How different are molecule with same connectivity Let us represent molecules by graphs (atoms are vertices and bonds are edges).
Let $A$ be the adjacency matrix of the molecular graph. $A_{ij} = b$ where $b$ is either binary: 0 when atom i and j are not bonded else 1 (Repr. 1). Or, further we may take b from the set $\{0,1,2,3\}$ depending on whether the no bond (0) or single(1) double(2) or triple (3) bond is present (Rper. 2). Here were are not distinguishing the atoms - for our purose they represent a node in a graph.

Let us consider only non-isomers and molecules with at least 10 atoms. How different will be two molecules with same graphs (with respect to Repr. 1 or Repr. 2) in terms of their properties or particularly about drug-like properties. My interest is to know how faithfully the topological indices characterizes the molecular activity. Specific or general examples of pairs of  non-isomer molecules with same graph but very different property could be useful. It would be helpful if I can find enough examples to convince myself about the usefulness or otherwise of topological indices which are calculated from the grpahical representation outlined above. Topological indices are popular in drug discovery research, but I am not very much convinced about their utility since most the results published are not statistically significant (use of very small data size - from 15 to some hundreds molecules, among other weakness) nor rigorously validated. 

Thank you.
In my experience, calculated parameters like topological indices are not viewed as significant if considered alone. It's a strategy for finding trends or correlations and, subsequently, candidates from  large volumes of chemical structures. Many studies will publish trends they find in a target chemical groups which are not meant to apply to every chemical group.

An example of how all these seemly useless data is useful - see the Merck Kaggle Challenge. 
https://www.kaggle.com/c/MerckActivity

http://blog.kaggle.com/2012/11/01/deep-learning-how-i-did-it-merck-1st-place-interview/

Statistical significance is harder to comment on. I work with between 8 and 20 million data points and the idea of how to define statistical significance is evolving in cheminformatics as it evolves in big data analytics. It gets much more complicated because standard methods, such as the t-test, fail.

NOTE
I just realized there might be some fundamental confusion. Mol files include atomic information as do most graph based chemical representations. They're stored as properties of the nodes. An algorithm can choose to use or ignore this information. Not all topological parameters ignore atomic information.