Eigenvalues of the normalized complex Laplacian on finite electrical networks

TL;DR

This paper analyzes the spectrum of the normalized complex Laplacian in electrical networks, revealing eigenvalues in larger regions, including some with negative real parts and large magnitudes, and provides bounds with sharp examples.

Contribution

It extends spectral analysis to complex Laplacians in electrical networks, offering new bounds and demonstrating the existence of eigenvalues with negative real parts.

Findings

Eigenvalues lie in a larger region than real Laplacian eigenvalues.

Existence of eigenvalues with negative real part and magnitude greater than 2.

Provides sharp lower bounds for the first non-vanishing eigenvalue.

Abstract

The spectrum of the normalized complex Laplacian for electrical networks is analyzed. We show that eigenvalues lie in a larger region compared to the case of the real Laplacian. We show the existence of eigenvalues with negative real part and absolute value greater than 2. An estimate from below for the first non-vanishing eigenvalue in modulus is provided. We supplement the estimates with examples, showing sharpness.