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

176 A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANCase (i) m = 1We define f : V[(P n ;S m )] → {1,2,3,...,3n} as follows:{3i 1 ≤ i ≤ n and i is oddf(u i ) =3i−2 1 ≤ i ≤ n and i is even,andandf(v (i)1 ) = 3i−1,for 1 ≤ i ≤ n{f(v (i) 3i−2 1 ≤ i ≤ n and i is odd2 ) =3i 1 ≤ i ≤ n and i is even.The induced edge labeling is as followsf ∗ (u i u i+1 ) = 3i, for 1 ≤ i ≤ n−1,{f ∗ (u i v (i) 3i−1 1 ≤ i ≤ n and i is odd1 ) =3i−2 1 ≤ i ≤ n and i is even{f ∗ (v (i)1 v (i) 3i−2 1 ≤ i ≤ n and i is odd2 ) =3i−1 1 ≤ i ≤ n and i is even.Case (ii) m = 2We define f : V[(P n ;S m )] → {1,2,3,...,4n} as follows:{4i 1 ≤ i ≤ n and i is oddf(u i ) =4i−2 1 ≤ i ≤ n and i is even,andandf(v (i)1 ) = 4i−1,for 1 ≤ i ≤ n,f(v (i)2 ) = 4i−3,for 1 ≤ i ≤ nf(v (i){4i−2 1 ≤ i ≤ n and i is odd3 ) = 4i 1 ≤ i ≤ n and i is even.The induced edge labeling is as follows:f ∗ (u i u i+1 ) = 4i, for1 ≤ i ≤ n−1,{f ∗ (u i v (i) 4i−1 1 ≤ i ≤ n and i is odd1 ) = 4i−2 1 ≤ i ≤ n and i is evenf ∗ (v (i)1 v (i)2 ) = 4i−3, for 1 ≤ i ≤ n{f ∗ (v (i)1 v (i) 4i−2 1 ≤ i ≤ n and i is odd3 ) =4i−1 1 ≤ i ≤ n and i is even.Hence, f is a F-Geometric mean labeling of (P n ;S m .) Thus the graph (P n ;S m ) isa F-Geometric mean graph, for m ≤ 2 and n ≥ 1.□A F-Geometric mean labeling of (P 7 ;S 1 ) and (P 8 ;S 2 ) is as shown in Figure 10.