International Journal of Mathematical Archive-8(1), 2017, 128-138 Available online through www.ijma.info ISSN 2229 – 5046 International Journal of Mathematical Archive- 8(1), Jan. – 2017 128 ON THE WIENER AND HYPER WIENER INDICES OF CERTAIN DOMINATING TRANSFORMATION GRAPHS OF KRAGUJEVAC TREES B. BASAVANAGOUD* 1 , SUJATA TIMMANAIKAR 2 1 Department of Mathematics, Karnatak University, Dharwad-580 003, India. 2 Department of Mathematics, Government Engineering College-581 110, Haveri, India. (Received On: 15-12-16; Revised & Accepted On: 23-01-17) ABSTRACT Let G be a graph. The distance ) , ( v u d G between the vertices u and v of the graph G is equal to the length of the shortest path that connects u and . v The Wiener index ) (G W is the sum of distances between all pairs of vertices of , G whereas the hyper-Wiener index ) (G WW is defined as ] ) , ( ) , ( [ 2 1 = ) ( 2 ) ( ) , ( v u d v u d G WW G G G V v u + ∑ ⊆ . In this paper Wiener and hyper-Wiener indices are obtained for dominating transformation(d-transformations) graphs of Kragujevac tree T . Keywords: distance, Wiener index, hyper-Wiener index, d-transformation graphs. Subject Classification: 05C05, 05C07, 05C69. 1. INTRODUCTION In this paper we are concerned with simple graphs, having no directed or weighted edges, and no self loops. Let ) , ( = E V G be such a graph. The number of vertices of G we denote by n and the number of edges we denote by , m thus n G V |= ) ( | and m G E |= ) ( | . By the open neighborhood of a vertex v of G we mean the set )} ( : ) ( { = ) ( G E uv G V u v N G ∈ ∈ . The distance ) , ( v u d G between the vertices u and v of the graph G is equal to the length of a shortest path that connects u and v . Let S be a finite set and let } ,..., , { = 2 1 n S S S F be a partition of . S Then the intersection graph ) ( F Ω of F is the graph whose vertices are the subsets in F and in which two vertices i S and j S are adjacent if and only if . Φ ≠ ∩ j i S S For terminology not defined here we refer the reader to [6]. A topological index of a graph is a single unique number characteristic of the graph and is mathematically invariant under graph automorphism. Usage of topological indices in biology and chemistry began in 1947 when H. Wiener [17, 18] introduced Wiener index which is denoted by W and is given by ) , ( = ) ( ) ( ) , ( v u d G W G G V v u ∑ ⊆ The hyper-Wiener index of acyclic graphs was introduced by Milan Randic in 1993. Then Klein et al. [11], generalized Randic’s definition for all connected graphs, as a generalization of the Wiener index. It is defined as ] ) , ( ) , ( [ 2 1 = ) ( 2 ) ( ) , ( v u d v u d G WW G G G V v u + ∑ ⊆ Further, information about Wiener index and hyper-Wiener index are given in [2, 3, 4, 9, 10, 19]. Corresponding Author: B. Basavanagoud* 1 1 Department of Mathematics, Karnatak University, Dharwad-580 003, India.