Power Mean Labeling of some Standard Graphs

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1 Power Mean Labeling of some Standard Graphs P. Mercy & S. Somasundaram Department of Mathematics Manomaniam Sundaranar University Tirunelveli Abstarct: A graph G = (V, E) is called a Power mean graph with p vertices and q edges, if it is possible to label the vertices x V with distinct elements f (x) from, 2, 3,, q + in such way that when each edge e = uv is labeled with or f (e = uv) = f (e = uv) = ( f (u) f (v) f f (u) (v) ) f (u)+ f (v) ( f (u) f (v) f (v) f (u) ) f (u) + f (v) then the edge labels are distinct. Here f is called Power mean labeling of G. We investigate on Power mean graph labeling on some standard graphs. 200 Mathematics Subject Classification : 05C38, 05C78 Keywords and Phrases: PO W E R M E A N G R A P H, TR I A N G U L A R SN A K E T n, ALT E R N AT E TR I A N G U L A R SN A K E A(T n ), QU A D R I L AT E R A L SN A K E Q n, ALT E R N AT E QU A D R I L AT E R A L SN A K E A(Q n ). INT RODUCT ION The graphs considered here are finite and undirected graphs. Let G = (V, E) be a graph with p vertices and q edges. For a detailed survey of graph labeling one may refer to Gallian[2] and also []. For all other standard terminologies and notations we follow Harary[3]. To cite some labeling techniques, we record a few of them. Somasundaram and Ponraj [6, 9] introduced and studied mean labeling for some standard graphs. Santhya et al. [4] and Sandhya and Somasundaram [5] introduced and studied Harmonic mean labeling of graphs. Somasundaram et al. [7, 8] introduced the concept of Geometric mean labeling of graphs and studied their behaviour. In this paper we investigate power mean labeling for some standard graphs like Triangular Snake T n, Alternate Triangular Snake A(T n ), Quadrilateral Snake Q n and Alternate Quadrilateral Snake A(Q n ). Corresponding Author: id: soorjimemphil@gmail.com id: soorjimemphil@gmail.com Vol: I. Issue LVIII, December

2 2 DEFINIT ION AND RESULT S Here we state our definition of Power mean labeling. Definition 2.. A graph G = (V, E) with p vertices and q edges is said to be a Power Mean Graph if it is possible to label the vertices x V with distinct labels f (x) from, 2, 3,..., q + is such a way that when each edge e = uv is labeled with f (e = uv) = ( f (u) f (v) f (v) f (u) ) f (u)+ f (v) or f (e = uv) = ( f (u) f (v) f (v) f (u) ) f (u) + f (v) then the resulting edge labels are distinct. In this case is called Power mean labelling of G. f Remark 2.. If G is a Power mean labeling graph, then must be a label for one of the vertices of G, since an edge should get label. Remark 2.2. If p > q +, then the graph G = (p, q) is not a Power mean graph, since it doesn t have sufficient labels from {, 2, 3,..., q + } for the vertices of G. The following Proposition will be used in the edge labelings of some standard graphs for Power mean labeling. Proposition 2.. Let a, b and i be positive integers with a < b. Then (i) a < (a b b a ) a+b < b, i < (i +2 (ii) (i + 2) i ) 2i+2 i < (i i+3 (iii) (i + 3) i ) 2i+3 i < (i i+4 (iv) (i + 4) i ) 2i+4 (v) ( i i ) i+ = i i+ < 2. < (i + ), < (i + 2), < (i + 2), and Proof. (i) Since a a+b = a a a b < b a a b < b a b b = b a+b, we get the inequality in Proposition 2..( i ). That is, the Power mean of two numbers lies between the numbers a and b. Thus we infer that if vertices u, v have labels i, i + respectively, then the edge uv may be labeled i or i + for Power mean labeling. Vol: I. Issue LVIII, December

3 (ii) As a proof of this inequality, we see i i+2 (i + 2) i < i 2 [i(i + 2)] i, < i 2 (i + ) 2i, since i(i + 2) < (i + ) 2, < (i + ) 2 (i + ) 2i, = (i + ) 2i+2. This leads to [(i i+2 (i + 2) i ) 2i+2 ] < i +. Therefore, if u, v have labels i, i + 2 respectively, then the edge uv may be labeled i and i +. (iii) Next we have i i+3 (i + 3) i = i 3 [i(i + 3)] i, < i 3 (i + 2) 2i, since i(i + 3) < (i + 2) 2, < (i + 2) 3 (i + 2) 2i, = (i + 2) 2i+3. This leads to [i i+3 (i + 3) i ] 2i+3 < (i + 2). Hence, if u, v have labels i, i + 3 respectively, then the edge uv may be labeled i + without ambiguity. (iv) Now i i+4 (i + 4) i = i 4 [i(i + 4)] i, < i 4 (i + 2) 2i, since i(i + 4) < (i + 2) 2, < (i + 2) 4 (i + 2) 2i, = (i + 2) 2i+4. Therefore [i i+4 (i + 4) i ] 2i+4 < i + 2. Hence if u, v have labels i, i + 4 respectively, then the edge uv may be labeled i +. (v) Now 2 i+ = (i + ) i+, = + (i+) C + + (i+) C i+, (i + 2) terms, i + 2 > i. Vol: I. Issue LVIII, December

