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

F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 175andf(v (i)2 ) = ⎧⎨⎩3i3i+1The induced edge labeling is as follows⎧⎨ 3if ∗ (u i u i+1 ) =⎩ 3i+1⌊√ ⌋f ∗ (u n u 1 ) = 6n ,andf ∗ (u i v (i)1 ) = ⎧⎨⎩⎧⎨f ∗ (u i v (i)2 ) =⎩3i−23i−13i−13i⌊ √6n ⌋1 ≤ i ≤3⌊ √6n ⌋+1 ≤ i ≤ n.3⌊ √6n⌋1 ≤ i ≤3⌊ √6n ⌋+1 ≤ i ≤ n−1,3⌊ √6n+1 ⌋1 ≤ i ≤3⌊ √6n+1 ⌋+1 ≤ i ≤ n3⌊ √6n ⌋1 ≤ i ≤3⌊ √6n ⌋+1 ≤ i ≤ n.Hence, f is a F-Geometric mean labeling of C n ⊙ S 2 . Thus the graph C n ⊙ S 2 is aF-Geometric mean graph, for n ≥ 3.□A F-Geometric mean labeling of C 6 ⊙S 2 is as shown in Figure 9.31 1 2196 2 3184417 175185 616716158 810 1514 149 913 12101112 1113Figure 9. A F-Geometric mean labeling of C 6 ⊙S 2Theorem 2.6. (P n ;S m ) is a F-Geometric mean graph, for m ≤ 2 and n ≥ 1.Proof. Let u 1 ,u 2 ,...,u n be the vertices of the path P n and Let v (i)1 ,v (i)2 ,...,v (i)m+1be the vertices of the star graph S m such that v (i)1 is the central vertex of S m , for1 ≤ i ≤ n.3