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Research Article Fibonacci Mean Anti-Magic Labeling Of Kong. Res. J. 5(1): 1-3, 2018 ISSN 2349-2694, All Rights Reserved, Publisher: Kongunadu Arts and Science College, Coimbatore. http://krjscience.com RESEARCH ARTICLE FIBONACCI MEAN ANTI-MAGIC LABELING OF SOME GRAPHS Ameenal Bibi, K. and T. Ranjani* P.G. and Research Department of Mathematics, D.K.M College for Women (Autonomous), Vellore - 632 001, Tamil Nadu, India. ABSTRACT In this paper, we introduced Fibonacci mean anti-magic labeling in graphs. A graph G with p vertices and q edges is said to have Fibonacci mean anti-magic labeling if there is an injective function 푓: 퐸(퐺) → 퐹푗 , ie, it is possible to label the edges with the Fibonacci number Fj where (j= 0,1,1,2…n) in such a way that the edge uv is labeled with ∣푓 푢 +푓 푣 ∣ 푖푓 ∣ 푓 푢 + 푓 푣 ∣ 푖푠 푒푣푒푛, 2 ∣ 푓 푢 +푓 푣 ∣+1 푖푓 ∣ 푓 푢 + 푓 푣 ∣ 푖푠 표푑푑 and the resulting vertex labels admit mean 2 anti-magic labeling. In this paper, we discussed the Fibonacci mean anti-magic labeling for some special classes of graphs. Keywords: Fibonacci mean labeling, circulant graph, Bistar, Petersen graph, Fibonacci mean anti-magic labeling. AMS Subject Classification (2010): 05c78. 1. INTRODUCTION The concept of Fibonacci labeling was Definition 1.2. introduced by David W. Bange and Anthony E. A graph G with p vertices and q edges Barkauskas in the form Fibonacci graceful (1). The admits mean anti-magic labeling if there is an concept of skolem difference mean labeling was injective function 푓from the edges 퐸 퐺 → introduced by Murugan and Subramanian (2). {0,1,1 … 푞} such that when each uv is labeled with Somasundaram and Ponraj have introduced the ∣푓 푢 +푓 푣 ∣ 푖푓 ∣ 푓 푢 + 푓 푣 ∣ 푖푠 푒푣푒푛 and notion of mean labeling of graphs. 2 ∣ 푓 푢 +푓 푣 ∣+1 Hartsfield and Ringel introduced the concept of 푖푓 ∣ 푓 푢 + 푓 푣 ∣ 푖푠 표푑푑 then the anti-magic labeling which is an assignment of 2 resulting vertices are distinctly labeled. distinct values to different vertices in a graph that in such a way that when taking the sums of the Note: A graph which admits mean anti –magic labels, all the sums will be having different labeling is called mean anti-magic graph. constants. Definition 1.3. For various graph theoretic notations and A graph G is called anti-magic if the q edges of terminology, we followed Gross and Yellen (3). G can be distinctly labeled in such a way that when Sridevi et al. (4) proved the path and cycle graphs taking the sum of the edge labels incident to each are Fibonacci divisor cordial graphs. A dynamic vertex, they all will have different (distinct) survey of graph labeling is updated by Gallian (5). constants. Rokad and Ghodasara (6) proved that Fibonacci cordial labelingexists for some special graphs. In 2. RESULTS this paper, we have discussed different families of Theorem 2.1. graphs which satisfy the conditions of Fibonacci mean anti-magic labeling. The circulant graph 퐶푛 (푛 ≥ 6) admits Fibonacci mean anti-magic labeling with the Definition 1.1. generating set (1,2). Fibonacci number can be defined by the Proof: linear recurrence relation 퐹푛 = 퐹푛−1 + 퐹푛−2, 푛 ≥ 2 where 퐹0 = 0,퐹1 = 1 .This generates the infinite Let 퐺 = 퐶푛 (1,2) be the 4-regular graph with sequence of integers in the form (푛 ≥ 6). 