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Trees and Graphs

By contracting an edge e = uv, we mean removing e and identifying the vertices u and v as a single new vertex. Let num_T(G) denote the number of spanning trees of the graph G.
a. Show that the following recursive formula holds:
num_T(G) = num_T(G - e) + num_T (G * e)
where G * e means the multigraph obtained from G by contracting the edge e.
b. Give a counterexample if multiple edges are identified to make G * e into a graph.

Solution Preview

(a)
Let us consider all the spanning trees of graph G and separate them into two groups:

One group contains all the spanning trees which do not include edge e.
These are the trees that would be all the spanning trees of (G-e) and so their number is

num_T(G-e).

Also note, that these trees are totally ...

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