International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol. 3, No. 1, February 2014 NEW FAMILIES OF ODD HARMONIOUS GRAPHS M. E. Abdel-Aal Department of Mathematics, Faculty of Science,Benha University, Benha 13518, Egypt ABSTRACT In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ 2, m ≡ 0 (mod 4) are odd harmonious graphs. Finally many new families of odd harmonious graphs are introduced. KEYWORDS Odd harmonious labeling, Eulernian graph, Cartesian product, Cyclic graphs. INTRODUCTION Graph labeling have often been motivated by practical problems is one of fascinating areas of research. A systematic study of various applications of graph labeling is carried out in Bloom and Golomb [1]. Labeled graph plays vital role to determine optimal circuit layouts for computers and for the representation of compressed data structure. Many of the results about graph labeling are collected and updated regularly in a survey by Gallian [2]. The reader can consult this survey for more information about the subject. We begin simple, finite, connected and undirected graph G = (V(G), E(G)) with p vertices and q edges. For all other standard terminology and notations we follow Harary [3]. Most graph labeling methods trace their origion to one introduced by Rosa [4] called such a labeling a β-valuation and Golomb [5] subsequently called graceful labeling, and one introduced by Graham and Sloane [6] called harmonious labeling. Several infinite families of graceful and harmonious graphs have been reported. Many illustrious works on graceful graphs brought a tide to different ways of labeling the elements of graph such as odd graceful. A graph G of size q is odd-graceful, if there is an injection f from V (G) to {0,1,2,…, 2q -1} such that, when each edge xy is assigned the label or weight f(x) - f(y) , the resulting edge labels are {1, 3, 5, . . . , 2q - 1}. This definition was introduced by Gnanajothi [7]. Many researchers have studied odd graceful labeling. Seoud and Abdel-Aal [8], [9] they give a survey of all connected graph of order ≤ 6 which are odd graceful, and they also introduce some families of odd graceful graphs. 1 International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol. 3, No. 1, February 2014 A graph G is said to be odd harmonious if there exists an injection f: V(G) → {0, 1, 2, …, 2q-1} such that the induced function f* : E(G) →{1, 3, . . . , 2q − 1} defined by f*(uv) = f(u) + f(v) is a bijection. Then f is said to be an odd harmonious labeling of G. Liang and Bai [10] introduced concept of odd harmonious labeling and they have obtained the necessary conditions for the existence of odd harmonious labeling of a graph. They proved that if G is an odd harmonious graph, then G is a bipartite graph. Also they claim that if a ( p, q) − graph G is odd harmonious, then 2 q ≤ p ≤ 2q − 1 , but this is not always correct. Take the path P2 as a counter example. In this paper, we show that the number of edges for any odd harmonious Eulernian graph must be congruent to 0 or 2 (mod 4), and we found a counter example to prove that, not necessary every Eulernian graph, with number of edges congruent to 0 or 2 (mod 4), to be an