I need to prove that a connected graph exists such that it has exactly $k$ spanning trees ( for $k \neq 2$)
Each cycle in a graph is connected, because a cycle is a set of edges such that one edge is connected to the other edge's tail, and the vertices are unique (no repetitions). Each cycle with length $k$ has exactly $k$ spanning trees and they are, all the $k$ options to start from ($k$ vertices in the cycle to start from) meaning that for any $k \neq 2$ I can find a connected graph with $k$ spanning trees.
I would like to hear your thoughts... Thank you!