On the Eigenvalues of Graphs with Mixed Algebraic Structure

TL;DR

This paper investigates the spectral properties of a matrix combining Laplacian and adjacency matrices of graphs, analyzing how these properties relate to graph structure and stability in dynamic networks across different parameter regimes.

Contribution

It introduces a novel matrix construction blending Laplacian and adjacency matrices and explores its spectral behavior in various regimes, connecting it to network stability.

Findings

Spectral properties depend on graph configuration and matrix parameters.

Connections established between spectral characteristics and network stability.

Conjecture proposed for the signature of the combined matrix.

Abstract

We study some spectral properties of a matrix that is constructed as a combination of a Laplacian and an adjacency matrix of simple graphs. The matrix considered depends on a positive parameter, as such we consider the implications in different regimes of such a parameter, perturbative and beyond. Our main goal is to relate spectral properties to the graph's configuration, or to basic properties of the Laplacian and adjacency matrices. We explain the connections with dynamic networks and their stability properties, which lead us to state a conjecture for the signature.