Super Fibonacci Graceful Labeling Of Some Cycle Related Graphs
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1 International Journal of Mathematics Trends and Technology (IJMTT) Volume 3 Number 3 - October06 Super Fibonacci Graceful Labeling Of Some Cycle Related Graphs D.Amuthavalli, M. Premkumar Assistant Professor of Mathematics, N.R. School of Architecture, Kinathukadavu Tamil Nadu,India. Assistant Professor of Mathematics, Mahendra Engineering college (Autonomous) Namakkal Tamil Nadu,India. ABSTRACT The function (where is the Fibonacci number) is said to be Super Fibonacci graceful if the induced edge labeling defined by is bijective. In this paper an analysis is made on Friendship graph, Butterfly graph, The graph and the graph obtained by attaching a triangle at each pendant vertex of a Star graph,(3,n)-kite graph under Super Fibonacci graceful labeling. AMS Classification 05C7 KEYWORDS Friendship graph, Butterfly graph, The graph, The graph, Fibonacci graceful labeling and Super Fibonacci graceful labeling.. INTRODUCTION In graph labeling vertices or edges or both assigned values subject to the conditions. The graceful labeling introduced by Rosa (967)[7]. J.A. Gallian [4] studied a complete survey on graph labeling.david.w. and Anthony. E. Baraaukas [] have investigated the cycle structure of Fibonacci graceful mapping. A Fibonacci graceful labeling and Super Fibonacci graceful labeling have been introduced by Kathiresan and Amutha [5] in 006.S. Meena and P. Kavitha [6] proved that the butterfly graphs are Prime graphs. R. Uma and D.Amuthavalli [] proved that some Star- related graphs under Fibonacci graceful. Dushyant Tanna [4] has proved Friendship graphs are Harmonious. Dr. M. Subbiah [9] was proved that windmill graphs are strong edge graceful.. DEFINITIONS Definition.. If the vertices or edges or both assigned values subject to certain condition(s) then it is known as graph labeling. Definition.. The function is called a graceful labeling of agraph G if is injective and the induced function defined as is bijective. Definition.3. is the The function (where Fibonacci number) is said to be Fibonacci graceful if the induced edge labeling defined by bijective. Definition.4. The function (where is the Fibonacci number) is said to be Super Fibonacci graceful if the induced edge labeling defined by Definition.4 is bijective. A Friendship graph is a graph which consists of n triangles with a common vertex. Definition.5 Two cycles of the same order vertex with an arbitrary number is sharing a common of pendant edges ISSN: Page 64
2 International Journal of Mathematics Trends and Technology (IJMTT) Volume 3 Number 3 - October06 attached at the common vertex called butterfly graph where are two positive integers. Definition.6 Then the above defined function admits Super Fibonacci graceful labeling. Hence Friendship graphs are Super Fibonacci graceful. The Friendship graph are shown in figure. The wind mill graphs to be the family of graphs consisting of copies of with a vertex in common. Definition.7 The graph windmill graph. is obtained from the, removing an edge in each Definition. A graph is obtained by attaching a triangle at each pendant vertex of the Star graphs. Definition.9 The kite graph is the graph obtained by joining a cycle graph to a path graph with a bridge. 3.RESULTS 3. Theorem : 3. Example: Figure. The Friendship graph The Friendship graph is shown in figure. In the order is and the size is Friendship graphs are Super graceful Fibonacci Proof: Let be the friendship graph. The order of and the size of is By the definition of, the vertex set,. Let be the second vertices of the triangles and let be the third vertices of the triangles, and let the apex vertex be the first vertex of all the triangles. The edge set where and Now let us define the function Figure. The Friendship graph 3.3 Theorem : 44 9 The Butterfly graphs are Super Fibonacci graceful graphs. Proof : ISSN: Page 65
