Sorry about the delay in responding.

If I understand you correctly, you are talking about multiplying the adjacency matrices of two different graphs. Whereas graph power is about multiplying the adjacency matrix by itself. So, in case of multiplying two different adjacency matrices, we need to be clear about the semantics of the operation.

In case of graph power, $G^p$, what we are basically doing is: (1) computing a transitive closure of node reachability up to a level $p$ and (2) adding whenever a node is reachable from another in p or less (transitive) steps, we add an edge to make it a direct path. This reduces the diameter.

Now, when you say multiplying two networks, are you talking about a graph product such as, say, graph composition (http://mathworld.wolfram.com/GraphComposition.html)? In that case, the product is a new network, with a different number of nodes. If not, can you explain how the multiplication works?

Kevin:

Thanks for the comments. Again, I am a bit stuck regarding the multiplication of two networks. Perhaps you can explain with an example how this works.

Your other point is also interesting — taking only $A^k$. In other words, adding an edge only if there is a k-length path (and not where, $1 < pathlength <= k$). What happens do diameter when you short-circuit all k-length paths in a network, is an interesting question. This has an associated question — can we determine bounds on the number of k-length paths that exist in a network? I had investigated these questions a little. I call it the power* operation. I only have very elementary results though.

Unfortunately, I’m not sure that just multiplying the adjacency matrices would lead to a diameter reduction.

For example, what would happen if both of the original networks were complete bipartite on the same partition of vertices? In this case both of the original graphs have diameter two, but the product of their adjacency matrices corresponds to two disjoint complete graphs, and doesn’t even have finite diameter.

In the original post this was dodged because the graph power was defined by taking $A^k+A^{k-1}+\dots+A$ instead of just $A^k$.

On a related question, what would happen if one of the networks has diameter less than m, like k < m. (I know that in this case the resulting diameter can not be greater than k.

Or if you know of a reference that I can go to for further information?

I sincerely appreciate your response/commets.

Shabnam

Let’s not go into citation mania and name dropping please.. *feels irritated*

With the above disclaimer in place, let me say that _I_ think that the most important (excuse me for using such a phrase here) skill needed for computer scientists — more than most other disciplines — is a thorough understanding of and the capability to think in and apply /logic/. Please note that I say that this is _needed_, I am not trying to imply that computer scientists actually do have this (or not, for that matter.)

As far as your question about the ‘core’ computer sciences areas is concerned, I think it would be incorrect to classify anything as core or not. Although somethings are clearly more core (sic) than others, I think any attempt to narrow down on the list of core things cannot be completely fair.