International Journal of Advanced Engineering Research and Science (IJAERS)
[Vol-3, Issue-9, Sep- 2016] ISSN: 2349-6495(P) — 2456-1908(O)
Snakes related strongly∗-graphs I. I. Jadav1 , G. V. Ghodasara2 1 Research Scholar, R. K. University, Rajkot−360020, India. jadaviram@gmail.com. , 2 H. & H. B. Kotak Institute of Science, Rajkot−360001, India. gaurang enjoy@yahoo.co.in Abstact —A graph with n vertices is said to be strongly∗ -graph if its vertices can be assigned the values {1, 2, . . . , n} in such a way that when an edge whose end vertices are labeled i and j, is labeled with the value i + j + ij such that all edges have distinct labels. Here we derive some snakes related strongly∗ -graphs. Keyword - Strongly∗ -labeling, Strongly∗ -graph. AMS Subject classification number: 05C78. I. INTRODUCTION Here graph G is considered as a simple, finite, undirected graph. Definition I.1. [1] A graph G with n vertices is said to be strongly∗ -graph if there is a bijection f : V (G) −→ {1, 2 . . . , n} such that the induced edge function f ∗ : E(G) −→ N defined as f ∗ (e = uv) = f (u) + f (v) + f (u) · f (v) is injective. Here f is called strongly∗ -labeling of graph G. II. MAIN RESULTS Definition II.1. [2] A triangular snake Tn is obtained from a path Pn by replacing each edge of the Pn by a cycle C3 . Definition II.2. [4] An alternate triangular snake A(Tn ) is obtained from a path Pn by replacing each alternate edge of Pn by a cycle C3 . Theorem II.1. Alternate triangular snake A(Tn ) is strongly∗ -graph.
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Proof. Let A(Tn ) be the alternate triangular snake obtained from a path Pn with consecutive vertices {u1 , u2 , . . . , un } by joining ui and ui+1 with new vertex vj alternatively, 1 ≤ i ≤ n − 1, 1 ≤ j ≤ b n2 c. To define vertex labeling function f : V (A(Tn )) → {1, 2, . . . , |V (A(Tn ))|} we consider the following cases. Case 1: Triangle in A(Tn ) starts from u1 . Note that in this case |V (A(Tn ))| = b(n2 c + n and |E(A(Tn ))| = 2n − 1, if n is even. 2(n − 1), if n is odd. f (ui ) = b n2 c + n, 1 ≤ i ≤ n. f (vj ) = 3j − 1, 1 ≤ j ≤ b n2 c. Strongly∗ -labeling of A(Tn ) in this case is shown in Figure 1. Case 2:Triangle in A(Tn ) starts from u2 . Subcase 1: n is odd. Defined as case-1. Subcase 2: n is even. Note that |V (A(Tn ))| = 3n − 1 and 2 |E(A(Tn ))| = 2n − 3 f (ui ) = 3i2 − 1, 1 ≤ i ≤ n. f (vj ) = 3j, 1 ≤ j ≤ b n2 c. Strongly∗ -labeling of A(Tn ) in this case is shown in Figure 2. Here the vertex labels produced by labeling f in A(Tn ) are strictly increasing. Page | 240
International Journal of Advanced Engineering Research and Science (IJAERS)
Hence obviously edge labels produced by f ∗ (uv) = f (u) + f (v) + f (u) ¡ f (v) are increasing for all pair of vertices u and v in A(Tn ). Hence the edge labels produced are all distinct. So, the labeling pattern defined above satisfies the conditions of strongly ∗ labeling. i.e. A(Tn ) is strongly∗ -graph. Definition II.3. [3] A double triangular snake DTn consists of two triangular snakes that have a common path. Theorem II.2. Double triangular snake DTn is strongly∗ -graph. Proof. Let {u1 , u2 , . . . , un } be successive vertices of Pn in DTn , where ui is adjacent to ui+1 , 1 ≤ i ≤ n − 1. Join ui and ui+1 with two new vertices vi and wi , 1 ≤ i ≤ n − 1. Here, |V (DTn )| = 3n − 2 and |E(DTn )| = 5(n − 1). We define vertex labeling function f : V (DTn ) → {1, 2, . . . , |V (DTn )|} as follows. f (ui ) = 3i − 2, 1 ≤ i ≤ n. f (vi ) = 3i − 1 1 ≤ i ≤ n − 1. f (wi ) = 3i Strongly∗ -labeling of DTn is shown in Figure 3. Since the vertex labels are strictly increasing, for any pair of adjacent vertices (ui , vj ) and (us , ut ), we have (f (ui ) + f (vj ) + f (ui )f (vj )) 6= (f (us ) + f (ut ) + f (us )f (ut )) ⇒ f ∗ (ui vj ) 6= f ∗ (us vt ). Hence above defined labeling pattern satisfies the conditions of strongly∗ -labeling. i.e. Double triangular snake is strongly∗ graph. Definition II.4. [4] An alternate double triangular snake DA(Tn ) consists of two alternate triangular snakes that have common path.
