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

Kragujevac Journal of MathematicsVolume 37(1) (2013), Pages 163–186.F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHSAND THORN GRAPHSA. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANAbstract. A function f is called a F-Geometric mean labeling of a graph G(V,E)if f : V(G) → {1,2,3,...,q+1} is injective ⌊ and the induced function f ∗ : E(G) →√f(u)f(v) ⌋{1,2,3,...,q} defined as f ∗ (uv) = , for all uv ∈ E(G), is bijective.A graph that admits a F-Geometric mean labelling is called a F-Geometric meangraph. In this paper, we have discussed the F-Geometric mean labeling of somechain graphs and thorn graphs.1. IntroductionThroughoutthispaper, byagraphwemeanafinite, undirectedandsimplegraph.Let G(V,E) be a graph with p vertices and q edges. For notations and terminology,we follow [3]. For a detailed survey on graph labeling, we refer [2].Path on n vertices is denoted by P n and a cycle on n vertices is denoted by C n .G ⊙ K 1 is the graph obtained from G by attaching a new pendant vertex to eachvertex of G. A star graph S m is the complete bipartite graph K 1,m . G ⊙ S 2 is thegraph obtained from G by attaching two pendant vertices at each vertex of G. Ifv (i)1 ,v(i) 2 ,v(i) 3 ,...,v(i) m+1 and u 1,u 2 ,u 3 ...,u n be the vertex of the star graph S m and thepath P n , then the graph (P n ;S m ) is obtained from n copies of S m and the path P n byjoining u i with the central vertex v (i)1 of the i th copy of S m by means of an edge, for1 ≤ i ≤ n. The H− graph is obtained from two paths u 1 ,u 2 ,...,u n and v 1 ,v 2 ,...,v nof equal length by joining an edge un+1vn+1when n is odd and un+2vnwhen n is2 22 2even. Let G 1 and G 2 be any two graphs with p 1 and p 2 vertices, respectively. Thenthe cartesian product G 1 ×G 2 has p 1 p 2 vertices which are {(u,v)/u ∈ G 1 ,v ∈ G 2 }.The edges are defined as follows: (u 1 ,v 1 ) and (u 2 ,v 2 ) are adjacent in G 1 ×G 2 if eitherKey words and phrases. Labeling, F-Geometric mean labeling, F-Geometric mean graph.2010 Mathematics Subject Classification. 05C78.Received: February 18, 2012.Revised: January 29, 2013.163

164 A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANu 1 = u 2 and v 1 and v 2 are adjacent in G 2 or u 1 and u 2 are adjacent in G 1 and v 1 = v 2 .The Ladder graph L n is obtained from P n ×P 2 .ThestudyofgracefulgraphsandgracefullabelingmethodsfirstintroducedbyRosa[5]. The concept of mean labeling was first introduced by S. Somasundaram and R.Ponraj [6] and it was developed in [4] and [7]. In [10], R. Vasuki et al. discussedthe mean labeling of cyclic snake and armed crown. In [8], S. Somasundaram et al.defined the geometric mean labeling as follows.AgraphG = (V,E)withpvertices andq edgesissaidtobeageometricmeangraphif it is possible to label the vertices x ∈ V with distinct labels f(x) from ⌊ 1,2,...,q+1 √f(u)f(v) ⌋in such way that when each edge e = uv is labeled with f(uv) = or⌈ √f(u)f(v) ⌋then the edge labels are distinct.In the above definition, the readers will get some confusion in finding the edgelabels which edge is assigned by flooring function and which edge is assigned byceiling function.In [9], they have given the geometric mean labeling of the graph C 5 ∪C 7 as in theFigure 1.11 266 78 u 72 512v83 512 9 w3 4 4 1111 10 10 9Figure 1. A Geometric mean labeling of C 5 ∪C 7 .⌊ √f(u)f(v) ⌋Fromtheabovefigure,fortheedgeuv,theyhaveusedflooringfunction⌈ √f(u)f(v) ⌉and for the edge vw, they have used ceiling function for fulfilling theirrequirement. To avoid the confusion of⌊assigning the edge labels in their definition,√f(u)f(v) ⌋we just consider the flooring function for our discussion. Based onour definition, the F-Geometric mean labeling of the same graph C 5 ∪C 7 is given inFigure 2.161 26 7893 78 93 5 1011 104 5 1112 13412Figure 2. A F-Geometric mean labeling of C 5 ∪C 7In [1], A. Durai Baskar et al. introduced Geometric mean graph.A function f is called a F-Geometric mean labeling of a graph G(V,E) iff : V(G) → {1,2,3,...,q +1} is injective and the induced function f ∗ : E(G) →