0,1,1,2,3,5,8,13,21,34,55,89,144… *Correspondence: T. Ranjani, P.G. and Research Department of Mathematics, D.K.M College for Women (Autonomous), Vellore - 1 632 001, Tamil Nadu, India. E.mail: [email protected] We define the labeling function 푓: 퐸(퐺) → Theorem 2.5. 퐹푗 where (j=0,1,1…n) The Wheel graph 푊푛 admits Fibonacci mean Then apply mean labeling for the edges so that anti-magic labeling. the sum of the labels of the vertices are all distinct. Proof: Thus, the above labeling pattern gives rise to a Let 푢 , 푢 , 푢 … . 푢 be the edges of 푊 and Fibonacci mean anti-magic labeling on the given 0 1 2 2푛 푛 graph 퐺 = 퐶푛 (1,2). Let 푣0, 푣1, 푣2 … . 푣푛 be the vertices of the Wheel graph 푊 . Example 2.2. 푛 We defined the labeling function 푓: 퐸(퐺) → 퐹푗 where (j=0,1,1…n) such that each uv is labeled ∣푓 푢 +푓 푣 ∣ with 푖푓 ∣ 푓 푢 + 푓 푣 ∣ 푖푠 푒푣푒푛 and 2 ∣ 푓 푢 +푓 푣 ∣+1 푖푓 ∣ 푓 푢 + 푓 푣 ∣ 푖푠 표푑푑 then the 2 resulting vertices are distinctly labeled. By applying mean labeling to the edges of 푊푛 we obtained the sum of the vertex labels are all Fig. 1. Fibonacci mean anti-magic labeling of distinct (different constants). circulant graph 퐶6 Hence the Wheel graph 푊푛 admits Fibonacci Theorem 2.3. mean anti-magic labeling. Petersen graph admits Fibonacci mean Example 2.6: anti-magic labeling. Proof. Petersen graph is a three regular graph with 10 vertices and 15 edges. Let 푢0, 푢1, 푢2 … . 푢14 be the edges and Let 푣0, 푣1, 푣2 … . 푣9 be the vertices of Petersen graph. We define the labeling function 푓: 퐸(퐺) → Fig. 3. Fibonacci mean anti-magic labeling of 퐹푗 where (j=0,1,1…n) such that each uv is labeled Wheel graph 푊6. ∣푓 푢 +푓 푣 ∣ with 푖푓 ∣ 푓 푢 + 푓 푣 ∣ 푖푠 푒푣푒푛 and 2 Theorem 2.7. ∣ 푓 푢 +푓 푣 ∣+1 푖푓 ∣ 푓 푢 + 푓 푣 ∣ 푖푠 표푑푑 then the 2 Bistar 퐵푛,푛 admits Fibonacci mean anti-magic resulting vertices are distinctly labeled. labeling. By applying the above mean labeling to the Proof: edges, we obtained the sum of the vertex labels are all distinct (different constants). Let 푣1,0 푎푛푑푣2,0 be the apex (central) vertices of 퐵푛,푛 . Hence Petersen graph admits Fibonacci mean anti-magic labeling. Let 푣1,1 … . 푣1,푛 be the pendent vertices adjacent to the vertex 푣1,0 . Example 2.4. Let 푣2,1 … . 푣2,푛 be the pendent vertices adjacent to the vertex 푣2,0 . Let 푢0 be the edge of the two apex vertices. Let 푢1,푢2 … be the edges of all the pendent vertices. We defined the labeling function 푓: 퐸(퐺) → 퐹푗 where (j=0,1,1…n) such that each uv is labeled Fig. 2. Fibonacci mean anti-magic labeling of ∣푓 푢 +푓 푣 ∣ with 푖푓 ∣ 푓 푢 + 푓 푣 ∣ 푖푠 푒푣푒푛 and Petersen graph. 2 2 ∣ 푓 푢 +푓 푣 ∣+1 푖푓 ∣ 푓 푢 + 푓 푣 ∣ 푖푠 표푑푑 then the REFERENCES 2 resulting vertices are distinctly labeled. 