Athor Ann Mary Philip, Department of Mathematics, Assmption College, Changanasserry , Kerala, India. anmaryphilip@gmail.com