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

166 A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANfor 2 ≤ j ≤ n,f(v (j)i ) =⎧⎪⎨⎪⎩j−1 ∑k=1j−1 ∑k=1j−1 ∑k=1j−1 ∑k=1p k +2i−1p k +2(i−1)p k +2(i−2)p k +2p j −2(i−2)2 ≤ i ≤ ⌊ p j⌋2 +1i = ⌊ p j⌋2 +2 and pj is oddi = ⌊ p j⌋2 +2 and pj is eveni = ⌊ p j⌋2 +3 ≤ i ≤ pjThe induced edge labeling is as follows:for 2 ≤ j ≤ n,⌋⌋+1 and p1 is odd2p 1 −2(i−1) i = ⌊ p 1⌋⌋ 2 +1 and p1 is even2p 1 −2(i−1) +2 ≤ i ≤ p1 −1,⎧⎪ 2i−1 1 ≤ i ≤ ⌊ p 1⎨2f ∗ (v (1)i v (1)i+1 ) = 2i−1 i = ⌊ p 12⎪ ⎩f ∗ (v (1)p 1v (1)1 ) = 2,⌊ p12f ∗ (v (j)i v (j)j−1 ∑p k +2i−1k=1j−1 ∑⎧⎪ ⎨ p k +2i−1i+1 ) = k=1⎪ ⎩j−1 ∑k=1j−1 ∑k=1p k +2p j −2(i−1)p k +2p j −2(i−1)1 ≤ i ≤ ⌊ p j⌋2i = ⌊ p j⌋2 +1 and pj is oddi = ⌊ p j⌋2 +1 and pj is even⌊ pj⌋2 +2 ≤ i ≤ pj −1andj−1f ∗ (v p (j)jv (j)1 ) = ∑p k +2.Hence, f is a F-Geometric mean labeling of G ∗ (p 1 ,p 2 ,...,p n ). Thus the graphG ∗ (p 1 ,p 2 ,...,p n ) is a F-Geometric mean graph.□A F-Geometric mean labeling of G ∗ (10,9,12,4,5) is as shown in Figure 4.k=1

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.

168 A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANThe induced edge labeling is as follows:{ ⌊f ∗ (v (1)i v (1) 2i 1 ≤ i ≤p1⌋i+1 ) = ⌊ 22p 1 +1−2ip1⌋+1 ≤ i ≤ p1 −1,and for 2 ≤ j ≤ n,f ∗ (v (1)p 1v (1)1 ) = 12f ∗ (v (j)i v (j)j−1 ∑p k −j +2i+2k=1j−1 ∑⎧⎪ ⎨ p k +2p j +1−j −2ii+1 ) = k=1⎪ ⎩j−1 ∑k=1j−1 ∑k=1p k −j +2i+1p k +2p j +2−j −2i1 ≤ i ≤ ⌊ p j⌋2 and j is even⌊ pj⌋2 +1 ≤ i ≤ pj −1 and j even1 ≤ i ≤ ⌊ p j⌋2 and j is odd⌊ pj⌋2 +1 ≤ i ≤ pj −1 and j odd.Case (ii) p j is even, for 1 ≤ j ≤ n.For 1 ≤ j ≤ n − 1, the j th and (j + 1) th cycles are identified by the edgesv (j) and v (j+1)1 v p (j+1)j+1.v (j)p j2p j +22We define f : V[G ′ (p 1 ,p 2 ,...,p n )] →⎧⎨f(v (1)i ) =⎩{1,2,3,...,}n∑p j −n+2 as follows:j=11 i = 12i 2 ≤ i ≤ ⌊ p 1⌋⌋ 22p 1 +3−2i +1 ≤ i ≤ p1⌊ p12and for 2 ≤ j ≤ n,f(v (j)i ) =⎧⎪⎨⎪⎩j−1 ∑k=1j−1 ∑k=1p k −j +2i+1p k +2p j +4−j −2i2 ≤ i ≤ ⌊ p j⌋2⌊ pj⌋2 +1 ≤ i ≤ pj −1.The induced edge labeling is as follows:{ ⌊f ∗ (v (1)i v (1) 2i 1 ≤ i ≤p1⌋i+1 ) = ⌊ 22p 1 +1−2ip1⌋+1 ≤ i ≤ p1 −1,f ∗ (v (1)p 1v (1)1 ) = 12

F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 169and for 2 ≤ j ≤ n,f ∗ (v (j)i v (j)j−1 ∑⎧⎪ ⎨i+1 ) = k=1⎪ ⎩j−1 ∑k=1p j −j +2i+1p k +2p j +2−j −2i1 ≤ i ≤ ⌊ p j⌋2⌊ pj⌋2 +1 ≤ i ≤ pj −1.Hence, f is a F-Geometric mean labeling of G ′ (p 1 ,p 2 ,...,p n ). Thus the graphG ′ (p 1 ,p 2 ,...,p n ) is a F-Geometric mean graph.□A F-Geometric mean labeling of G ′ (7,5,9,13) and G ′ (4,8,10,6) is as shown inFigure 5.442113 36565877891014141311 11121210 15 1516 1725 27 16 18 25 27 2923 2318 21 2920 3119193120 17 30 3222 22 2824 302426282616 232 77 91414 16 1821 235 9 12 18 1 4 4 1120 2511 205 12 211256 10 13 19 33 8 8 1015 2215 17 19 24 2617 24Figure 5. F-Geometric mean labeling of G ′ (7,5,9,13) and G ′ (4,8,10,6)The graph Ĝ(p 1,m 1 ,p 2 ,m 2 ,...,m n−1 ,p n ) is obtained from n cycles of lengthp 1 ,p 2 ,...,p n and (n − 1) paths on m 1 ,m 2 ,...,m n−1 vertices respectively by identifyinga cycle and a path at a vertex alternatively as follows: If the j th cycles is ofodd length, then its(pj +32and if the j th cycle is of even length, then its) thvertex is identified with a pendant vertex of j th path( ) th pj +22 vertex is identified with apendant vertex of j th path while the other pendant vertex of the j th path is identifiedwith the first vertex of the (j +1) th cycle.Theorem 2.3. Ĝ(p 1,m 1 ,p 2 ,m 2 ,...,m n−1 p n ) is a F-Geometric mean graph for anyp j ’s and m j ’s.Proof. Let {v (j)i ;1 ≤ j ≤ n,1 ≤ i ≤ p j } and {u (j)i ;1 ≤ j ≤ n−1,1 ≤ i ≤ m j } be then number of cycles and (n−1) number of paths respectively.

