Let G = (V , E) be a simple connected graph with vertex set V and edge set E. The Wiener index of G is defined by Wwhere n eu (e|G) (resp. n ev (e|G)) is the number of edges of G closer to u (resp. v) than v (resp. u). In this paper we find the above indices for various graphs using the group of automorphisms of G. This is an efficient method of finding these indices especially when the automorphism group of G has a few orbits on V or E. We also find the Wiener indices of a few graphs which frequently arise in mathematical chemistry using inductive methods.

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A finite group whose irreducible complex characters are rational is called a Q-group. In this paper we will find the structure of Frobenius Q-groups.

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