Arithmetic Graphs by Dileepkumar R

By Dileepkumar R

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Let there be two pendant vertices u and v in T such that the length of u0 − u path is equal to the length of v0 − v path. If the edge u0 v0 is deleted from T and u and v are joined by an edge uv, then such a transformation of T is called an elementary parallel transformation (or an ept) and the edge u0 v0 is called a transformable edge. If by a sequence of ept’s T can be reduced to a path then T is called a Tp -tree (transformed tree) and any such sequence regarded as a composition of mappings (ept’s) denoted by P, is called a parallel transformation of T.

By the definition of a Tp -tree there exists a parallel transformation P of T so that we get P(T). , 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.

6 In this section, we prove that for all odd n, the cycle related graphs Wn , Hn , W(2,n), W(t,n), Cn K1 , W0 (t, n) + K p are (k, d)-arithmetic for k = (n−1) d and k = (3n−1) d. 1 For all positive integer d and odd n, the generalized web graph W(t, n) is (k, d)-arithmetic for k = (n−1) d. v1,n . vt,n . vt+1,n respectively and the centre of the web as v0,0 . Define a labeling f : V (W (t, n)) −→ N such that F (u) = f (u) if u ∈ V (G) 0 if u = v 30 f (vm,i ) =  i−1 ( 2 )d + (m − 1)nd     n−1 i    ( 2 )d + ( 2 )d + (m − 1)nd ( 3n−1 )d + ( i−1 )d + (m − 2)nd 2 2     (n − 1)d + ( 2i )d + (m − 2)nd    n−1 ( 2 )d + 2tnd m odd, i odd; 1 ≤ m ≤ t + 1, 1≤i≤n m odd, i even; 1 ≤ m ≤ t + 1, 2≤i≤n−1 m even, i odd; 2 ≤ m ≤ t + 1, 1≤i≤n m even, i even; 2 ≤ m ≤ t + 1, 2≤i≤n−1 (4) m = i = 0.

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