170 A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANFor 1 ≤ j ≤ n−1, the j th cycle and j th path are identified by a vertex v (j)p j +22u (j)1 while p j is even and v (j)p j +32andand u (j)1 while p j is odd. And the j th path and (j+1) thcycle are identified by a vertex u (j)m jand v (j+1)1 .We define f : V[Ĝ(p 1,m 1 ,p 2 ,m 2 ,...,m n−1 ,p n )] →{1,2,3,..., n−1 ∑(p j +m j )+p n −j=1}n+2 as follows:{ ⌊f(v (1) 2i−1 1 ≤ i ≤p1⌋i ) = ⌋ 2 +12p 1 +4−2i +2 ≤ i ≤ p1 ,⌊ p12f(u (1)i ) = p 1 +i, for 2 ≤ i ≤ m 1 ,for 2 ≤ j ≤ n,f(v (j)i ) =and for 3 ≤ j ≤ n,⎧⎪⎨⎪⎩f(u (j−1)i ) =j−1 ∑(p k +m k )+2i−j 2 ≤ i ≤ ⌊ p j⌋+1k=1j−1 ∑(p k +m k )+2i−j −1 i = ⌊ p j⌋+2k=122and p j is oddj−1 ∑(p k +m k )+2i−j −3 i = ⌊ p j⌋+2 andk=1j−1 ∑(p k +m k )+2p j −2i−j +5k=12p j is even⌊ pj⌋2 +3 ≤ i ≤ pjj−2∑(p k +m k )+p j−1 +i+2−j, for 2 ≤ i ≤ m j−1k=1The induced edge labeling is as follows:2i−1 1 ≤ i ≤⎧⎪ ⌊ p 1⌋22i−1 i = ⌊ p 1⌋⎨2 +1 andf ∗ (v (1)i v (1)i+1 ) = p 1 is odd2p 1 −2i+2 i = ⌊ p 1⌋+1 and⎪ ⎩f ∗ (v (1)p 1v (1)1 ) = 2,2p 1 −2i+22p 1 is ⌋ even+2 ≤ i ≤ p1 −1,⌊ p12f ∗ (u (1)i u (1)i+1 ) = p 1 +i, for 1 ≤ i ≤ m 1 −1,

F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 171for 2 ≤ j ≤ n,j−1 ∑(p k +m k )+2i−j 1 ≤ i ≤⎧⎪ ⌊ p j⌋2k=1j−1 ∑(p k +m k )+2i−j i = ⌊ p j⌋2 +1 andk=1⎨f ∗ (v (j)i v (j)i+1 ) = p j is oddj−1 ∑(p k +m k )+2p j −2i−j +3 i = ⌊ p j⌋+1 and⎪ ⎩f ∗ (v (j)p jv (j)1 ) =and for 3 ≤ j ≤ n,k=1j−1 ∑(p k +m k )+2p j −2i−j +3k=1j−1∑(p k +m k )−j +3k=1k=12p j is even⌊ pj⌋2 +2 ≤ i ≤ pj −1,j−2f ∗ (u (j−1)i u (j−1)i+1 ) = ∑(p k +mk)+p j−1 +i+2−j, for 1 ≤ i ≤ m j−1 −1.Hence, f is a F-Geometric mean labeling of Ĝ(p 1,m 1 ,p 2 ,m 2 ...,m n−1 ,p n ). Thus thegraph Ĝ(p 1,m 1 ,p 2 ,m 2 ...,m n−1 ,p n ) is a F-Geometric mean graph. □A F-Geometric mean labeling of Ĝ(8,4,5,6,10) is as shown in Figure 6.516 28 3 3 5 71414162626 28 121 724 24 301 9 9 10 11 12171718 210 1118 19 19 2020 21 21 223022 32823 31 13 15 4 4 6 825 25 29 31272961527Figure 6. A F-Geometric mean labeling of Ĝ(8,4,5,6,10)Theorem 2.4. C n ⊙K 1 is a F-Geometric mean graph, for n ≥ 3.Proof. Let v 1 ,v 2 ,...,v n be the vertices of the cycle C n and let u i be the pendantvertices attached⌊√at each⌋v i , for 1 ≤ i ≤ n. Consider the graph C n ⊙K 1 , for n ≥ 4.Case (i) 2n+1 is odd.

