# Properties of a generalized graph

I'll start with formulating my problem and then ask my question:

To generalize a graph $Ga = (Va,Ea)$, we partition its nodes into disjoint sets. The elements of a partitioning $V$ are subsets of $Va$. They can be thought of as supernodes since they contain nodes from $Ga$, but are themselves the nodes of a undirected generalized graph $G = (V, E)$. The superedges of E include self-loops and are labeled with non-negative weights by the function $d : E \rightarrow Z^*$. for example, let $X$ and $Y$ be two supernodes in $V$. $d(X,Y)$=|{$(x,y)\in Ea$ such that $x\in X, y\in Y$}|. ie, the number of edges between nodes x and y that encapsulated by supernodes $X$ and $Y$.

My question is to get bounds on some properties of the generalized graph $G$? these properties are: the mean degree(G), diameter(G), radius(G), global clustering coefficient (G), ...!

Thank you.

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Where did this problem come from? Did you just make it up? –  Graphth Apr 20 '12 at 14:53