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[Vol-3, Issue-9, Sep- 2016] ISSN: 2349-6495(P) — 2456-1908(O)
Theorem II.3. Alternate double triangular snake DA(Tn ) is strongly∗ -graph. Proof. Let DA(Tn ) be the alternate double triangular snake obtained from a path Pn with consecutive vertices {u1 , u2 , . . . , un } by joining ui and ui+1 with two new vertices vj and wj alternatively, 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ b n2 c. In this case we define vertex labeling function f : V (DA(Tn )) → {1, 2, . . . , |V (DA(Tn ))|} as follows. Case 1: Triangle in DA(Tn ) starts from u1 . Note that |V (DA(Tn ))| = ( 2n, if n is even. 2n − 1, if n is odd. and |E(DA(Tn ))| = ( 3n − 1, if n is even. 3n − 3, if n is odd. f (ui ) = 2i − 1, 1 ≤ i ≤ n. f (vj ) = 4j − 2, n 1 ≤ j ≤ b c. 2 f (wj ) = 4j, In this case strongly∗ -labeling of DA(Tn ) is shown in Figure 4. Case 2:Triangle in DA(Tn ) starts from u2 . Subcase 1: n is odd. Defined as case-1. Subcase 2: n is even. Note that |V (DA(Tn ))| = 4n − 2 and |E(DA(Tn ))| = 3n − 5. f (u1 ) = 1. f (ui ) = 2(i − 1), 2 ≤ i ≤ n. f (vj ) = 4j − 1, n 1 ≤ j ≤ b c. 2 f (wj ) = 4j + 1, In this case strongly∗ -labeling of DA(Tn ) is shown in Figure 5. It can be easily observed that if vertex labels are in increasing order so, the produced edge labels are also increasing. So, they produced different edge labeling.
