By Dileepkumar R
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Extra resources for Arithmetic Graphs
Vn starting from one pendant vertex of P(T) right up to other and preserve the same for T. Now construct the sundivision tree S(T) of T by introducing exactly one vertex between every edge vi vj of T and denote the vertex as vi,j . z be the z transformable edges of T with mx < mx + 1 for all x. Let tx be the path length from the vertex vm x to the corresplnding pendant vertex decided by the transformable edge vmx vhx of T. z where k and d are positive integers and 2q is the number of edges of S(T).
4 Let (a, b, k, d)=(3, 4, 10, 2). Then the partitions (k1 , k2 ) of k with 0 ≤ k1 < k2 are (0, 10), (1, 9), (2, 8), (3, 7), (4, 6). Then d | (k2 − k1 ) for each of these partitions (k1 , K2 ), but only for the first three of these do we have r ≥ 3 = a, k2 − k1 = 2r. 1 As another example, (a, b, k, d)=(3, 4, 12, 3), then the partitions (k1 , k2 ) of k with 0 ≤ k1 < k2 are (0, 12), (1, 11), (2, 10), (3, 9), (4, 8), (5, 7). Then only (0, 12) satisfies C2 and hence when (k1 , k2 ) is any of these partitions of k, the graph K3,4 has a (k,d)arithmetic numbering f with k1 , k2 ∈ f (K3,4 ).
3] Chartrand, Gray and Lesniak, Linda, Graphs and Digraphs, Chapman and Hall, London, 1996.  Clark, John and Holton, D. A, A First Look At Graph Theory, Allied Publishers Ltd, Singapore, 1995.  Gallian, Joseph. A, A Dynamic Survey of Graph Labeling, Electron. J. Combinatorics, 10, 2007, DS6. M and Shetty, Sudhakar, On Arithmetic Graphs, Indian J. , 33(8), 2002, 1275-1283.