Tetrahedral and Pentatopic Sum Labeling of Graphs

International Journal of Applied Information Systems (IJAIS) ISSN : 2249-0868 Foundation of Computer Science FCS, New York, USA Volume 5 - No. 6, April 2013 - www.ijais.org

S. Murugesan D. Jayaraman J. Shiama Department of Mathematics Department of Mathematics Department of Mathematics, C. B. M. College, Kovaipudur, C. B. M. College, Kovaipudur, C. M. S. College of Science and Commerce, Coimbatore - 641 042, Coimbatore - 641 042, Tamil Nadu, INDIA. Coimbatore- 641 049,Tamil Nadu, INDIA. Tamil Nadu, INDIA.

ABSTRACT DEFINITION 4. A tetrahedral number(or triangular pyrami- dal number) is a figurative number that represents a pyramid A (p, q) graph G is said to admit tetrahedral or pentatopic sum la- with a triangular base and three sides, called a tetrahedron. beling if its vertices can be labeled by non negative integers such The nth tetrahedral number is the sum of the first n triangu- that the induced edge labels obtained by the sum of the labels of end th lar numbers. If the n tetrahedral number is denoted by Bn, vertices are the first q tetrahedral or pentatope numbers. A graph G 1 then Bn = 6 n(n + 1)(n + 2). The tetrahedral numbers are which admits tetrahedral or pentatopic sum labeling is called tetra- 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, hedral or pentatopic sum graph. In this paper we prove that paths, 816, 969,... combs, stars, subdivision of stars and Bm,n admit tetrahedral and pentatopic sum labeling. DEFINITION 5. A tetrahedral sum labeling of a graph G is a one-to-one function f : V (G) → N that induces a bijection Keywords: + f : E(G) → {B1,B2, ..., Bq} of the edges of G defined by Tetrahedral sum labeling, pentatopic sum labeling. f +(uv) = f(u) + f(v), ∀e = uv ∈ E(G). The graph which admits such labeling is called a tetrahedral sum graph. 1. INTRODUCTION EXAMPLE 1. A tetrahedral sum graph with 8 vertices is The graphs considered here are finite, connected, undirected and shown below. simple. The vertex set and edge set of a graph G are denoted by V (G) and E(G) respectively. For various graph theoretic no- tations and terminology we follow Harary [1] and for number theory we follow Burton [2]. We will give the brief summary of definitions which are useful for the present investigations. 1 4 DEFINITION 1. If the vertices of the graph are assigned val- r ues subject to certain conditions it is known as graph labeling. 1 4 r 10 10 A dynamic survey on graph labeling is regularly updated by Gal- r lian [3] and it is published by Electronic Journal of Combina- 20 020 35 torics. Vast amount of literature is available on different types of 35 graph labeling and more than 1000 research papers have been r r r published so far in last four decades. 8456 Most important labeling problems have three important ingredi- ents. * A set of numbers from which vertex labels are chosen; * A rule that assigns a value to each edge; 36120 84 * A condition that these values must satisfy. r r The present work is to aimed to discuss two such labelings known as tetrahedral sum labeling and pentatopic sum labeling.

DEFINITION 2. A triangular number is a number obtained DEFINITION 6. A pentatope number is a figurative num- 1 by adding all positive integers less than or equal to a given pos- ber given by P topn = 4 Bn(n + 3) where Bn is th th th itive integer n. If the n triangular number is denoted by An, the n tetrahedral number. The n pentatope number 1 then An = n(n + 1). is the sum of the first n tetrahedral numbers. If the 2 th n pentatope number is denoted by Cn, then Cn = The triangular numbers are 1,3,6,10,15,21,28,36,45,55,... 1 24 n(n + 1)(n + 2)(n + 3). The pentatopic numbers are 1,5,15,35,70,126,210,330,495,715,1001,1365,1820,2380,3060, DEFINITION 3. A triangular sum labeling of a graph G is a 3876,4845,... one-to-one function f : V (G) → N (where N is the set of all + non-negative integers) that induces a bijection f : E(G) → DEFINITION 7. A pentatopic sum labeling of a graph G is + {A1,A2, ..., Aq} of the edges of G defined by f (uv) = f(u)+ a one-to-one function f : V (G) → N that induces a bijection + f(v), ∀e = uv ∈ E(G). The graph which admits such labeling f : E(G) → {C1,C2, ..., Cq} of the edges of G defined by is called a triangular sum graph. f +(uv) = f(u) + f(v), ∀e = uv ∈ E(G). The graph which admits such labeling is called a pentatopic sum graph. This concept was introduced by Hegde and Shankaran [4]. Mo- tivated by the concept of triangular sum labeling, we introduce two new types of labeling called tetrahedral sum labeling and EXAMPLE 2. A pentatopic sum graph with 13 vertices is pentatopic sum labeling. shown below.