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Hence above defined labeling pattern satisfies the conditions of strongly ∗ -labeling. i.e. DA(Tn ) is strongly∗ -graph. Definition II.5. [3] A quadrilateral snake Qn is obtained from a path Pn by replacing each edge of Pn by a cycle C4 . Theorem II.4. Quadrilateral snake Qn is strongly∗ -graph. Proof. Let Pn be the path of {u1 , u2 , . . . , un } to construct Qn , join ui and ui+1 (alternatively) to two new vertices vi and wi by the edges ui vi , ui+1 wi and vi wi , 1 ≤ i ≤ n − 1. Here |V (Qn )| = 3n − 2 and |E(Qn )| = 4(n − 1). We define vertex labeling function f : V (Qn ) → {1, 2, . . . , |V (Qn )|} as follows. f (ui ) = 3i − 2, 1 ≤ i ≤ n. f (vi ) = 3i − 1, 1 ≤ i ≤ n − 1. f (wi ) = 3i, Strongly∗ -labeling of Qn is shown in Figure 6. Here every edge of Pn is an edge of cycle C4 having the vertices ui , vi , wi and ui+1 which are labeled in strictly increasing. So, it is obvious that for any pair of adjacent vertices produce different edge labeling for defined function. So, they satisfies the conditions of strongly ∗ graph. i.e. Qn is strongly∗ -graph. Definition II.6. [4] An alternate quadrilateral snake A(Qn ) is obtained from a path Pn , each alternate edge of Pn is replaced by a cycle C4 . Theorem II.5. Alternate quadrilateral snake A(Qn ) is strongly∗ -graph. Proof. Let A(Qn ) be the alternate quadrilateral snake obtained from a
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[Vol-3, Issue-9, Sep- 2016] ISSN: 2349-6495(P) — 2456-1908(O)
path Pn with consecutive vertices {u1 , u2 , . . . , un } by joining ui and ui+1 (alternatively) with new vertices vj and wj , 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ b n2 c. To define vertex labeling function f : V (A(Qn )) → {1, 2, . . . , |V (A(Qn ))|} we consider the following cases. Case 1: Quadrilateral in A(Qn ) starts from u1 . Note that |V (A(Qn ))| = ( 2n, if n is even. 2n − 1, if n is odd. and |E(A(Qn ))| = ( 5n − 1, if n is even. 2 5(n − 1), if n is odd. ( 2i, if i is even. f (ui ) = 2i − 1, if i is odd. f (vj ) = 4j − 2, n 1 ≤ j ≤ b c. 2 f (wj ) = 4j − 1, In this case strongly∗ -labeling of A(Qn ) is shown in Figure 7. Case 2:Quadrilateral in A(Qn ) starts from u2 . Subcase 1: n is odd. Defined as case-1. Subcase 2: n is even. Note that |V (A(Qn ))| = 2(n − 1) and |E(A(Qn ))| = 5n − 4. 2 f (ui ) = 2(i − 1), 1 ≤ i ≤ n. f (vj ) = 4j − 1, n 1 ≤ j ≤ b c. 2 f (wj ) = 4j, In this case strongly∗ -labeling of A(Qn ) is shown in Figure 8. In above cases every alternate edge of path is related to cycle C4 and defined labeling function f for A(Qn ) are strictly increasing. So, the induced edge labels by f ∗ (ui vj ) = f (ui ) + f (vj ) + f (ui ) ¡ f (vj ), f ∗ (wj vj ) = f (wj ) + f (vj ) + f (wj ) ¡ f (vj ), f ∗ (ui wj ) = f (ui ) + f (wj ) + f (ui ) ¡ f (wj ) and f ∗ (ui ui+1 ) = f (ui ) + f (ui+1 ) + f (ui ) ¡ Page | 242
International Journal of Advanced Engineering Research and Science (IJAERS)
f (ui+1 ) are increasing and distinct for all pair of adjacent vertices (ui , ui+1 ), (ui , vj ), (wj , vj ) and (wj , ui+1 ) in A(Qn ). Hence above defined labeling pattern satisfies the conditions of strongly∗ -labeling. i.e. A(Qn ) is strongly∗ -graph Definition II.7. [3] A double quadrilateral snake DQn is consist of two quadrilateral snakes that have common path. Theorem II.6. Double quadrilateral snake DQn is strongly∗ -graph. Proof. Let Pn be the path of {u1 , u2 , . . . , un } to construct DQn , join ui and ui+1 (alternatively) to four new vertices vi , wi , vi0 and wi0 by the edges ui vi , ui+1 wi , vi wi , ui vi0 , ui+1 wi0 and vi0 wi0 , 1 ≤ i ≤ n − 1. Here |V (DQn )| = 5n − 4 and |E(DQn )| = 7(n − 1). We define vertex labeling function f : V (DQn ) → {1, 2, . . . , |V (DQn )|} as follows. f (u1 ) = 1. f (ui ) = 5i − 6, 2 ≤ i ≤ n. f (vi ) = 5i − 3, f (wi ) = 5i − 2, 1 ≤ i ≤ n − 1. f (vi0 ) = 5i + 1, f (wi0 ) = 5i, Strongly∗ -labeling of DQn is shown in Figure 9. Here every edge of Pn is an edge of cycle C6 with one chord having the vertices ui , vi , wi , ui+1 , wi0 and vi0 which are labeled in strictly increasing. So, it is obvious that for any pair of adjacent vertices (ui , vi ), (vi , wi ), (ui , ui+1 ), (wi0 , vi0 ), (wi0 , ui+1 ) and (ui , vi0 ) produce different edge labeling for defined function. So, they satisfies the conditions of strongly ∗ -graph. i.e. DQn is strongly∗ -graph.
