By C. E. Eckersley, Margaret Macaulay, Phebean A. Ogundipe

ISBN-10: 0582558980

ISBN-13: 9780582558984

This ever renowned four-book sequence Brighter Grammar is helping scholars comprehend the major issues of English grammar, utilizing basically crucial technical phrases. Graded routines permit academics to observe improvement, and vigorous illustrations increase the textual content and keep the scholars' intrest.

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**Additional info for Brighter Grammar 4: An English Grammar with Exercises**

**Sample text**

2 Every Tp -tree T is (k+(q-1)d, d)-arithmetic for all positive integers k and d. Proof Let T be a Tp -tree with n+1 vertices. Pk ) of the ept’s Pi used to arrive at the path P(T). Clearly Ed and Ep have the same number of edges. Then denote the vertices of P(T) successively as v1 , v2 ,... vp starting from one pendent vertex of P(T) right up to other. Define f : V (P (T )) −→N by f (vi ) = (i−1)d 2 for odd i, k + (q − 1)d + (i−2) 2 1≤i≤n+1 d for even i, 2 ≤ i ≤ n + 1 where k and d are positive integers and q is the number of edges of T.

6 For any nonnegative integer r and positive integer d, C4t+3 is ((2t+1)d+2r, d)-arithmetic. Proof Under the hypotheses, the map f : V (C4t+3 ) −→ N defined by 46 f (ui ) = r + ( i−1 )d, if i is odd 2 1 (2t + 1)d + r + 2 id, if i is even (19) is a required arithmetic numbering of C4t+3 . This completes the proof. 8 by taking d=2 and r=0. 5, that for an odd cycle to be (k, d)-arithmetic it is necessary that k and d to be parity. This prompts us to the following: Conjecture 3 (1) If C4t+1 is (k, d)-arithmetic then k=2dt+2r for some integer r ≥ 0.

5, that for an odd cycle to be (k, d)-arithmetic it is necessary that k and d to be parity. This prompts us to the following: Conjecture 3 (1) If C4t+1 is (k, d)-arithmetic then k=2dt+2r for some integer r ≥ 0. (2) If C4t+3 is (k, d)-arithmetic then k=(2t+1)d+2r for some integer r ≥ 0. 47 REFERENCES [1] Acharya, B. D and Hegde, S. Graph Theory, 14(3), 1989, 275-299. [2] Balakrishnan, R and Ranganathan, K, A Text Book of Graph Theory, Springer, Newyork, 1996. [3] Chartrand, Gray and Lesniak, Linda, Graphs and Digraphs, Chapman and Hall, London, 1996.

### Brighter Grammar 4: An English Grammar with Exercises by C. E. Eckersley, Margaret Macaulay, Phebean A. Ogundipe

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