31 International Journal of Applied Information Systems (IJAIS) ISSN : 2249-0868 Foundation of Computer Science FCS, New York, USA Volume 5 - No. 6, April 2013 - www.ijais.org

Case(ii): i is even For 1 ≤ i ≤ n − 1, 35 126 1 1 f(u ) + f(u ) = i(i + 2)(2i − 1) + i(i + 2)(2i + 5) 15 210 i i+1 24 24 1 r r = i(i + 1)(i + 2) 535 495 r15 126 r 6 15 210 1001 = Bi r r + 1 495 = f (vi) r r 1001 Thus the induced edge labels are the ﬁrst n − 1 tetrahedral num- 0 bers. 70r 1365 Hence path Pn admits tetrahedral sum labeling.

70 1365 EXAMPLE 3. The tetrahedral sum labeling of P11 is shown below. r r 330 1820 0 1 3 7 13 22 34 50 70 95 125 260715 455 1 4 10 20 35 56 84 120 165 220 r r rrrrrrrrrrr

THEOREM 2. The comb Pn K1 admits tetrahedral sum 2. MAIN RESULTS labeling.

Here we prove that paths, combs, stars, double stars and bistars PROOF. Let Pn : u1u2...un be the path and let vi = admit tetradedral and pentatopic sum labeling. uiui+1(1 ≤i≤n-1) be the edges. Let w1, w2, ..., wn be the pendant vertices adjacent to u1, u2, ..., un respectively and ti = uiwi(1 ≤i≤n-1) be the edges. THEOREM 1. The path Pn admits tetrahedral sum labeling. For i=1,2,...,n, deﬁne

( PROOF. Let Pn : u1u2...un be the path and let vi = 1 (i − 1)(i + 1)(2i + 3) if i is odd u u (1 ≤i≤n-1) be the edges. f(u ) = 24 i i+1 i 1 i(i + 2)(2i − 1) if i is even For i=1,2,...,n, deﬁne 24 and  h   1 2i3 + (12n − 3)i2 1  24  (i − 1)(i + 1)(2i + 3) if i is odd  i  24 +(12n2 − 2)i + (4n3 − 4n + 3) , if i is odd f(ui) = f(wi) = h 1 2i3 + (12n − 3)i2  1  24  i(i + 2)(2i − 1) if i is even  24  2 2 i +(12n − 2)i + 4n(n − 1) , if i is even.

+ We will prove that the induced edge labels obtained by the Then f(ui) + f(ui+1) = f (vi) for 1 ≤ i ≤ n − 1 sum of the labels of end vertices are the ﬁrst n − 1 tetrahedral + numbers. and f(ui) + f(wi) = f (ti) for 1 ≤ i ≤ n.

Case(i): i is odd Thus the induced edge labels are the ﬁrst 2n−1 tetrahedral num- For 1 ≤ i ≤ n − 1, bers. Hence comb admits tetrahedral sum labeling.

1 EXAMPLE 4. The tetrahedral sum labeling of P K is f(u ) + f(u ) = (i − 1)(i + 1)(2i + 3) 6 1 i i+1 24 shown below. 1 + (i + 1)(i + 3)(2i + 1) 24 1 h = (i + 1) (i − 1)(2i + 3) 0 1 3 7 13 22 24 i 1 4 10 20 35 + (i + 3)(2i + 1) r r r r r r 1 56 84 120 165 220 286 = i(i + 1)(i + 2) 6 = Bi = f +(v 56 83 117 158 207 264 i) r r r r r r

32 International Journal of Applied Information Systems (IJAIS) ISSN : 2249-0868 Foundation of Computer Science FCS, New York, USA Volume 5 - No. 6, April 2013 - www.ijais.org

THEOREM 3. The star graph K1,n admits tetrahedral sum labeling. 83

84 r PROOF. Let v be the apex vertex and let v1, v2, ..., vn be the pendant vertices of the star K1,n. Deﬁne 1 120 308 364 116 r 1 r r 4 f(v) = 0 56 56 4 1 rr andf(v ) = i(i + 1)(i + 2), 1 ≤ i ≤ n. i 6 35 0 10 r 286 20 35 10 165 rr 20 We see that the induced edge labels are the ﬁrst n tetrahedral 251 155 numbers. r r r 220 Hence K1,n admits tetrahedral sum labeling.

200 EXAMPLE 5. The tetrahedral sum labeling of K1,8 is shown r below.

THEOREM 5. The bistar Bm,n admits tetrahedral sum labeling.

