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

F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 1773 3 4 6 9 9 10 12 15 15 16 18212 4 8 10 14 16 20 2 5 8 11 14 17 201 5 7 11 13 17 19 1 6 7 12 13 18 194 4 6 8 12 12 14 16 20 20 22 24 28 28 3036111419222730113 7 11151923 27 315 7 9 10 13 15 17 18 21223 25 26 29 31 5 9 13 17 21 25 292 8 10 16 18 24 26Figure 10. A F-Geometric mean labeling of (P 7 ;S 1 ) and (P 8 ;S 2 )32Theorem 2.7. For a H−graph G, G⊙K 1 is a F-Geometric mean graph.Proof. Let u 1 ,u 2 ,...,u n and v 1 ,v 2 ,...,v n be the vertices of G. ThereforeandV(G⊙K 1 ) = V(G)∪{u ′ i,v ′ i;1 ≤ i ≤ n}E(G⊙K 1 ) = E(G)∪{u i u ′ i ,v iv i ′ ;1 ≤ i ≤ n}.Case (i) n ≡ 0(mod 4).We define f : V(G⊙K 1 ) → {1,2,3,...,4n} as follows:{2i−1 1 ≤ i ≤ n and i is oddf(u i ) =2i 1 ≤ i ≤ n and i is even,{2i 1 ≤ i ≤ n and i is oddf(u ′ i ) = 2i−1 1 ≤ i ≤ n and i is even,{ ⌊2n−3+4i 1 ≤ i ≤n⌋f(v i ) = 2 −1 and i is odd2n−1+4i 1 ≤ i ≤ ⌊ n2⌋and i is even,{ ⌊2n−2+4i 1 ≤ i ≤n⌋f(v n+1−i ) = 2 −1 and i is odd2n+4i 1 ≤ i ≤ ⌊ n2⌋and i is even,