We give an upper bound for the uniqueness transition on an arbitrary locally finite graph ${ \cal G}$ in terms of the limit of the spectral radii $\rho\left[ H({ \cal G}_t)\right]$ of the non-backtracking (Hashimoto) matrices for an increasing sequence of subgraphs ${ \cal G}_t\subset{ \cal G}_{ t+1}$ which converge to ${ \cal G}$. With the added assumption of strong local connectivity for the oriented line graph (OLG) of ${ \cal G}$, connectivity on any finite subgraph ${ \cal G}'\subset{ \cal G}$ decays exponentially for $p<(\rho\left[ H({ \cal G}^{ \prime})\right])^{ -1}$.
Categories: math-ph cond-mat.dis-nn math.MP