1 PROOF. Let V (Bm,n) = {u, v, ui, vj : 1 ≤i≤m, 1 ≤j≤n} and 4 E(B ) = {uv, uu , vv : 1 ≤i≤m, 1 ≤j≤n}. 120 r m,n i j @ Deﬁne f by 1 r @120 r @ 4 f(u) = 0,

84 @ f(v) = 1, 84 0 10 @ 10 1 f(u ) = (i + 1)(i + 2)(i + 3), 1 ≤ i ≤ m 56 rr@ r i 6 @ 20 1 35 @ andf(vj ) = (m + j + 1)(m + j + 2)(m + j + 3) − 1, 1 ≤ j ≤ n. @ 6 56 20 r r We see that the induced edge labels are the ﬁrst m + n + 1 35 tetrahedral numbers. r Hence Bm,n admits tetrahedral sum labeling.

EXAMPLE 7. The tetrahedral sum labeling of B6,4 is shown below.

THEOREM 4. S(K1,n), the subdivision of the star K1,n, admits tetrahedral sum labeling.

4 PROOF. Let V (S(K1,n)) = {v, vi, ui : 1 ≤ i ≤ n} and E(S(K )) = {vv , v u : 1 ≤ i ≤ n}. 119 1,n i i i 10 4 Deﬁne f by r 10 120 r r 164 20 20 165 1 f(v) = 0, r r 35 0 1 220 1 35 56 r r 219 f(vi) = i(i + 1)(i + 2), 1 ≤ i ≤ n 286 6 r 84 r 1 3 2 56 andf(ui) = [n + 3n + 2n + 3ni(i + n + 2)], 1 ≤ i ≤ n. 285 6 r 84 r r We see that the induced edge labels are the ﬁrst 2n tetrahedral numbers. Hence S(K1,n) admits tetrahedral sum labeling. THEOREM 6. The path Pn admits pentatopic sum labeling.

PROOF. Let Pn : u1u2...un be the path and let vi = EXAMPLE 6. The tetrahedral sum labeling of S(K1,6) is uiui+1(1 ≤i≤n-1) be the edges. shown below. For i=1,2,...,n, deﬁne

33 International Journal of Applied Information Systems (IJAIS) ISSN : 2249-0868 Foundation of Computer Science FCS, New York, USA Volume 5 - No. 6, April 2013 - www.ijais.org

For i=1,2,...,n, deﬁne ( 1 3 2 48 (i − 1)(i + 5i + 7i + 3) if i is odd f(ui) = 1 3 2 48 i(i + 4i + 2i − 4) if i is even ( 1 3 2 48 (i − 1)(i + 5i + 7i + 3) if i is odd We will prove that the induced edge labels obtained by the sum f(ui) = 1 i(i3 + 4i2 + 2i − 4) if i is even of the labels of end vertices are the ﬁrst n−1 pentatope numbers. 48

Case(i): i is odd For 1 ≤ i ≤ n − 1, and

1 3 2 f(u ) + f(u ) = (i − 1)(i + 5i + 7i + 3)  1 4 3 2 2 i i+1 48 [i + 8ni + (12n + 12n − 4)i  48 1 +(8n3 + 12n2 − 4n)i + (i + 1)[(i + 1)3 + 4(i + 1)2  48 +(2n4 + 4n3 − 2n2 − 4n + 3)] if i is odd f(w ) = + 2(i + 1) − 4] i 1 4 3 2 2  48 [i + 8ni + (12n + 12n − 4)i  3 2 4 1 3 2 +(8n + 12n − 4n)i + (2n = (i − 1)(i + 5i + 7i + 3)  48 +4n3 − 2n2 − 4n)] if i is even. 1 + (i + 1)(i3 + 7i2 + 13i + 3) 48 + 1 Then f(ui) + f(ui+1) = f (vi) for 1 ≤ i ≤ n − 1 = i(i3 + 6i2 + 11i + 6) 24 + 1 and f(ui) + f(wi) = f (ti) for 1 ≤ i ≤ n. = i(i + 1)(i + 2)(i + 3) 24 Thus the induced edge labels are the first 2n − 1 pentatope num- = Ci bers. + = f (vi) Hence comb admits pentatopic sum labeling. Case(ii): i is even For 1 ≤ i ≤ n − 1, EXAMPLE 9. The pentatopic sum labeling of P5 K1 is 1 shown below. f(u ) + f(u ) = i(i3 + 4i2 + 2i − 4) i i+1 48 1 + i[(i + 1)3 + 5(i + 1)2 48 + 7(i + 1) + 3] 0 1 4 11 24 1 = i(i3 + 4i2 + 2i − 4) 1 5 15 35 48 r r r r r 1 + i(i3 + 8i2 + 20i + 16) 70 126 210 330 495 48 1 = i(i3 + 6i2 + 11i + 6) 24 1 70 125 206 319 471 = i(i + 1)(i + 2)(i + 3) r r r r r 24 = Ci + = f vi THEOREM 8. The star graph K1,n admits pentatopic sum Thus the induced edge labels are the first n − 1 pentatope num- labeling. bers. Hence path Pn admits pentatopic sum labeling. PROOF. Let v be the apex vertex and let v1, v2, ..., vn be the EXAMPLE 8. The pentatopic sum labeling of P is shown 9 pendant vertices of the star K1,n. below. Define