4 Therefore ( i i ) i+ = i i+ < 2. Thus we observe that if u, v uv may be labeled or 2. are labeled, i respectively, then the edge 2. Power Mean labeling for Triangular Snake T n Triangular Snake: A triangular Snake is a graph in which every edge of a path is replaced by C 3. It is denoted by T n. Theorem 2.. Any triangular Snake T n is a Power mean graph. Proof. Let T n be a triangular Snake with 2n vertices and 3n 3 edges. Define a function f : V(G) {, 2, 3,, q + = 3n 2} as (i) f (v ) =, (ii) f (v i ) = 3i 2 ; 2 i n, (iii) f (w i ) = 3i ; i n. 2..( i ), ( ii ) and ( iii ) the edge are labeled (i) E(v i v i+ ) = 3i ; i n, (ii) E(v i w i ) = 3i 2 ; i n, (iii) E(v i+ w i ) = 3i ; i n. As the edges are distinct, the graph T n is a Power mean labeled graph. Example 2.. Illustrative example for T 6 in Figure 2.. w =3 w2 =6 w3 =9 w4 =2 w5 = v = v2 =4 v3 =7 v4 =0 v5 =3 v6 =6 Figure 2.: T Power Mean labeling for Alternate Triangular Snake A(T n ) Alternate Triangular Snake: An alternate Triangular Snake A(T n ) is obtained from a path u, u 2, u 3,, u n by joining u i and u i+ (alternatively) to a new vertex w i. That is, every alternate edge of a path is replaced by C 3. Vol: I. Issue LVIII, December

5 Theorem 2.2. Alternate Triangular Snake is a Power mean graph. Proof. Let A(T n ) be an Alternate Triangular Snake. The following two cases are to be discussed. Case (i) : If the triangle starts from v, define a function f : V(A(T n )) {, 2, 3,..., q + } as (i) f (v i ) = 2i ; i n, (ii) f (u i ) = 2i ; i =, 3, 5, 7,..., n. 2..( i ), ( ii ) and ( iii ), the edge labels are labeled (i) E(v i v i+ ) = 2i ; i n, (ii) E(u i v i ) = 2i ; i =, 3, 5,..., n, (iii) E(u i v i+ ) = 2i + i =, 3, 5,..., n. Case (ii) : If the triangle starts from v 2, define a function f : V(A(T n )) {, 2, 3,..., q + } as (i) f (v ) =, (ii) f (v 2 ) = 2, (iii) f (v i ) = 2i 2 ; i = 3 i n, (iv) f (ui ) = 2i ; i = 2, 4, 6,..., n ( i ), ( ii ) and ( iii ), the edges are labeled (i) E(v i v i+ ) = 2i ; i l eqn, (ii) E(ui vi ) = 2i 2 i = 2, 4, 6,..., n 2, (iii) E(u i v i+ ) = 2i ; i = 2, 4, 6,..., n 2. As the edges are different, any alternate triangular snake A(T n ) cases. is a Power mean labeled graph in both Example 2.2. Illustrative example for Case (i): A(T 6 ). in Figure 2.2. u =2 u3 =6 u5 = v = 2 v2 =34 v3 =5 6v4 =7 8v5 =9 0v6 = Figure 2.2: Case(i): A(T 6 ) ) Vol: I. Issue LVIII, December

6 Example 2.3. Illustrative example for Case (ii): A(T 6 ) in Figure 2.3. u2 =3 u4 = v = v2 =2 3v3 =4 v4 =6 v5 =8 v6 =0 Figure 2.3: Case(ii): A(T 6 ) 2.3 Power Mean labeling for Quadrilateral snake Q n Quadrilateral snake: A Quadrilateral snake Q n is obtained from a path u, u 2, u 3,..., u n by joining u i and u i+ to v i and w i respectively and joining v i and w i that is every edge of a path is replaced by a cycle C 4. Theorem 2.3. Any Quadrilateral snake Q n is a Power mean graph. Proof. Let Q n be the Quadrilateral snake with 3n 2 vertices and 4n 4 edges(where n is the number of vertices of the path). Define a function f : V(G) {, 2, 3,..., q + } by as (i) f (u i ) = 4i 3 ; i n, (ii) f (v i ) = 4i 2 ; i n, (iii) f (w i ) = 4i ; i n. 2..( i ), ( ii ) and ( iii ), the edges are labeled (i) E(u i u i+ ) = 4i 2 ; i n, (ii) E(u i v i ) = 4i 3 ; i n, (iii) E(v i w i ) = 4i ; i n, (iv) E(w i u i+ ) = 4i ; i n. As the edges are distinct, the Quadrilateral snake Q n is a Power mean labeled graph. Vol: I. Issue LVIII, December

