Spectral properties of Sturm-Liouville operators on infinite metric graphs

This paper mainly deals with the Sturm-Liouville operator \begin{equation*} $\mathbf{H}$=\frac{1}{w(x)}\left( -\frac{$\mathrm{d}$}{$\mathrm{d}$x}p(x)\frac{ $\mathrm{d}$}{$\mathrm{d}$x}+q(x)\right) ,$$x\in Γ\end{equation*} acting in $L_w^2\left(\mathrm{\Gamma }\right),$ where $\mathrm{\Gamma }$ is a metric graph. We establish a relationship between the bottom of the spectrum and the positive solutions of quantum graphs, which is a generalization of the classical Allegretto-Piepenbrink theorem. Moreover, we prove the Persson-type theorem, which characterizes the infimum of the essential spectrum.