  The Perron-Frobenius theorem is applied to identify the superfluid transition
of a two-component Fermi gas with a zero-range s-wave interaction. According to
the quantum cluster expansion method of Lee and Yang, the grand partition
function is expressed by the Lee-Yang contracted 0-graphs. A singularity of an
infinite series of ladder-type Lee-Yang contracted 0-graphs is analyzed. We
point out that the singularity is governed by the Perron-Frobenius eigenvalue
of a certain primitive matrix which is defined in terms of the two-body cluster
functions and the Fermi distribution functions. As a consequence, it is found
that there exists a unique fugacity at the phase transition point, which
implies that there is no fragmentation of Bose-Einstein condensates of dimers
and Cooper pairs at the ladder-approximation level of Lee-Yang contracted
0-graphs. An application to a Bose-Einstein condensate of strongly bounded
dimers is also made.
