By Dileepkumar R

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Книга arithmetic for the foreign scholar: arithmetic HL - overseas. .. arithmetic for the foreign pupil: arithmetic HL - overseas Baccalaureate degree Programme Книги Математика Автор: Paul city, John Owen, Robert Haese Год издания: 2004 Формат: pdf Издат. :Haese & Harris courses Страниц: 960 Размер: 9,6 ISBN: 1876543094 Язык: Английский0 (голосов: zero) Оценка:Mathematics for the foreign scholar: arithmetic HL (International Baccalaureate degree Programme) is our interpretation of the strategies defined within the IBO degree Programme arithmetic HL consultant (first examinations 2006).

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Since vi+t vj−t is an edge in the path P(T) it follows that i+t+1=j-t ⇒ j = i + 2t + 1. The value of the edge vi vj is f + (vi vj ) = f + (vi vi+2t+1 ) = f (vi ) + f (vi+2t+1 ) If i is odd and 1 ≤ i ≤ n, then d + k + (q − 1)d + f (vi ) + f (vi+2t+1 ) = (i−2) 2 (i+2t+1−2) 2 (5) d = k + (q − 1)d + (i + t − 1)d. If i is even and 2 ≤ i ≤ n, then f (vi ) + f (vi+2t+1 ) = k + (q − 1)d + (i−2) 2 d+ (i+2t+1−1) 2 = k + (q − 1)d + (i + t − 1)d 35 (6) d (7) Therefore, from (5), (6), (7), we get f + (vi vj ) = k + (q − 1)d + (i + t − 1)d ∀i, 1 ≤ i ≤ n (8) The value of the edge vi+t vj−t is f + (vi+t vj−t ) = f (vi+t ) + f (vj−t ) = f (vi+t ) + f (vi+t+1 ) If i+t is odd, then f (vi+t ) + f (vi+t+1 ) = (i+t−1) 2 d + k + (q − 1)d + (i+t+1−2) 2 d = k + (q − 1)d + (i + t − 1)d If i+t is even, then f (vi+t ) + f (vi+t+1 ) = (i+t+1−1) 2 d + k + (q − 1)d + (i+t−2) 2 (9) (10) d = k + (q − 1)d + (i + t − 1)d (11) Therefore, from (9), (10), (11), we get f + (vi+t vj−t ) = k + (q − 1)d + (i + t − 1)d (12) From (8) and(12), f + (vi vj ) = f + (vi+t vj−t ).

2. k + (a + b − 2)d}. Hence f (ui ) = f (vj ) yields k2 − k1 = (i − j)d (16) Therefore, if d | (k2 − k1 ), then (16) is a contradiction, and on the other hand, if k2 − k1 = rd, r ≥ a, then (16) yields rd = k2 − k1 = (i − j)d ⇒ i−j =r ≥a ⇒ i≥a+j a contradiction to the fact that 1 ≤ i ≤ a and 1 ≤ j ≤ b. Thus f must be a required (k, d)-arithmetic numbering of Ca,b . 2. 3 displays two such instance. 3 It seems a hard problem to determine in general the full range of values of k and d and possible partitions (k1 , k2 ) of k, 0 ≤ k1 < k2 , with k2 − k1 = rd and r < a such that Ca,b , 2 ≤ a ≤ b, has a (k, d)-arithmetic numbering f for which k1 , k2 ∈ f (Ca,b ).

Thus, K1,b is “arbitrarily arithmetic” in the sense that it is (k, d)-arithmetic for all values of k and d. 6, there are no other connected bipartite graphs with this property. 7, there is no Eulerian graph with this property. It would be interesting to know if there is any graph different from K1,b that is arbitrarily arithmetic. 2 For any positive integer k and d and for any partition of k into two parts k1 and k2 with 0 ≤ k1 < k2 , the complete bipartite graph Ka,b , 2 ≤ a ≤ b has a (k, d)-arithmetic numbering f such that k1 , k2 ∈ f (G) where either C1: d | (k2 − k1 ) or C2: k2 − k1 = rd for some r ≥ a.

### Arithmetic Graphs by Dileepkumar R

by Paul

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