odd harmonious graph. This result corresponds to the result in case G is graceful and Eulerian, which had been stated and proved by rosa [4]. Also we show that many new families of graphs are odd harmonious. For instance, we obtained the odd harmonious labelings for joining two copies of even cycles with a common edge or with a common vertex and C n × Pm . Brief, new families of odd harmonious graphs are introduced. MAIN RESULTS Theorem 2.1 If G is an odd harmonious Eulerian graph with q edges, then q ≡ 0 or 2 (mod 4). Proof Let G be an odd harmonious Eulerian graph, and let f : V(G) → {0, 1, 2, …, 2q-1} be an odd harmonious labeling for G. Since G is an Eulerian graph then ( f (vi ) + f (v j )) = 2k , k is a ∑ constant. For each vi , vj ∈ V(G), implies that ∑ f (v ) + f (v ) = 2k , so 1 + 3 + 5 + ... + 2q − 1 = 2k , this ' i ' j q (1 + 2q − 1) = 2k ' . Hence q ≡ 0 or 2 (mod 4). 2 The inverse of the last result is not true. Remark 2.2 Not necessary every Eulernian graph with number of edges congruent to 0 or 2 (mod 4) to be an odd harmonious graph. Proof By counter example: let G = C6 where C6 is Eulerian graph with q = 6, q ≡ 2 (mod 4), while C6 is not odd harmonious follows from, every Cn is odd harmonious if and only if n ≡ 0 (mod 4), [10]. 2 International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol. 3, No. 1, February 2014 Theorem 2.3 Two copies of even cycle Cn sharing a common edge is an odd harmonious graph. Proof Let v1 , v 2 , ... , v n be the vertices of cycle Cn of even order. Consider two copies of cycle Cn. Let G denotes the graph of two copies with even cycle Cn sharing a common edge, clearly G has V (G ) = 2n − 2 and E (G ) = 2n − 1 . Without loss of generality let this edge be e = v n v 3n . 2 +1 2 We define the labeling function: f : V(G) → {0, 1, 2, 3, …, 4n -3} as follows, we consider two cases: Case (i): n ≡ 0 (mod 4), n + 1: 2 f (vi ) = (i − 1) , for 1 ≤ i ≤ n + 2 ≤ i ≤ n − 1 (we ignore this step when n = 4): 2 n n i + 1, (i even), i = 2 + 2, 2 + 4, ..., n − 2 f ( vi ) = i − 1, (i odd), i = n + 3, n + 5, ..., n − 1, 2 2 3n for n ≤ i ≤ : 2 f (vi ) = 3n − i − 1, , for and for 3n + 1 ≤ i ≤ 2n − 2 (we ignore this step when n = 4): 2 3n 3n 3n − i − 3, (i odd ), i = 2 + 1, 2 + 3, ..., 2 n − 1 f ( vi ) = 3n − i − 1, (i even ), i = 3n + 2, 3n + 4, ..., 2 n − 2 . 2 2 Case (ii): n ≡ 2 (mod 4), n + 1: 2 f (vi ) = (i − 1) for 1 ≤ i ≤ for , n + 2 ≤ i ≤ n + 1: 2 3 International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol. 3, No. 1, February 2014 i + 1, f ( vi ) = i − 1, (i odd), (i even), n n + 2, + 4, ..., n + 1 2 2 n n i = + 3, + 5, ..., n, 2 2 i= for n + 2 ≤ i ≤ 2n − 2 : 3n − i − 1, (i even), i = n + 2, n + 4, ..., 2n − 2 , 3n f (vi ) = 3n − i + 3, (i even), i = n + 3, n + 5, ..., , 2 3n 3n 3n − i + 1, (i odd), i = 2 + 2, 2 + 4, ..., 2n − 1. We observe that f is injective. The edge labels will be as follows: Case (i): n ≡ 0 (mod 4), n induce the edge labels: 2 n f * (vi vi +1 ) = f (vi ) + f (vi+1 ) = {2i − 1, 1 ≤ i ≤ } = {1,3,5, …, n - 1}. 2 • The vertices v1 and v 2 n − 2 , induce the edge labels: • The vertices vi and vi +1 , 1 ≤ i ≤ f * (v1 v2 n −2 ) = f (v1 ) + f (v2 n −2 ) = n + 1. • The vertices v n 2 +1 and v n , induce the edge labels: 2 +2 f * (v n v n ) = f (v n ) + f (v n ) = n + 3. 