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[Vol-3, Issue-9, Sep- 2016] ISSN: 2349-6495(P) — 2456-1908(O)
Definition II.8. [4] An alternate double quadrilateral snake A(DQn ) is consist of two alternate double quadrilateral snakes that have common path. Theorem II.7. Alternate double quadrilateral snake DA(Qn ) is strongly∗ -graph. Proof. Let DA(Qn ) be the alternate double quadrilateral snake obtained from a path Pn with consecutive vertices {u1 , u2 , . . . , un } by joining ui and ui+1 (alternatively) with four new vertices vj , wj , vj0 and wj0 by the edges ui vj , ui+1 wj , vj wj , ui vj0 , ui+1 wj0 and vj0 wj0 , 1 ≤ i ≤ n and 1 ≤ j ≤ b n2 c. To define vertex labeling function f : V (DA(Qn )) → {1, 2, . . . , |V (DA(Qn ))|} we consider the following cases. Case 1: Quadrilateral in DA(Qn ) starts from u1 . Note that |V (DA(Qn ))| = ( 3n, if n is even. 3n − 2, if n is odd. and |E(DA(Qn ))| = ( 4n − 5, if n is even. 4(n − 1), if n is odd. f (ui ) = 3i − 2, 1 ≤ i ≤ n. f (vj ) = 6j − 4, f (wj ) = 6j − 3, n 1 ≤ j ≤ b c. 0 f (vj ) = 6j, 2 0 f (wj ) = 6j − 1, In this case strongly∗ -labeling of DA(Qn ) is shown in Figure 10. Case 2: Quadrilateral in DA(Qn ) starts from u2 . Subcase 1: n is odd. Defined as case-1. Subcase 2: n is even. Note that |V (DA(Qn ))| = 3n − 4 and |E(DA(Qn ))| = 4n − 7. f (u1 ) = 1. f (ui ) = 3i − 4, 2 ≤ i ≤ n. Page | 243
[Vol-3, Issue-9, Sep- 2016] ISSN: 2349-6495(P) — 2456-1908(O)
International Journal of Advanced Engineering Research and Science (IJAERS)
f (vj ) = 6j − 3, f (wj ) = 6j − 2, n 1 ≤ j ≤ b c. 0 f (vj ) = 6j + 1, 2 0 f (wj ) = 6j,
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International Journal of Advanced Engineering Research and Science (IJAERS)
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[Vol-3, Issue-9, Sep- 2016] ISSN: 2349-6495(P) — 2456-1908(O)
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IV. CONCLUSION It has been proved that all cycles and paths admit labeling specific manner produces specific snakes-type graphs. It is intrusting to see whether these combination admit strong ∗ labeling or not. REFERENCES [1] C. Adiga and D. Somnashekara, ”Strongly∗ -graphs”, Math. Forum, 13(1999), 31-36. [2] J. A. Gallian, ”A dynemic survey of graph labeling”, The Electronics Journal of Combinatorics, (2015), ]DS6 1 − 389. [3] R. Ponraj, S. Sathish Narayanan, R. Kala, ”Difference cordial labeling of graphs obtained from Double snakes”, International Journal of Mathematics Research, 5(2013), 317-322. [4] R. Ponraj, S. Sathish Narayanan, ”Difference cordiality of some graphs obtained from Double Alternate Snake Graphs”, Global Journal of Mathematical Sciences: Theory and Practical, 5(2013), 167-175.
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