172 A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANWe define f : V[C n ⊙K 1 ] → {1,2,3,...,2n+1} as follows:⎧1 i = 1 ⎪⎨ ⌊ √2n+1 ⌋f(v i ) = 2i 2 ≤ i ≤ +12⌊ √2n+1 ⌋ and⎪⎩ 2i+1 +2 ≤ i ≤ n2⎧2 i = 1 ⌊ √2n+1 ⌋⎪⎨ 2i−1 2 ≤ i ≤2⌊f(u i ) =√2n+1 ⌋2i+1 i = +12⌊ √2n+1 ⌋⎪⎩ 2i +2 ≤ i ≤ n.The induced edge labeling is as follows:⎧ ⌊ √2n+1 ⌋⎨ 2i 1 ≤ i ≤f ∗ 2(v i v i+1 ) = ⌊ √2n+1 ⌋⎩ 2i+1 +1 ≤ i ≤ n−1,2⌊ √2n+1 ⌋f ∗ (v 1 v n ) = and⎧ ⌊ √2n+1 ⌋⎨ 2i−1 1 ≤ i ≤f ∗ 2(u i v i ) = ⌊ √2n+1 ⌋⎩ 2i +1 ≤ i ≤ n.2⌊√ ⌋Case (ii) 2n+1 is even.We define f : V[C n ⊙K 1 ] → {1,2,3,...,2n+1} as follows:⎧1 i = 1 ⎪⎨ ⌊ √2n+1 ⌋f(v i ) = 2i 2 ≤ i ≤2⌊ √2n+1 ⌋ and⎪⎩ 2i+1 +1 ≤ i ≤ i ≤ n2⎧2 i = 1 ⎪⎨ ⌊ √2n+1 ⌋f(u i ) = 2i−1 2 ≤ i ≤2⌊ √2n+1 ⌋⎪⎩ 2i +1 ≤ i ≤ n.The induced edge labeling is as follows:⎧ ⌊ √2n+1 ⌋⎨ 2i 1 ≤ i ≤ −1f ∗ 2(v i v i+1 ) = ⌊ √2n+1 ⌋⎩ 2i+1 ≤ i ≤ n−1,2⌊ √2n+1 ⌋f ∗ (v 1 v n ) = and⎧ ⌊ √2n+1 ⌋⎨ 2i−1 1 ≤ i ≤f ∗ 2(u i v i ) = ⌊ √2n+1 ⌋⎩ 2i +1 ≤ i ≤ n.222

F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 173Hence, the graph C n ⊙K 1 , for n ≥ 4 admits F-Geometric mean labeling.For n = 3, a F-Geometric mean labeling of C 3 ⊙K 1 is as shown in Figure 7.112□23 6 54 4 5 7 6Figure 7. A F - Geometric mean labeling of C 3 ⊙K 1A F-Geometric mean labeling of C 12 ⊙K 1 is as shown in Figure 8.21 2432 5 243 1 25227 4 4 23226 6 237218 92120 8209191119181010111317171816 12 15 13 1512161414Figure 8. A F-Geometric mean labeling of C 12 ⊙K 1Theorem 2.5. C n ⊙S 2 is a F-Geometric mean graph for n ≥ 3.Proof. Let u 1 ,u 2 ,...,u n be the vertices of the cycle C n . Let v (i)1 be the pendantvertices at each vertex u i , for 1 ≤ i ≤ n. Therefore,andV[C n ⊙S 2 ] = V(C n )∪{v (i)1 ,v(i) 2;1 ≤ i ≤ n}E[C n ⊙S 2 ] = E(C n )∪{u i v (i)1 ,u i v (i)2 ;1 ≤ i ≤ n}.

174 A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANCase (i) ⌊√ 6n ⌋ is a multiple of 3.We define f : V[C n ⊙S 2 ] → {1,2,3,...,3n+1} as follows:⎧ ⌊ √6n ⌋3i−1 1 ≤ i ≤3⎪⎨ ⌊ √6n ⌋f(u i ) = 3i+1 i = +13⌊ √6n ⌋ ⎪⎩3i +2 ≤ i ≤ n,3⎧ ⌊ √6n ⌋⎪⎨ 3i−2 1 ≤ i ≤f(v (i)31 ) = ⌊ √6n ⌋⎪⎩ 3i−1 +1 ≤ i ≤ n3andf(v (i)2 ) = ⎧⎨⎩3i3i+1⌊ √6n ⌋1 ≤ i ≤ +13⌊ √6n ⌋+2 ≤ i ≤ n.3The induced edge labeling is as follows⎧⎨ 3if ∗ (u i u i+1 ) =⎩ 3i+1⌊√ ⌋f ∗ (u n u 1 ) = 6n ,⎧⎨ 3i−2f ∗ (u i v (i)1 ) =⎩ 3i−1⌊ √6n⌋1 ≤ i ≤ −13⌊ √6n ⌋≤ i ≤ n−1,3⌊ √6n⌋1 ≤ i ≤3⌊ √6n ⌋+1 ≤ i ≤ n3and⎧⎨f ∗ (u i v (i)2 ) =⎩3i−13i⌊ √6n ⌋1 ≤ i ≤3⌊ √6n ⌋+1 ≤ i ≤ n.3Case (ii) ⌊√ 6n ⌋ is not a multiple of 3.We define f : V[C n ⊙S 2 ] → {1,2,3,...,3n+1} as follows:⎧ ⌊ √6n ⌋⎨ 3i−1 1 ≤ i ≤3f(u i ) = ⌊ √6n ⌋⎩ 3i +1 ≤ i ≤ n,3⎧ ⌊ √6n+1 ⌋⎨ 3i−2 1 ≤ i ≤f(v (i)31 ) = ⌊ √6n+1 ⌋⎩ 3i−1 +1 ≤ i ≤ n3

