On (Sub)stochastic and Transient Weightings of Infinite Strong Digraphs
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 $λ_{S}(\cal{F})$ be the Perron root (spectral radius) of its weighted adjacency matrix. We prove that the infimum of $\ell_{max}(\cal{F})\bigl(1-λ_{S}(\cal{F})\bigr)$ 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.