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

F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 1675 717 3 5243 7 915 15 17 19 242626 28 34 40 1 1 1213 22 22 28 30 32 34 38 38409 19211 20 30 364 11203236421 3110 12 18 232329 31 33 35 37 396 6 8 16 10 14 14 18 2525 27 29 8 16 27 35 39Figure 4. A F-Geometric mean labeling of G ∗ (10,9,12,4,5)The graph G ′ (p 1 ,p 2 ,...,p n ) is obtained from n cycles of length p 1 ,p 2 ,...,p n by identifyingconsecutive cycles at an edge as follows: The( ) th pj +32 edge of j th cycle is( ) thidentified with the first edge of (j +1) th pj +1cycle when j is odd and the2 edgeof j th cycle is identified with the first edge of (j +1) th cycle when j is even.Theorem 2.2. G ′ (p 1 ,p 2 ,...,p n ) is a F-Geometric mean graph if all p j ’s are odd orall p j ’s are even, for 1 ≤ j ≤ n.Proof. Let {v (j)i ;1 ≤ j ≤ n,1 ≤ i ≤ p j } be the vertices of the n number of cycles.Case (i) p j is odd, for 1 ≤ j ≤ n.For 1 ≤ j ≤ n − 1, the j th and (j + 1) th cycles are identified by the edgesv (j)p j+1 v (j)p j+3 and v (j+1)1 v p (j+1)j+1while j is odd and v (j)p j−1 v (j)p j+1 and v (j+1)1 v p (j+1)j+1while j is2 22 2even.{}n∑We define f : V[G ′ (p 1 ,p 2 ,...,p n )] → 1,2,3,..., p j −n+2 as follows:⎧⎨f(v (1)i ) =⎩j=11 i = 12i 2 ≤ i ≤ ⌊ p 1⌋⌋ 2 +12p 1 +3−2i +2 ≤ i ≤ p1⌊ p1241and for 2 ≤ j ≤ n,f(v (j)i ) =⎧⎪⎨⎪⎩j−1 ∑k=1j−1 ∑k=1j−1 ∑k=1j−1 ∑k=1p k −j +2i+2p k +2p j +3−j −2ip k −j +2i+1p k +2p j +4−j −2i2 ≤ i ≤ ⌊ p j⌋2 and j is even⌊ pj⌋2 +1 ≤ i ≤ pj −1 and j even2 ≤ i ≤ ⌊ p j⌋2 +1 and j is odd⌊ pj⌋2 +2 ≤ i ≤ pj −1 and j odd.