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

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.

F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 1773 3 4 6 9 9 10 12 15 15 16 18212 4 8 10 14 16 20 2 5 8 11 14 17 201 5 7 11 13 17 19 1 6 7 12 13 18 194 4 6 8 12 12 14 16 20 20 22 24 28 28 3036111419222730113 7 11151923 27 315 7 9 10 13 15 17 18 21223 25 26 29 31 5 9 13 17 21 25 292 8 10 16 18 24 26Figure 10. A F-Geometric mean labeling of (P 7 ;S 1 ) and (P 8 ;S 2 )32Theorem 2.7. For a H−graph G, G⊙K 1 is a F-Geometric mean graph.Proof. Let u 1 ,u 2 ,...,u n and v 1 ,v 2 ,...,v n be the vertices of G. ThereforeandV(G⊙K 1 ) = V(G)∪{u ′ i,v ′ i;1 ≤ i ≤ n}E(G⊙K 1 ) = E(G)∪{u i u ′ i ,v iv i ′ ;1 ≤ i ≤ n}.Case (i) n ≡ 0(mod 4).We define f : V(G⊙K 1 ) → {1,2,3,...,4n} as follows:{2i−1 1 ≤ i ≤ n and i is oddf(u i ) =2i 1 ≤ i ≤ n and i is even,{2i 1 ≤ i ≤ n and i is oddf(u ′ i ) = 2i−1 1 ≤ i ≤ n and i is even,{ ⌊2n−3+4i 1 ≤ i ≤n⌋f(v i ) = 2 −1 and i is odd2n−1+4i 1 ≤ i ≤ ⌊ n2⌋and i is even,{ ⌊2n−2+4i 1 ≤ i ≤n⌋f(v n+1−i ) = 2 −1 and i is odd2n+4i 1 ≤ i ≤ ⌊ n2⌋and i is even,

178 A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANand⎧⎪ f(v i )+2 ⎨f(v i ′ ) = f(v i )−2⎪ f(v i )−2 ⎩f(v i )+2The induced edge labeling is as follows:1 ≤ i ≤ ⌊ n⌊ 2⌋−1 and i is oddn⌋2 +1 ≤ i ≤ n and i is odd1 ≤ i ≤ ⌊ n⌋ 2⌋and i is even+2 ≤ i ≤ n and i is even.⌊ n2f ∗ (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⌋and⎧⎪ f(v i ) ⎨f ∗ (v i v i ′ ) = f(v i )−2⎪ f(v i )−2 ⎩f(v i )1 ≤ i ≤ ⌊ n⌊ 2⌋−1 and i is oddn⌋2 +1 ≤ i ≤ n and i is odd1 ≤ i ≤ ⌊ n⌋ 2⌋and i is even+2 ≤ i ≤ n and i is even⌊ n2f ∗ (u i+1 v i ) = 2n, for i =⌊ n2⌋.Case (ii) n ≡ 1(mod 4).We define f : V(G⊙K 1 ) → {1,2,3,...,4n} as follows:andf(u i ) = 2i, for 1 ≤ i ≤ n,f(u ′ i) = 2i−1, for 1 ≤ i ≤ n,{ ⌊2n−3+4i 1 ≤ i ≤n⌋f(v i ) = 2 +1 and i is odd2n−1+4i 1 ≤ i ≤ ⌊ n2⌋and i is even,{ ⌊2n−2+4i 1 ≤ i ≤n⌋f(v n+1 −i) = 2 and i is odd2n+4i 1 ≤ i ≤ ⌊ n2⌋and i is even⎧⎪ f(v ⎨ i )+2f(v i ′ ) = f(v i )+1⎪ f(v ⎩ i )+2f(v i )−2The induced edge labeling is as follows:1 ≤ i ≤ ⌊ n2⌋and i is oddi = ⌊ n⌊ 2⌋+1 and i is oddn⌋2 +3 ≤ i ≤ n and i is odd1 ≤ i ≤ n and i is even.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 ≤ ,2⌋

