 By Dileepkumar R

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Extra resources for Arithmetic Graphs

Example text

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.