# Possible numbers of cycles in a connected graph $H$ when $e(H) = |H| + 1$

Let $$H$$ be a connected graph such that $$e(H) = |H| + 1$$. How many cycles may $$H$$ have?

I can only show that this number is at least $$2$$ (by more generally showing, e.g. by induction, that if $$e(H) = |H| + s$$ for any $$s\geq - 1$$, then $$H$$ has at least $$s+1$$ cycles; I can also show that for $$s=0, -1$$ there are exactly $$s+1$$ cycles).

Any help appreciated!

Hint Pick $$T$$ to be a spanning tree for $$H$$. Then $$e(T)=|T|-1=|H|-1=e(H)-2$$
Therefore, $$H$$ is a spanning tree with two extra edges.