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⌋

180 A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANCase (iv) n ≡ 3(mod 4).We define f : V(G⊙K 1 ) → {1,2,3,...,4n} as follows:andf(u i ) = 2i, for 1 ≤ i ≤ n,f(u ′ i) = 2i−1, for 1 ≤ i ≤ n,{ ⌊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⌋and i is even,⎧⎪ f(v ⎨ i )−2f(v i ′ ) = f(v i )+1⎪ f(v ⎩ i )−2f(v i )+2The induced edge labeling is as follows:f ∗ (u i u i+1 ) = 2i, for 1 ≤ i ≤ n−1,1 ≤ i ≤ ⌊ n2⌋and i is oddi = ⌊ n⌊ 2⌋+1 and i is oddn⌋2 +3 ≤ i ≤ n and i is odd1 ≤ i ≤ n and i is even.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 ≤ ,2⌋and{f ∗ (v i v i ′ ) = f(vi )−2 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⌋Hence, f is a F-Geometric mean labeling of G ⊙ K 1 . Thus the graph G ⊙ K 1 is aF-Geometric mean graph.□A F-Geometric mean labeling of G⊙K 1 is as shown in Figure 11.Theorem 2.8. For a H−graph G, G⊙S 2 is a F-Geometric mean graph.Proof. Let u 1 ,u 2 ,...,u n and v 1 ,v 2 ,...,v n be the vertices of G. Therefore,andV[G⊙S 2 ] = V(G)∪{u ′ i ,u′′i ,v′ i ,v′′ i;1 ≤ i ≤ n}E[G⊙S 2 ] = E(G)∪{u i u ′ i ,u iu ′′i ,v iv i ′ ,v iv i ′′ ;1 ≤ i ≤ n}.

F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 181135791113 1 2 15 17 115 1 2 13 15 132 172 15 3 4 1919 21 3 4 31717 194 214 19 5 6 23 25 23 55 6 21 23 216 256 23 7 814 27 2827 8 7 ★ ★★★ 12 22 2287 24 9 26810 2426 24 9 209 10 1818 2010 2210 16 11 122020 22 11 12 11 16141412 18 13 14 18 16 16Figure 11. A F-Geometric mean labeling of H graph G⊙K 1Case (i) n is odd.We define f : V(G⊙S 2 ) → {1,2,3,...,6n} as follows:f(u i ) = 3i−1, for 1 ≤ i ≤ n,f(u ′ i) = 3i−2, for 1 ≤ i ≤ n,f(u ′′i ) = 3i, for 1 ≤ i ≤ n,⌊ nf(v i ) = 3n−2+6i, for 1 ≤ i ≤ ,⌊2⌋nf(v n+1−i ) = 3n−3+6i, for 1 ≤ i ≤ +1,⎧ 2⌋⎨ f(v i )−2 1 ≤ i ≤ ⌊ ⌋nf(v i ′ ) = 2f(v i )−1 i = ⌊ n⎩ ⌋ 2⌋+1f(v i )+2 +2 ≤ i ≤ n⌊ n2and{f(v i ′′ ) = f(vi )+2f(v i )−21 ≤ i ≤ ⌊ ⌋n⌋ 2+1 ≤ i ≤ n.⌊ n2The induced edge labeling is as follows:f ∗ (u i u i+1 ) = 3i, for 1 ≤ i ≤ n−1,f ∗ (u i u ′ i ) = 3i−2, for 1 ≤ i ≤ n,f ∗ (u i u ′′i) = 3i−1, for 1 ≤ i ≤ n,⌊ nf ∗ (u i v i ) = 3n, for i = +1,2⌋

182 A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANand⌊ nf ∗ (v i v i+1 ) = 3n+6i, for 1 ≤ i ≤ ,2⌋⌊ nf ∗ (v n+1−i v n−i ) = 3n−1+6i, for 1 ≤ i ≤ ,⎧2⌋⎨ f(v i )−2 1 ≤ i ≤ ⌊ ⌋nf ∗ (v i v i) ′ 2= f(v i )−1 i = ⌊ n⎩ ⌋ 2⌋+1f(v i ) +2 ≤ i ≤ n{f ∗ (v i v i ′′ ) = f(vi )f(v i )−2⌊ n21 ≤ i ≤ ⌊ ⌋n⌋ 2+1 ≤ i ≤ n.⌊ n2Case (ii) n is even.We define f : V(G⊙S 2 ) → {1,2,3,...,6n} as follows:andf(u i ) = 3i−1, for 1 ≤ i ≤ n,f(u ′ i ) = 3i−2, for 1 ≤ i ≤ n,f(u ′′i) = 3i, for 1 ≤ i ≤ n−1,f(u ′′ n ) = 3n+1,⌊ nf(v i ) = 3n+1+6i, for 1 ≤ i ≤ −1,2⌋⌊ nf(v n+1−i ) = 3n+6(i−1), for 2 ≤ i ≤ +1,2⌋f(v n ) = 3n+2,{f(v i) ′ f(vi )−2 1 ≤ i ≤ ⌊ ⌋n= ⌋ 2f(v i )+2 +1 ≤ i ≤ n−1,f(v ′ n) = f(v n )+1f(v ′′i ) = ⎧⎨⎩f(v i )+2f(v i )−1f(v i )−2The induced edge labeling is as follows:⌊ n21 ≤ i ≤ ⌊ n2⌋−1i = ⌊ ⌋n⌋ 2+1 ≤ i ≤ n.⌊ n2f ∗ (u i u i+1 ) = 3i, for 1 ≤ i ≤ n−1,f ∗ (u i u ′ i ) = 3i−2, for 1 ≤ i ≤ n,f ∗ (u i u ′′i ) = 3i−1, for 1 ≤ i ≤ n,⌊ nf ∗ (u i+1 v i ) = 3n+1, for i = ,2⌋f ∗ (v i v i+1 ) = 3n+3+6i, for 1 ≤ i ≤⌊ n2⌋,