2 +1 2 +2 2 +1 2 +2 n + 2 ≤ i ≤ n − 2 induce the edge labels: 2 n f * (vi vi +1 ) = f (vi ) + f (vi +1 ) = {2i + 1, + 2 ≤ i ≤ n − 2} 2 • The vertices vi and vi +1 , • The vertices v n 2 +1 = {n+5, n+7,…, 2n - 3}. and v 3n , induce the edge labels: 2 * f (v n v 3n ) = f (v n ) + f (v 3n ) = n – 1. 2 +1 2 2 +1 • The vertices vi and vi +1 , 2 3n + 1 ≤ i ≤ 2n − 3 induce the edge labels: 2 4 International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol. 3, No. 1, February 2014 f * (vi vi +1 ) = f (vi ) + f (vi +1 ) = {6n − 2i − 5, 3n + 1 ≤ i ≤ 2n − 3} 2 ={3n-7, 3n-9,…, 2n +1}. • The vertices v 3n 2 and v 3n , induce the edge labels: +1 2 * f (v n v 3n ) = f (v n ) + f (v 3n ) = 3n – 5. 2 +1 2 2 +1 2 • The vertices vn −1 and vn , induce the edge labels: f * (vn−1 vn ) = f (vn−1 ) + f (vn ) = 3n – 3. 3n • The vertices vi and vi +1 , n ≤ i ≤ − 1 induce the edge labels: 2 3n f * (vi vi +1 ) = f (vi ) + f (vi +1 ) = {6n − 2i − 3, n ≤ i ≤ − 1} 2 ={4n-3, 4n-5,…, 3n - 1}. Case (ii): n ≡ 2 (mod 4), n induce the edge labels: 2 n f * (vi vi +1 ) = f (vi ) + f (vi +1 ) = {2i − 1, 1 ≤ i ≤ } = {1,3,5, …, n - 1}. 2 • The vertices v1 and v2 n − 2 , induce the edge labels: • The vertices vi and vi +1 , 1 ≤ i ≤ f * (v1 v2 n− 2 ) = f (v1 ) + f (v 2 n− 2 ) = n + 1. • The vertices v n 2 +1 and v n , induce the edge labels: 2 +2 f * (v n v n ) = f (v n ) + f (v n ) = n + 3. 2 +1 2 +2 2 +1 2 +2 n + 2 ≤ i ≤ n induce the edge labels: 2 n f * (vi vi +1 ) = f (vi ) + f (vi +1 ) = {2i + 1, + 2 ≤ i ≤ n} 2 • The vertices vi and vi +1 , • The vertices v n 2 +1 = {n+5, n+7,…, 2n +1}. and v 3n , induce the edge labels: 2 * f (v n v 3n ) = f (v n ) + f (v 3n ) = 2n + 3. 2 +1 2 2 +1 2 3n + 1 ≤ i ≤ 2n − 3 induce the edge labels: 2 3n f * (vi vi +1 ) = f (vi ) + f (vi +1 ) = {6n − 2i − 1, + 1 ≤ i ≤ 2n − 3} 2 • The vertices vi and vi +1 , ={3n-3, 3n-5,…, 2n +5}. • The vertices vn +1 and vn + 2 , induce the edge labels: f * (vn+1 vn ) = f (vn+1 ) + f (vn ) = 3n-1. 5 International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol. 3, No. 1, February 2014 • The vertices vi and vi +1 , n + 2 ≤ i ≤ 3n induce the edge labels: 2 f * (vi vi +1 ) = f (vi ) + f (vi +1 ) = {6n − 2i + 1, n + 2 ≤ i ≤ 3n } 2 ={4n-3, 4n-5,…, 3n +1}. Now, we obtained all the edge labels {1, 3, 5,…, 4n-3} in each case, so f * is injective as required. Hence G admits odd harmonious labeling. Example 2.4. An odd harmonious labeling of two copies of cycles C12 sharing a common edge, and an odd harmonious labeling of two copies of cycles C10 sharing a common edge are shown in figure(1) and figure (2) respectively: Figure (1): Two copies of cycle C12 sharing a common edge with its odd harmonious labeling. Figure (2): Two copies of cycle C10 sharing a common edge with its odd harmonious labeling. In the following theorems we mention only the vertices labels, the reader can fulfill the proof as we did in the previous theorems. Theorem 2.5 Two copies of even cycle Cn sharing a common vertex is an odd harmonious graph when n ≡ 0 (mod 4). Proof Let v1 , v 2 , ... , v n be the vertices of cycle Cn , n ≡ 0 (mod 