Eigenvalue Sums of Combinatorial Magnetic Laplacians on Finite Graphs

TL;DR

This paper constructs magnetic Laplacians on finite directed graphs, studies their properties, and derives sharp bounds on the sums of their eigenvalues using variational principles, extending known inequalities.

Contribution

It introduces a class of magnetic Laplacians on directed graphs and establishes new eigenvalue sum bounds applicable to these operators.

Findings

Derived bounds on eigenvalue sums for magnetic Laplacians

Extended inequalities to directed graphs from undirected cases

Provided conditions under which bounds hold

Abstract

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 \[\frac{1}{k}\sum_{j=0}^{k-1}\lambda_j \leq d-1.\]