F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 183f ∗ (v n+1−i v n−i ) = 3n−4+6i, for 2 ≤ i ≤⌊ n2⌋,f ∗ (v n v n−1 ) = 3n+3,{f ∗ (v i v i) ′ f(vi )−2 1 ≤ i ≤ ⌊ ⌋n= ⌋ 2f(v i ) +1 ≤ i ≤ n⌊ n2and⎧⎨f ∗ (v i v i) ′′ =⎩f(v i )f(v i )−1f(v i )−21 ≤ i ≤ ⌊ n2⌋−1i = ⌊ ⌋n⌋ 2+1 ≤ i ≤ n.⌊ n2Hence, f is a F-Geometric mean labeling of G ⊙ S 2 . Thus the graph G ⊙S 2 is aF-Geometric mean graph.□A F-Geometric mean labeling of G⊙S 2 is as shown in Figure 12.1 13 2 2 31 29291 14 43 33 31 333 2 2 25 2323 3565 5 37❍ 354 3427 25 27 7 ❍❍❍6 29 5 5 31❍ 29 8 7 6 39 3739 7 ❍❍ 9 8 4341❍ 8 7 6 33 31413310 9 8 3735 11 10 9 4543 4511 48 461035 1246 11 10 9 3913 37 3913 1225 44 47 471121 42 41124113 1514 144242 4413 12 38 40 4016 1615 38 40 1514 14364036 3817 3636 3816 1516 32 34 18 173418 32 341730 30 32 19 193418 3017✟ 2018 26 28✟ 19 1928 ✟ 30 32 20 21 27 28 28✟20 2421✟✟ 24 26 22 26 22 23 26 27 20 22 22 23 24 212524Figure 12. A F-Geometric mean labeling of H graph G⊙S 2Theorem 2.9. L n ⊙K 1 is a F-Geometric mean graph, for n ≥ 2.Proof. Let V(L n ) = {u i ,v i ;1 ≤ i ≤ n} be the vertex set of the ladder L n andE(L n ) = {u i v i ;1 ≤ i ≤ n} ∪ {u i u i+1 ,v i v i+1 ;1 ≤ i ≤ n − 1} be the edge set of theladder L n . Let w i be the pendent vertex adjacent to u i , and x i be the pendent vertexadjacent to v i , for 1 ≤ i ≤ n.

184 A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANandWe define f : V(L n ⊙K 1 ) → {1,2,3,...,5n−1} as follows:f(u 1 ) = 3,f(v 1 ) = 4,f(x 1 ) = 2,The induced edge labeling is as followsf(u i ) = 5i−3, for 2 ≤ i ≤ n,f(v i ) = 5i−2, for 2 ≤ i ≤ n,f(w i ) = 5i−4, for 1 ≤ i ≤ nf(x i ) = 5i−1, for 2 ≤ i ≤ n.f ∗ (u 1 v 1 ) = 3,f ∗ (v 1 x 1 ) = 2,f ∗ (u i v i ) = 5i−3, for 2 ≤ i ≤ n,f ∗ (w i u i ) = 5i−4, for 1 ≤ i ≤ n,f ∗ (v i x i ) = 5i−2, for 2 ≤ i ≤ n,f ∗ (u i u i+1 ) = 5i−1, for 1 ≤ i ≤ n−1,f ∗ (v i v i+1 ) = 5i, for 1 ≤ i ≤ n−1.Hence f is a F-Geometric mean labeling of L n ⊙K 1 . Thus the graph L n ⊙ K 1 is aF-Geometric mean graph, for n ≥ 2.□A F-Geometric mean labeling of L 8 ⊙K 1 is as shown in Figure 13. 116611111616 43 79 1214 1719 2224 27293234 373 7 12 17 22 27 32 374 5 10 15 20 25 30 35 8 13 18 23 28 33382 8 13 18 23 28 33 38 2 9 14 19 24293439Figure 13. A F-Geometric mean labeling of L 8 ⊙K 1Theorem 2.10. L n ⊙S 2 is a F-Geometric mean graph, for n ≥ 2.Proof. Let V(L n ) = {u i ,v i ;1 ≤ i ≤ n} be the vertex set of the ladder L n andE(L n ) = {u i v i ;1 ≤ i ≤ n} ∪ {u i u i+1 ,v i v i+1 ;1 ≤ i ≤ n − 1} be the edge set of theladder L n .2121262631313636

