# What is the maximun possible number of simple cycles in an undirected connected garph with n nodes and n+1 edges?

Considering we have an undirected connected graph with n nodes and n+1 edges , what is the maximum number of simple cycles the graph can have ? what is the proof ?

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Remove two edges while keeping the graph connected. A connected graph with $$n$$ nodes and $$n-1$$ edges is a tree (hence without cycles). Add one of the edges, back, and you create one cycle. Add the other edge back, and you create at most $$2$$ more simple cycles, so the total is at most three.