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

F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 171for 2 ≤ j ≤ n,j−1 ∑(p k +m k )+2i−j 1 ≤ i ≤⎧⎪ ⌊ p j⌋2k=1j−1 ∑(p k +m k )+2i−j i = ⌊ p j⌋2 +1 andk=1⎨f ∗ (v (j)i v (j)i+1 ) = p j is oddj−1 ∑(p k +m k )+2p j −2i−j +3 i = ⌊ p j⌋+1 and⎪ ⎩f ∗ (v (j)p jv (j)1 ) =and for 3 ≤ j ≤ n,k=1j−1 ∑(p k +m k )+2p j −2i−j +3k=1j−1∑(p k +m k )−j +3k=1k=12p j is even⌊ pj⌋2 +2 ≤ i ≤ pj −1,j−2f ∗ (u (j−1)i u (j−1)i+1 ) = ∑(p k +mk)+p j−1 +i+2−j, for 1 ≤ i ≤ m j−1 −1.Hence, f is a F-Geometric mean labeling of Ĝ(p 1,m 1 ,p 2 ,m 2 ...,m n−1 ,p n ). Thus thegraph Ĝ(p 1,m 1 ,p 2 ,m 2 ...,m n−1 ,p n ) is a F-Geometric mean graph. □A F-Geometric mean labeling of Ĝ(8,4,5,6,10) is as shown in Figure 6.516 28 3 3 5 71414162626 28 121 724 24 301 9 9 10 11 12171718 210 1118 19 19 2020 21 21 223022 32823 31 13 15 4 4 6 825 25 29 31272961527Figure 6. A F-Geometric mean labeling of Ĝ(8,4,5,6,10)Theorem 2.4. C n ⊙K 1 is a F-Geometric mean graph, for n ≥ 3.Proof. Let v 1 ,v 2 ,...,v n be the vertices of the cycle C n and let u i be the pendantvertices attached⌊√at each⌋v i , for 1 ≤ i ≤ n. Consider the graph C n ⊙K 1 , for n ≥ 4.Case (i) 2n+1 is odd.