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An explicitly-gauge-invariant expansion in powers of $e/\hbar$ times themagnetic flux density is formally obtained for the density of states (ascharacterized by the trace of the resolvent $\widehat G$ = $(\omega - \hath)^{-1}$) of a charged particle moving on a Hermitian quantum network that isembedded in a Euclidean background that supports a uniform magnetic fluxdensity. The explicit expressions, given here up to third order in the fluxdensity, are also valid for the ``local trace'' (the trace of $\widehat P_i\widehat G$, where $\widehat P_i$ is the projector on a network node), and donot appear to have been previously given.

journal of mathematical physics

Gauge-invariant perturbation expansion in powers of electric charge for  the density-of-states of a network model for charged-particle motion in a  uniform background magnetic flux density