F-GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 185Let w (i)1 and w (i)2 be the pendant vertices at each vertex u i , for 1 ≤ i ≤ n and x (i)1and x (i)2 be the pendant vertices at each vertex v i , for 1 ≤ i ≤ n. ThereforeV(L n ⊙S 2 ) = V(L n )∪{w (i)1 ,w (i)2 ,x (i)1 ,x (i)2 ;1 ≤ i ≤ n}andE(L n ⊙S 2 ) = E(L n )∪{u i w (i)1 ,u iw (i)2 ,v ix (i)1 ,v ix (i)2 ;1 ≤ i ≤ n}.We define f : V(L n ⊙S 2 ) → {1,2,3,...,7n−1} as follows :⎧⎨ 3 i = 1f(u i ) = 7i−2 2 ≤ i ≤ n and i is even⎩7i−5 3 ≤ i ≤ n and i is odd,⎧⎨ 5 i = 1f(v i ) = 7i−4 2 ≤ i ≤ n and i is even⎩7i−1 3 ≤ i ≤ n and i is odd,andf(w (i){7i−3 1 ≤ i ≤ n and i is even1 ) = 7i−6 1 ≤ i ≤ n and i is odd,f(w (i)2 ) = ⎧⎨⎩f(x (i)1 ) =f(x (i)2 ) =2 i = 17i−1 2 ≤ i ≤ n and i is even7i−4 3 ≤ i ≤ n and i is odd,{7i−6 1 ≤ i ≤ n and i is even7i−3 1 ≤ i ≤ n and i is odd,⎧⎨ 6 i = 1⎩7i−57i−2The induced edge labeling is as follows:2 ≤ i ≤ n and i is even3 ≤ i ≤ n and i is odd.f ∗ (u i u i+1 ) = 7i−1, for 1 ≤ i ≤ n−1,f ∗ (u i v i ) = 7i−4, for 1 ≤ i ≤ n,f ∗ (v i v i+1 ) = 7i, for 1 ≤ i ≤ n−1,{f ∗ (u i w (i) 7i−3 1 ≤ i ≤ n and i is even1 ) = 7i−6 1 ≤ i ≤ n and i is odd,{f ∗ (u i w (i) 7i−2 1 ≤ i ≤ n and i is even2 ) =7i−5 1 ≤ i ≤ n and i is odd,{f ∗ (v i x (i) 7i−6 1 ≤ i ≤ n and i is even1 ) = 7i−3 1 ≤ i ≤ n and i is odd,{f ∗ (v i x (i) 7i−5 1 ≤ i ≤ n and i is even2 ) = 7i−2 1 ≤ i ≤ n and i is odd.

186 A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANHence, f is a F-Geometric mean labeling of L n ⊙ S 2 . Thus the graph L n ⊙ S 2 is aF-Geometric mean graph, for n ≥ 2.□A F-Geometric mean labeling of L 7 ⊙S 2 is as shown in Figure 14. 211 13 15 17 25 27 29 31 39 41 43 45 11 2 11 12 15 16 25 26 29 30 39 40 43 443 6 13 20 27 34 41 12 16 26 30 40 44310 17 24 31 38 45 7 10 14 20 21 24 28 345 35 38 42 484 5 8 9 ☞ ✂ ❏18 19 22 23 32 33 36 37 46 ☞ ❏❏ 47✂ 4 6 8 9 18 19 22 23 32 33 36 37 46 47Figure 14. A F-Geometric mean labeling of L 7 ⊙S 2References[1] A. Durai Baskar, S. Arockiaraj and B. Rajendran, Geometric Mean Graphs, (Communicated).[2] J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics,17(2011).[3] F. Harary, Graph theory, Addison Wesely, Reading Mass., 1972.[4] R. Ponraj and S. Somasundaram, Further results on mean graphs, Proceedings of Sacoeference,(2005), 443–448.[5] A. Rosa, On certain valuation of the vertices of graph, International Symposium, Rome, July1966, Gordon and Breach, N. Y. and Dunod Paris (1967), 349–355.[6] S. Somasundaram and R. Ponraj, Mean labeling of graphs, National Academy Science Letter,26(2003), 210–213.[7] S. Somasundaramand R. Ponraj,Some results on mean graphs, Pureand Applied MathematikaSciences, 58(2003), 29–35.[8] S. Somasundaram, P. Vidhyarani and R. Ponraj, Geometric mean labeling of graphs, Bulletinof Pure and Applied sciences, 30E (2011), 153–160[9] S. Somasundaram, P. Vidhyarani and S. S. Sandhya, Some results on Geometric mean graphs,International Mathematical Form, 7 (2012), 1381–1391.[10] R. Vasuki and A. Nagarajan, Further results on mean graphs, Scientia Magna, 6(3) (2010),1–14.Department of Mathematics,Mepco Schlenk Engineering College,Mepco Engineering College (Po) - 626 005Sivakasi, Tamilnadu, INDIA.E-mail address: a.duraibaskar@gmail.com, sarockiaraj 77@yahoo.com,drbr58msec@hotmail.com