7 Example 2.4. Illustrative example for Q 5 in Figure 2.4 v =2 w=3 v2 =6 w2=7 v3 =0 w3 = v4 =4 w4 = u = u2 =5 u3 =9 u4 =3 u5 =7 Figure 2.4: Q Power Mean labeling for Alternate Quadrilateral snake A(Q n ) Alternate Quadrilateral snake: An alternate Quadrilateral snake A(Q n ) is obtained from a path u, u 2, u 3,..., u n by joining u i u i+ (alternatively) to new vertices v i v i+ respectively and then joining v i and v i+ ie. Every alternate edge of a path is replaced by a cycle C 4. Theorem 2.4. Any Alternate Quadrilateral Snake is a Power mean graph. Proof. Let A(Q n ) be the Alternate Quadrilateral Snake. The following two cases are to be obtained. Case (i) : Let the Quadrilateral start from v. Define a function f : V(A(Q n )) {, 2, 3,..., q + } as (i) f (v ) =, (ii) f (v 2 ) = 4, (iii) f (v i ) = f (v i 2 ) + 5 ; i = 3, 4, 5,..., n, (iv) f (u i ) = f (v i ) ; i =, 3, 5,..., n, (v) f (u i ) = f (v i ) ; i = 2, 4, 6,..., n. 2..( i ), ( ii ) and ( iii ), the edges are labeled (i) f (v v 2 ) = 2, (ii) f (v 2 v 3 ) = 5, (iii) f (v i v i+ ) = f (v i 2 v i ) + 5 ; i = 3, 4, 5,..., n, Vol: I. Issue LVIII, December

8 (iv) f (v i u i ) = f (v i ) ; i =, 2, 3,..., n, (v) f (u i u i+ ) = f (u i+ ) i =, 3, 5,..., n. Example 2.5. Illustrative example for be the Alternate Quadrilateral Snake A(Q n ) starting from v in Figure 2.5 u =2 u2 =3 u3 =7 u4 =8 u5 =2 u6 = v = v2 =45v3 =6 v4 =90v5 = v6 =4 Figure 2.5: A(Q n ) starts from v Case (ii) :Let the Quadrilateral start from v 2. Define a function f : V(A(Q n )) {, 2, 3,..., q + }. as (i) f (v ) =, (ii) f (v 2 ) = 2, (iii) f (v 3 ) = 5, (iv) f (v i ) = f (v i 2 ) + 5 ; i = 4 i n, (v) f (u i ) = f (v i ) + ; i = 2, 4, 6,..., n 2, (vi) f (u i ) = f (v i ) ; i = 3, 5, 7,..., n. 2..( i ), ( ii ) and ( iii ), the edges are labeled (i) f (v v 2 ) =, (ii) f (v 2 v 3 ) = 4, (iii) f (v i v i+ ) = f (v i 2 v i ) + 5 ; 3 i n, (iv) f (v i u i ) = f (v i ) ; 2 i n, (v) f (u i u i+ ) = f (u i ) ; i =, 3, 5,..., n 3. As the edges labels are distinct, any alternate Quadrilateral Snake is a Power mean labeled graph in both cases. Vol: I. Issue LVIII, December

9 Example 2.6. Illustrative example for be the Alternate Quadrilateral Snake A(Q n ) starting from v 2 in Figure 2.6 u =3u2 =4 u3 =8u4 = v = v2 =2 v3 =5 v4 =7 v5 =0 v6 =2 Figure 2.6: A(Q n ) starts from v 2 3 CONCLUSION In this paper we have proved that Triangular Snake T n, Alternate Triangular Snake A(T n ), Quadrilateral Snake Q n and Alternate Quadrilateral Snake A(Q n ) graphs are amenable for Power Mean labeling and provided illustrative examples to support our investigation. References [] B.D. Acharya, S. Arumugam and A. Rosa, Labeling of Discrete Structures and Applications, Narosa Publishing House, New Delhi, 2008, 4. [2] J.A. Gallian. A dynamic Survey of graph labeling. The electronic Journal of Combinatories, 7 DS6, (202). [3] F. Harary, Graph Theory, Narosa Publishing House, New Delhi, 988. [4] S.S. Sandhya, S. Somasundaram and R.Ponraj, Some More Results on Harmonic Mean Graphs, Journal of Mathematics Research 4() (202), [5] S. S. Sandhya and S. Somasundaram, Harmonic Mean Labeling for Some Special Graphs, International Journal of Mathematics Research, 5() (203), [6] S. Somasundaram and R. Ponraj, Mean Labelings of Graphs, National Academy Science Letters, 26 (2003), [7] S. Somasundaram, P. Vidhyarani and R. Ponraj, Geometric Mean Labeling of Graphs, Bullettin of Pure and Applied Sciences, 30E(2) (20), [8] S. Somasundaram, P. Vidhyarani and R. Ponraj, Some Results on Geometric Mean Graphs, International Journal of Mathematical Forum, 7(28) (202), [9] S. Somasundaram and R. Ponraj, Some Results on Mean Graphs, Pure and Applied Mathematika Sciences, 53 (2003), Vol: I. Issue LVIII, December