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!