1. David W. Bange and Anthony E. Barkauskas, (1980). Fibonacci graceful graphs. Hence Bistar 퐵푛,푛 admits Fibonacci mean anti-magic labeling. 2. Murugan, K. and A. Subramanian, (2011). Skolem difference mean labeling of H- graphs. Example 2.8. Int. J. J. Math. Soft Compu. 1(1): 115-129. 3. Gross, J. and J. Yellan, (1999). Graph Theory and its Applications, CRC Press. 4. Sridevi, R., K. Nagarajan, A. Nellaimurugan and S. Navaneethakrishnan, (2013). Fibonacci divisor cordial graphs. Int. J. Math. Soft Compu. 3(3): 33-39. 5. Gallian, J.A. (2013).A dynamic survey of graph labeling. Electronics J. Combinatorics. 16: 61- 3. CONCLUSION 308. In this paper, we have obtained Fibonacci 6. Rokad, A.H. and G.V. Ghodasara, (2016). mean anti-magic labeling for the circulant graph, Fibonacci cordial labeling of some special the Wheel graph, Petersen graph and the Bistar. graphs. Ann. Pure Appl. Math. 11(1): 133-144. Further study on some more special graphs is under progress. 3 Kong. Res. J. 5(1): 4-7, 2018 ISSN 2349-2694, All Rights Reserved, Publisher: Kongunadu Arts and Science College, Coimbatore. http://krjscience.com RESEARCH ARTICLE DOMINATION AND TOTAL DOMINATION IN INTUITIONISTIC TRIPLE LAYERED SIMPLE FUZZY GRAPH Jehruth Emelda Mary, L.a* and K. Ameenal Bibib aP.G. and Research Department of Mathematics, St. Joseph’s College of Arts and Science College (Autonomous), Cuddalore - 607 001, Tamil Nadu, India. bP.G. and Research Department of Mathematics, D.K.M College for Women (Autonomous), Vellore - 632 001,Tamil Nadu. India. ABSTRACT In this paper, we discussed domination and total domination in Intuitionistic fuzzy graph. We determined the domination number 훾 and total domination number 훾푡 in Intuitionistic Triple Layered Simple fuzzy graph (ITLFG) and also verified the existence of 2-domination in intuitionistic Triple Layered simple fuzzy graph. Keywords: Intuitionistic Triple Layered Simple fuzzy graphs, Domination, Total domination, 2- Domination. 1. INTRODUCTION graph 퐺∗: ∗, 휇∗ . The pair 푇퐿 퐺 : 푣 , 휇 , 휈 , 푒 , 휇 , 휈 is called the Atanassov introduced the concept of 푖 푇퐿1 푇퐿1 푖푗 푇퐿2 푇퐿2 intuitionistic fuzzy graphs (1). Parvathi and Intuitionistic Triple Layered Fuzzy graph and is Karunambigai are also studied the concept of defined as follows. The Vertex set of ITL (G) be 휇 휈 . and The fuzzy subset 휇 휈 is intuitionistic fuzzy graphs and its properties (2). 푇퐿1, 푇퐿1 푇퐿1, 푇퐿1 Some of the properties of intuitionistic fuzzy graphs defined as 휇 푢 , 휈 푢 푖푓 푢 ∈ ∗ are introduced by Nagoorgani and Shajitha Begum 1 1 휇푇퐿1,휈푇퐿1 = ∗ (3). 휇2 푢푣 , 휈2 푢푣 푖푓 푢푣 ∈ 휇 Pathinathan and Jesintha Roseline defined Where, 0 ≤ 휇푇퐿1 + 휈푇퐿1 ≤ 1. the Triple Layered fuzzy graph and its properties The fuzzy relation 휇 휈 표푛 ∗ ∪ 휇∗ ∪ 휇∗ (4,5). In this paper, Jethurth Emelda Mary and 푇퐿2, 푇퐿2 is defined as Ameenalbibi introduced the concept of domination 훾 and total domination 훾푡 in Intuitionistic Triple Layered Simple fuzzy graph (ITLFG) under certain conditions and illustrated with some examples.
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