3 International Journal of Mathematics Trends and Technology (IJMTT) Volume 3 Number 3 - October06 Let be the Butterfly graph. The order of the Butterfly graph is and the size of the Butterfly graph is. By the definition of the Butterfly graph, the vertex set Let be the vertices of the two cycles and be the apex vertex of the two cycles. The edge set where and Now let us define the function Theorem : Figure 4. The Butterfly graph Then the above defined function Fibonacci graceful labeling. admits Super Hence the Butterfly graphs are Super Fibonacci graceful. The generalized graph is shown in figure 3. The graphs are Super Fibonacci graceful. Proof: Let be a graph. The order of the graph is and the size of the graph is Thenthe vertex set. Let be the second vertices of the, let be the third vertices of the, let be the fourth vertices of the And let the central vertex be the first vertex of all the. The edge set where Now let us defined the function Figure 3. The Butterfly graph 3.4 Example: The Butterfly graph is shown in figure 4. The order and size of the graph is and respectively Then the above defined function Fibonacci graceful labeling. admits Super ISSN: Page 66
4 International Journal of Mathematics Trends and Technology (IJMTT) Volume 3 Number 3 - October06 Figure 5. The graph 3.6 Example : Consider the graph is shown in figure 6.The order and size of the graph is and respectively. Hence the graphs are Super Fibonacci graceful. The generalized graph is shown in figure5 ISSN: Page 67
5 International Journal of Mathematics Trends and Technology (IJMTT) Volume 3 Number 3 - October Figure 6. The graph 3.7 Theorem : A graph obtained by attaching a triangle at each pendant vertex of a Star graph is Super Fibonacci graceful labeling. Proof : Let be a graph. The order of the graph is and the size of the graph is By the definition of the graph the vertex set. Let be the pendant vertices of the star and these are the first vertices of the triangles, let be the second vertices of the triangles, Let be the third vertices of the triangle. And let the central vertex be the apex vertex of the star. The edge set where Now let us defined the function Then the above defined function admits Super Fibonacci graceful labeling. Hence the graphs are Super Fibonacci graceful. The generalized graph is shown in figure 7. ISSN: Page 6
6 International Journal of Mathematics Trends and Technology (IJMTT) Volume 3 Number 3 - October06 Figure 7. The graph 3. Example : Consider the graph is shown in figure.the order and size of the graph is and respectively Figure. The graph 3 3 ISSN: Page 69
7 International Journal of Mathematics Trends and Technology (IJMTT) Volume 3 Number 3 - October Theorem : The Kite graphs graceful graphs. Proof: are Super Fibonacci Now let us define the function Let be a graph, the order of a graph is and the size. The vertex set and the edge set Let be the vertices of cycle. And let,, be the vertices of path.the edge set where and. Then the above defined function admits Super Fibonacci graceful labeling. Hence graphs are Super Fibonacci graceful graphs. 3. Example : Consider the graph is shown in figure 0. The order and size of the graph is and respectively Figure 0 ISSN: Page 70
8 International Journal of Mathematics Trends and Technology (IJMTT) Volume 3 Number 3 - October06 REFERENCES: [] Bondy J. A. and Murty U. S. R., Graph Theory with applications, Newyork Macmillan Ltd. Press (976). [] David.W and Anthony.E. Barakaukas, Fibonacci graceful graphs. [3] Dushyant Tanna, Harmonious labeling of certain graphs. International journal of advanced Engineering and Research and Studies.E-ISSN [4] Joseph A. Gallian, A Dynamic survey of graph labeling, the electronic journal of combinatorics,(03). [5] Kathiresan K. and Amutha S., Fibonacci graceful graphs, Ph.D. Thesis, Madurai Kamaraj University, (October 006). [6] S. Meena and P. Kavitha, Prime labeling of some butterfly graphs, International Journal of Mathematical Archive- 5(0), 04, 5-5. [7] S. Sandhiya and S.Somasundharam, Harmonic mean labeling of some new graphs, International Journal of Mathematical Archive- 5(0), 04, 5-5. [] Rosa.A, On certain valuation of vertices of graph,(967). [9] R.Uma and D. Amuthavalli Fibonacci graceful labeling of some star related graphs International Journal of Computer Applications,Vol.34,( January 06). [0] Dr. M. Subbiah Strong edge graceful labeling of Wind mill graphs Int. J. Cont. Math. Science,Vol.7,(0). ISSN: Page 7
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