4). Consider G be the graph of two copies of Cn sharing a common vertex with V (G ) = 2n − 1 and E (G ) = 2n . Without loss of generality let this vertex be v1 . Let G be described as indicated in Figure (3). v3 v2 v2 n−1 vn + 4 v1 v4 v5 v n +3 vn +1 vn vn + 2 Figure (3) 6 International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol. 3, No. 1, February 2014 We define the labeling function f : V(G) → {0, 1, 2, 3, …, 4n -1} as follows: n + 1: 2 f (vi ) = (i − 1) , for 1 ≤ i ≤ for n + 2 ≤ i ≤ n: 2 i + 1, f (v i ) = i − 1, for i = n + 1 : n n + 2 , + 4, ..., n 2 2 n 3n i = + 3, + 5, ..., n − 1, 2 2 (i even ), i = (i odd), f (v n +1 ) = 2n + 3, for n + 2 ≤ i ≤ 2n − 1 : i − 2 , f (v i ) = i, i + 2 , when when when i = n + 2 , n + 4, ..., 3n , 2 3n 3n + 2, + 4, ..., 2 n − 1, 2 2 i = n + 3, n + 5, ..., 2 n − 1 . i= Above defined labeling pattern exhausts all possibilities and in each case the graph under consideration admits odd harmonious labeling. Example 2.6. An odd harmonious labeling of two copies of C12 sharing a common vertex is shown in figure(4). Figure (4): Two copies of cycle C12 sharing a common vertex with its odd harmonious labeling. Remark 2.7 In theorems 2.3, 2.5 when n = 4, these theorems are coincided with corollary 3.12 (2) in [10] when i = 2,1 respectively. In 1980 Graham and Sloane [6], proved that Cm × Pn is harmonious when n is odd. We generalized this result for odd harmonious labeling in the following theorem. 7 International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol. 3, No. 1, February 2014 Theorem 2.8 The graphs C 4 m × Pn , for each m ≥ 1, n ≥ 2 are odd harmonious. Proof Let G = C 4 m × Pn be described as indicated in Figure (5): u 14 m u 12 u 22 u 11 u 12 u n4 m u n4 −m1− 1 u34 m −1 u24 m −1 u14 m −1 u n4 −m1 u 34 m u 24 m u 32 u 31 u n2 −1 u 1n −1 u n4 m − 1 u n2 u 1n Figure (5) The number of edges of the graph G is 4m(2n-1). We define the labeling function: f : V(G) → {0, 1, 2, 3, …, 8m(2n-1)-1} as follows: 8m(i − 1), f (u i1 ) = 8mi − 4m + 1 i = 1, 3, 5, ..., n or n − 1, i = 2, 4, 6, ..., n − 1 or n. We consider the following three cases: 4m , i = 1, 3, 5, ... , n or n − 1 2 4m j = 2, 4, 6, ..., 8mi − 4m − j + 3, , 2 f (u ij ) = 4m 8mi − 4m − j + 1, j = 3, 5, 7, ..., − 1. 2 Case(i): 1 < j ≤ 4m , i = 2, 4, 6, ... , n or n − 1 2 4m 8mi − 4m − j + 2, j = 3, 5, ..., − 1, 2 f (u ij ) = 4m 8mi − 4m − j , . j = 2, 4, 6, ..., 2 Case(ii): 1 < j ≤ Case(iii): 4m + 1 ≤ j ≤ 4 m, 1 ≤ i ≤ n 2 8mi − 4m − j + 1, f (u ij ) = 8mi − 4m − j i = 1, 3, 5, ..., n or n − 1, i = 2, 4, 6, ..., n − 1 or n. 8 International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol. 3, No. 1, February 2014 Above defined labeling pattern exhausts all possibilities and in each case the graph under onsideration admits odd harmonious labeling. Example 2.9. An odd harmonious labeling of the graph C 4 × P5 is shown in Figure (6). Figure (6): The graph C 4 × P5 with its odd harmonious labeling. Let G1 and G2 be two disjoint graphs. The corona ( G1Θ G2 ) of G1 and G2 is the graph obtained by taking one copy of G1 (which has q1 edges) and q1 copies of G2, and then sharing common edge between the ith edge of G1 and one edge in