  In the present paper, for a given (possibly, infinite) strongly connected
digraph $\cal{D},$ we consider the class $\cal{S}_{<}({\cal D})$ of all truthly
substochastic weightings of ${\cal D}$ (here, the word "truthly" means that
there exists a vertex whose out-weight is strictly less than $1$). For a finite
subdigraph $\cal{F}$ of $\cal{D}$ weighted by $S\in {\cal S}_{<}({\cal D}),$
let $\ell_{max}(\cal{F})$ be the length of its longest directed cycle and
$\lambda_{S}(\cal{F})$ be the Perron root (spectral radius) of its weighted
adjacency matrix. We prove that the infimum of
$\ell_{max}(\cal{F})\bigl(1-\lambda_{S}(\cal{F})\bigr)$ taken over all
$\cal{F}$ is positive for every $S\in \cal{S}_{<}({\cal D})$ if and only if
$\cal{D}$ admits a finite cycle transversal. The result obtained provides
general theorems on the set ${\cal T}({\cal D})$ of transient weightings of
${\cal D}.$ In particular, we present a theorem of alternatives for finite
approximations to elements of ${\cal T}({\cal D})$ and simply reprove V. Cyr's
criterion for ${\cal T}({\cal D})$ to be empty.
