  Given a digraph $G$, a lot of attention has been deserved on the maximum
number $\phi(G)$ of fixed points in a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$
with $G$ as interaction graph. In particular, a central problem in network
coding consists in studying the optimality of the classical upper bound
$\phi(G)\leq 2^{\tau}$, where $\tau$ is the minimum size of a feedback vertex
set of $G$. In this paper, we study the maximum number $\phi_m(G)$ of fixed
points in a {\em monotone} Boolean network with interaction graph $G$. We
establish new upper and lower bounds on $\phi_m(G)$ that depends on the cycle
structure of $G$. In addition to $\tau$, the involved parameters are the
maximum number $\nu$ of vertex-disjoint cycles, and the maximum number
$\nu^{*}$ of vertex-disjoint cycles verifying some additional technical
conditions. We improve the classical upper bound $2^\tau$ by proving that
$\phi_m(G)$ is at most the largest sub-lattice of $\{0,1\}^\tau$ without chain
of size $\nu+1$, and without another forbidden-pattern of size $2\nu^{*}$.
Then, we prove two optimal lower bounds: $\phi_m(G)\geq \nu+1$ and
$\phi_m(G)\geq 2^{\nu^{*}}$. As a consequence, we get the following
characterization: $\phi_m(G)=2^\tau$ if and only if $\nu^{*}=\tau$. As another
consequence, we get that if $c$ is the maximum length of a chordless cycle of
$G$ then $2^{\nu/3^c}\leq\phi_m(G)\leq 2^{c\nu}$. Finally, with the technics
introduced, we establish an upper bound on the number of fixed points of any
Boolean network according to its signed interaction graph.