the ith copy of G2. Theorem 2.10 The graphs C 4 m Θ C 4 for each m ≥ 1 are odd harmonious. Proof Let G = C 4 m Θ C 4 be described as indicated in Figure (7) u 24 m u14 m u 24 m−1 u14 m−1 v 4m 1 1 u u 1 2 v1 v2 v4 m v4 m−1 v3 v4 u12 u 22 u13 u 23 2 v 4m 2 v 4m 2 4m 2 1 u +2 4m u22 4m +1 u1 2 u +1 +1 4m 2 2 Figure (7) It is clear that the number of edges of the graph C 4 m Θ C 4 is 16m. We define the labeling function f : V(G) → {0, 1, 2, …, 32m - 1} as follows: 4 ( i − 1), f ( v i ) = 4 i − 3, 4 i + 1, ( i odd ) 1 ≤ i ≤ 4m − 1 4m ( i even ) 2 ≤ i ≤ , 2 4m ( i even ) + 2 ≤ i ≤ 4m , 2 Now, for labeling the remaining vertices u i1 , u i2 , 1 ≤ i ≤ 4m we consider the following cases: 9 International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol. 3, No. 1, February 2014 Case(i): when 1 ≤ i ≤ 4m 2 4 i − 3, f (u ) = 4i − 4, 4m − 1, 2 4m i = 2 , 4 , 6 , ..., , 2 4i − 2, f (u ) = 4 i − 1, 4m − 1, 2 4m i = 2 , 4 , 6 , ..., , 2 1 i 2 i Case(ii): when i = 1, 3 , 5 , ..., 4m + 1 ≤ i ≤ 4m 2 4 i + 1, f (u ) = 4i − 4, 1 i 4 i − 2 , f (u ) = 4 i + 3, 2 i i = 1 , 3 , 5 , ..., 4m 4m + 1, + 3 , ..., 4 m − 1, 2 2 4m 4m i= + 2, + 4 , ..., 4 m , 2 2 i= 4m 4m + 1, + 3 , ..., 4 m − 1, 2 2 4m 4m i= + 2, + 4 , ..., 4 m , 2 2 i= It follows that f is an odd harmonious labeling for C 4 m Θ C 4 . Hence C 4 m Θ C 4 is an odd harmonious graph. Example 2.11. An odd harmonious labeling of the C8 Θ C 4 , is shown in Figure (8). Figure (8): The graph C 8 Θ C 4 with its odd harmonious labeling. Let G1 and G2 be two disjoint graphs. The corona (G1 ʘ G2) of G1 and G2 is the graph obtained by taking one copy of G1 (which has p1 vertices ) and p1 copies of G2 , and then joining the ith vertex of G1 to every vertex in the ith copy of G2. 10 International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol. 3, No. 1, February 2014 Theorem 2.12 The graphs Sn ʘ K m , for n, m ≥ 1 are odd harmonious graph. Proof Let Sn ʘ K m , be described as indicated in Figure(9): u0 u1 u11 u12 u2 u1m u 12 un u 22 u 2m u 1n u n2 u nm Figure(9) It is clear that the number of edges of the graph Sn ʘ K m , is q = n(m+1). We define the labeling function f : V(G) → {0, 1, 2, …, 2n(m+1) - 1} as follows: f (u 0 ) = 0, f (u i ) = 2i − 1, 1≤ i ≤ n, j f (u i ) = 2q − (2m + 2)(i − 1) − 2 j , 1 ≤ i ≤ n, 1 ≤ j ≤ m, Above defined labeling pattern exhausts all possibilities and the graph under consideration admits odd harmonious labeling. Example 2.13. An odd harmonious labeling of the graph S3 ʘ K 3 , is shown in Figure (10). Figure (10): The graph S3 ʘ K 3 with its odd harmonious labeling. The graph K2,n (r, s) obtained from K2,n ,( n ≥ 2) by adding r and s (r, s ≥ 1) pendent edges out from the two vertices of degree n. Theorem 2.14 The graphs K2,n (r, s) are odd harmonious for all n, r, s ≥ 1. Proof Let K2,n (r, s) be described as indicated in Figure(11) 11 International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol. 3, No. 1, February 2014 w1 w2 w3 w r −1 wr v0 v1 v2 v3 vn u0 u1 u2 u3 u s −1 us Figure(11) The number of edges of the graph K2,n (r, s) is 2n + r + s. We define the labeling function : f : V(K2,n (r, s)) → {0, 1, 2, …2(2n + r + s) - 1} as follows f( wi ) = 2 i – 1 , 1≤i≤r