0 1 4 11 24 46 80 130 200 f(v) = 0 1 5 15 35 70 126 210 330 1 andf(v ) = i(i + 1)(i + 2)(i + 3), 1 ≤ i ≤ n. r r r r r r r r r i 24

We see that the induced edge labels are the ﬁrst n pentatope num- THEOREM 7. The comb Pn K1 admits pentatopic sum labeling. bers. Hence K1,n admits pentatopic sum labeling. PROOF. Let Pn : u1u2...un be the path and let vi = uiui+1(1 ≤i≤n-1) be the edges. Let w1, w2, ..., wn be the pendant vertices adjacent to u1, u2, ..., un respectively and ti = EXAMPLE 10. The pentatopic sum labeling of K1,7 is uiwi(1 ≤i≤n-1) be the edges. shown below.

34 International Journal of Applied Information Systems (IJAIS) ISSN : 2249-0868 Foundation of Computer Science FCS, New York, USA Volume 5 - No. 6, April 2013 - www.ijais.org

PROOF. Let V (Bm,n) = {u, v, ui, vj : 1 ≤i≤m, 1 ≤j≤n} and 1 E(Bm,n) = {uv, uui, vvj : 1 ≤i≤m, 1 ≤j≤n}. Deﬁne f by r 5210 f(u) = 0, 1 210 f(v) = 1, r 5 r 0 1 f(ui) = (i + 1)(i + 2)(i + 3)(i + 4), 1 ≤ i ≤ m 126 24 15 1 r 15126 andf(vj ) = (m + j + 1)(m + j + 2)(m + j + 3) 70 6 r 35 r (×)(m + j + 4) − 1, 1 ≤ j ≤ n. We see that the induced edge labels are the ﬁrst m + n + 1 3570 pentatope numbers. rr Hence Bm,n admits pentatopic sum labeling.

EXAMPLE 12. The pentatopic sum labeling of B4,3 is THEOREM 9. S(K1,n), the subdivision of the star K1,n, ad- shown below. mits pentatopic sum labeling.

PROOF. Let V (S(K1,n)) = {v, vi, ui : 1 ≤ i ≤ n} and 5 E(S(K1,n)) = {vvi, viui : 1 ≤ i ≤ n}. 125 Deﬁne f by 5 r 126 15 f(v) = 0, 15 r 1 1 210 f(v ) = i(i + 1)(i + 2)(i + 3), 1 ≤ i ≤ n r 209 i 24 35 0 1 35 330 and r r r 70 1 h 4 3 2 r f(ui) = n + 6n + 11n + 6n 329 24 70 i r + 2ni(2n2 + 3ni + 2i2 + 9n + 9i + 11) , 1 ≤ i ≤ n. r

We see that the induced edge labels are the ﬁrst 2n pentatope numbers. Hence S(K1,n) admits pentatopic sum labeling. 3. CONCLUDING REMARKS EXAMPLE 11. The pentatopic sum labeling of S(K1,5) is shown below. As product of three consecutive integers is divisible by 3! and product of four consecutive integers is divisible by 4! we have introduced tetrahedral sum labeling and pentatopic sum labeling 125 of graphs. These labelings can be extended by using the fact that product of k consecutive integers is divisible by k!. Also analo- r gous results for different graphs can be investigated. 126 1 4. ACKNOWLEDGEMENT 645 The authors would like to thank Dr.M.Mallikaarjunan, Assistant 1 r 210 70 205 Professor, Department of Mathematics, C.B.M.College, Coim- 715 5 r 5 r batore, India, for helping in the proof reading of this article. 70 r 0 r 5. REFERENCES 35 r 15 [1] F. Harary, Graph Theory, Addition-Wesley, Reading, 35 15 Mass, 1972. 495r r 330 [2] David M. Burton, Elementary Number Theory, Second Edition,Wm. C. Brown Company Publishers, 1980. 460 315 r r [3] J. A. Gallian, A dynamic survey of graph labeling, Elec- tronic Journal of Combinatorics, 17 (2010), DS6. [4] S.M.Hegde and P.Shankaran , On triangular sum labeling of graphs, in Labeling of Discrete Structures and Appli- cations, Narosa Publishing House, New Delhi, 2008, 109- THEOREM 10. The bistar Bm,n admits pentatopic sum la- 115. beling.