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

F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 179and{f ∗ (v i v i ′ ) = f(vi ) 1 ≤ i ≤ n and i is oddf(v i )−2 1 ≤ i ≤ n and i is even⌊ nf ∗ (u i v i ) = 2n, for i = +1.2⌋Case (iii) n ≡ 2(mod 4).We define f : V(G⊙K 1 ) → {1,2,3,...,4n} as follows:{2i 1 ≤ i ≤ n and i is oddf(u i ) =2i−1 1 ≤ i ≤ n and i is even,{2i−1 1 ≤ i ≤ n and i is oddf(u ′ i) =2i 1 ≤ i ≤ n and i is even,{ ⌊2n−1+4i 1 ≤ i ≤n⌋f(v i ) = 2 and i is odd2n−3+4i 1 ≤ i ≤ ⌊ n2⌋−1 and i is even,{ ⌊2n+4i 1 ≤ i ≤n⌋f(v n+1−i ) = 2 and i is odd2n−2+4i 1 ≤ i ≤ ⌊ n2⌋−1 and i is even,and⎧⎪ f(v i )−2 1 ≤ i ≤ ⌊ n⎨ ⌊ 2⌋and i is oddf(v i ′ ) = f(v i )+2 n⌋2 +2 ≤ i ≤ n and i is odd⎪ f(v i )+2 1 ≤ i ≤ ⌊ n⎩ ⌋ 2⌋−1 and i is evenf(v i )−2 +1 ≤ i ≤ n and i is evenand⌊ n2The induced edge labeling is as follows:f ∗ (u i u i+1 ) = 2i, for 1 ≤ i ≤ n−1,f ∗ (u i u ′ i ) = 2i−1, for 1 ≤ i ≤ n,⌊ nf ∗ (v i v i+1 ) = 2n−1+4i, for 1 ≤ i ≤ ,⌊2⌋nf ∗ (v n+1−i v n−i ) = 2n+4i, for 1 ≤ i ≤ −1,⎧ 2⌋f(v i )−2 1 ≤ i ≤ ⌊ n⎪⎨ ⌊ 2⌋and i is oddf ∗ (v i v i) ′ f(v= i ) n⌋2 +2 ≤ i ≤ n and i is oddf(v ⎪⎩ i ) 1 ≤ i ≤ ⌊ n⌋ 2⌋−1 and i is evenf(v i )−2 +1 ≤ i ≤ n and i is even⌊ n2⌊ nf ∗ (u i+1 v i ) = 2n, for i = +1.2⌋