Find $\tau(G)$ for the graph $G$ below.
This is what I tried so far: Let $e$ denote the horizontal edge between the two vertices as shown below. I wanted to use $\tau(G) = \tau(G-e) + \tau(G \circ e)$
$(G \circ e)$ denotes the contraction of edge $e$.
I got 4 as the answer. Is this correct?
In order for the upper two vertices to be connected to the remaining vertices, the two edges incident to them must be chosen. The remaining part of the graph is a 4-cycle and we need to obtain a spanning tree on these four vertices. Here any one of the four edges in the cycle can be removed. Thus the graph has 4 spanning trees.