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

F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 165{1,2,3,...,q} defined asf ∗ (uv) =⌊√f(u)f(v)⌋, for all uv ∈ E(G),isbijective. AgraphthatadmitsaF-GeometricmeanlabelingiscalledaF-Geometricmean graph.The graph shown in Figure 3 is a F-Geometric mean graph.174 115 16 114 5 14 16 212 13 98 829 6 11 10 107 5 3 312 11Figure 3. A F-Geometric mean graphIn this paper, we have discussed the F-Geometric mean labeling of some chaingraphs and thorn graphs.2. Main ResultsThe graph G ∗ (p 1 ,p 2 ,...,p n ) is obtained from n cycles of length p 1 ,p 2 ,...,p n byidentifying consecutive cycles at a vertex as follows. If the j th cycle is of odd length,( ) th pj +3then its2 vertex is identified with the first vertex of (j+1) th cycle and if the( ) thj th pj +2cycle is of even length, then its2 vertex is identified with the first vertexof (j +1) th cycle.Theorem 2.1. G ∗ (p 1 ,p 2 ,...,p n ) is a F-Geometric mean graph for any p j , for1 ≤ j ≤ n.Proof. Let {v (j)i ;1 ≤ j ≤ n,1 ≤ i ≤ p j } be the vertices of the n number of cycles.For 1 ≤ j ≤ n−1, the j th and (j +1) th cycles are identified by a vertex v (j)p j +3 and2v (j+1)1 while p j is odd and v (j)p j +2 and v (j+1)1 while p j is even.2 { }n∑We define f : V[G ∗ (p 1 ,p 2 ,...,p n ] → 1,2,3,..., p j +1 as follows:j=1f(v (1)i ) ={ ⌊2i−1 1 ≤ i ≤p1⌋⌋ 2 +12p 1 −2(i−2) +2 ≤ i ≤ p1⌊ p12and