f( v0 ) = 0 f( vi ) = 2( i + r ) - 1 , 1≤i≤n f ( u0 ) = 2n f ( ui ) = 2 ( r + n + i ) - 1 , 1≤i≤s The edge labels will be as follows: • The vertices v0 and wi , 1 ≤ i ≤ r , induce the edge labels {1, 3, 5, …, 2r - 1}. • The vertices v0 and vi , 1 ≤ i ≤ n , induce the edge labels {2r +1, 2n + 2r + 3, …, 2(r +n)–1}. • The vertices u0 and vi , 1 ≤ i ≤ n, induce the edge labels { 2(r + n) +1, 2(r + n) +3, …, 2(2n+r) - 1}. • The vertices u0 and ui, 1 ≤ i ≤ s, induce the edge labels {2(2n+r) + 1, 2(2n+r) + 3,… , 2(2n+r+s) - 1}. So we obtain the edge labels {1, 3, 5, …, 2(2n + r + s) - 1}. Hence the graph G is odd harmonious. Example 2.15. An odd harmonious labeling of the K2,5 (3, 4), is shown in Figure (12). Figure (12): The graph K2,5 (3, 4) with its odd harmonious labeling. CONCLUSION Since labelled graphs serve as practically useful models for wide-ranging applications such as communications network, circuit design, coding theory, radar, astronomy, X-ray and crystallography, it is desired to have generalized results or results for a whole class, if possible. This work has presented several families of odd harmonious graphs. To investigate similar results for other graph families and in the context of different labeling techniques is open area of research. 12 International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol. 3, No. 1, February 2014 REFERENCES [1] G. S. Bloom and S. W. Golomb,(1977) “ Applications of numbered undirected graphs”, Proc. IEEE, 65 , pp 562-570. [2] J. A. Gallian, (2011) A Dynamic Survey of Graph Labeling, Electronic J. Combin. Fourteenth edition. [3] F. Harary. (1969) GpaphTheory, Addison-Wesley, Reading MA. [4] A. Rosa. On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome, July 1966), Gordon and Breach, N. Y. and Dunod Paris (1967) 349-355. [5] S. W. Golomb, (1972) “How to number a graph, in Graph Theory and Computing”, R. C. Read, ed., Academic Press, New York, pp 23-37. [6] R. L. Graham, N. J. A. Sloane. On additive bases and harmonious graphs, SIAM J. Algebr. Disc. Math., Vol 1, No 4 (1980), pp 382-404. [7] R.B. Gnanajothi. (1991) Topics in graph theory, Ph.D. thesis, Madurai Kamaraj University, India. [8] M.A. Seoud and M.E. Abdel-Aal, (2013) “On odd graceful graphs”, Ars Combin., 108, pp 161-185. [9] M.E. Abdel-Aal. New Classes of Odd Graceful Graphs, International journal on applications of graph theory in wireless ad hoc networks and sensor networks (GRAPH-HOC) Vol.5, No.2, June 2013. [10] Z. Liang, Z . Bai, (2009) “ On the odd harmonious graphs with applications”, J. Appl. Math. Comput., 29, 105-116. Author Mohamed Elsayed Abdel-Aal received the B.Sc. (Mathematics) the M.Sc. (Pure Mathematics- Abstract Algebra) degree from Benha University, Benha, Egypt in 1999, 2005 respectively. Also, he received Ph.D. (Pure Mathematics) degree from Faculty of Mathematics, Tajik National University, Tajikistan, in 2011. He is a University lecturer of Pure Mathematics with the Benha University, Faculty of Science, Department of Pure Mathematics. His current research is Ordinary – partial differential equations Graph Theory and Abstract Algebra. 13

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