On the measure of spectra for discrete Schr\"odinger operators on periodic graphs

We consider discrete Schr"odinger operators $H_{\mu Q}=\mathrm{\Delta }+\mu Q$ with real periodic potentials $Q$ on periodic graphs, where $\mathrm{\Delta }$ is the adjacency operator and $\mu \in \mathbb{R}$ is a coupling constant. The spectra of the operators consist of a finite number of closed intervals (bands). In the large coupling regime, we obtain an asymptotic upper bound for the measure of the spectrum of $H_{\mu Q}$ which depends essentially on a “degeneracy degree” of the potential $Q$. This result extends the result of Y. Last obtained for the one-dimensional lattice $\mathbb{Z}$ to the case of general periodic graphs. It also may serve as a certain quantitative complement to the recent criterion of J. Fillman for the measure of the spectrum of $H_{\mu Q}$ to go to zero as $\mu \to \mathrm{\infty }$.