Eigenvalue Sums of Combinatorial Magnetic Laplacians on Finite Graphs

We give a construction of a class of magnetic Laplacian operators on finite directed graphs. We study some general combinatorial and algebraic properties of operators in this class before applying the Harrell-Stubbe Averaged Variational Principle to derive several sharp bounds on sums of eigenvalues of such operators. In particular, among other inequalities, we show that if $G$ is a directed graph on $n$ vertices arising from orienting a connected subgraph of $d$-regular loopless graph on $n$ vertices, then if $\Delta_\theta$ is any magnetic Laplacian on $G$, of which the standard combinatorial Laplacian is a special case, and $\lambda_0\leq \lambda_1\leq ...\leq\lambda_{n-1}$ are the eigenvalues of $\Delta_{\theta},$ then for $k\leq \frac{n}{2},$ we have $$\frac{1}{k}\sum_{j=0}^{k-1}\lambda_j \leq d-1.$$