F-GEOMETRIC MEAN LABELING OF SOME ... - Kjm.pmf.kg.ac.rs

More documents

Recommendations

Info

172 A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANWe define f : V[C n ⊙K 1 ] → {1,2,3,...,2n+1} as follows:⎧1 i = 1 ⎪⎨ ⌊ √2n+1 ⌋f(v i ) = 2i 2 ≤ i ≤ +12⌊ √2n+1 ⌋ and⎪⎩ 2i+1 +2 ≤ i ≤ n2⎧2 i = 1 ⌊ √2n+1 ⌋⎪⎨ 2i−1 2 ≤ i ≤2⌊f(u i ) =√2n+1 ⌋2i+1 i = +12⌊ √2n+1 ⌋⎪⎩ 2i +2 ≤ i ≤ n.The induced edge labeling is as follows:⎧ ⌊ √2n+1 ⌋⎨ 2i 1 ≤ i ≤f ∗ 2(v i v i+1 ) = ⌊ √2n+1 ⌋⎩ 2i+1 +1 ≤ i ≤ n−1,2⌊ √2n+1 ⌋f ∗ (v 1 v n ) = and⎧ ⌊ √2n+1 ⌋⎨ 2i−1 1 ≤ i ≤f ∗ 2(u i v i ) = ⌊ √2n+1 ⌋⎩ 2i +1 ≤ i ≤ n.2⌊√ ⌋Case (ii) 2n+1 is even.We define f : V[C n ⊙K 1 ] → {1,2,3,...,2n+1} as follows:⎧1 i = 1 ⎪⎨ ⌊ √2n+1 ⌋f(v i ) = 2i 2 ≤ i ≤2⌊ √2n+1 ⌋ and⎪⎩ 2i+1 +1 ≤ i ≤ i ≤ n2⎧2 i = 1 ⎪⎨ ⌊ √2n+1 ⌋f(u i ) = 2i−1 2 ≤ i ≤2⌊ √2n+1 ⌋⎪⎩ 2i +1 ≤ i ≤ n.The induced edge labeling is as follows:⎧ ⌊ √2n+1 ⌋⎨ 2i 1 ≤ i ≤ −1f ∗ 2(v i v i+1 ) = ⌊ √2n+1 ⌋⎩ 2i+1 ≤ i ≤ n−1,2⌊ √2n+1 ⌋f ∗ (v 1 v n ) = and⎧ ⌊ √2n+1 ⌋⎨ 2i−1 1 ≤ i ≤f ∗ 2(u i v i ) = ⌊ √2n+1 ⌋⎩ 2i +1 ≤ i ≤ n.222
F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 173Hence, the graph C n ⊙K 1 , for n ≥ 4 admits F-Geometric mean labeling.For n = 3, a F-Geometric mean labeling of C 3 ⊙K 1 is as shown in Figure 7.112□23 6 54 4 5 7 6Figure 7. A F - Geometric mean labeling of C 3 ⊙K 1A F-Geometric mean labeling of C 12 ⊙K 1 is as shown in Figure 8.21 2432 5 243 1 25227 4 4 23226 6 237218 92120 8209191119181010111317171816 12 15 13 1512161414Figure 8. A F-Geometric mean labeling of C 12 ⊙K 1Theorem 2.5. C n ⊙S 2 is a F-Geometric mean graph for n ≥ 3.Proof. Let u 1 ,u 2 ,...,u n be the vertices of the cycle C n . Let v (i)1 be the pendantvertices at each vertex u i , for 1 ≤ i ≤ n. Therefore,andV[C n ⊙S 2 ] = V(C n )∪{v (i)1 ,v(i) 2;1 ≤ i ≤ n}E[C n ⊙S 2 ] = E(C n )∪{u i v (i)1 ,u i v (i)2 ;1 ≤ i ≤ n}.
Page 1 and 2: Kragujevac Journal of MathematicsVo
Page 3 and 4: F-GEOMETRIC MEAN LABELING OF SOME C
Page 9: F-GEOMETRIC MEAN LABELING OF SOME C

F-GEOMETRIC MEAN LABELING OF SOME ... - Kjm.pmf.kg.ac.rs

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?