On Products and Line Graphs of Signed Graphs, their Eigenvalues and Energy

TL;DR

This paper investigates the spectral properties and energies of products and line graphs of signed graphs, providing explicit formulas and characterizations for various graph classes including Cartesian products and signed grids.

Contribution

It introduces new expressions for adjacency and Laplacian matrices of signed graph products, and analyzes their eigenvalues and energies, extending spectral graph theory to signed graphs.

Findings

Eigenvalues and energies expressed via Kronecker products

Characterization of balance in Cartesian products

Exact spectral results for signed grids and line graphs

Abstract

In this article we examine the adjacency and Laplacian matrices and their eigenvalues and energies of the general product (non-complete extended $p$-sum, or NEPS) of signed graphs. We express the adjacency matrix of the product in terms of the Kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor signed graphs. For the Cartesian product we characterize balance and compute expressions for the Laplacian eigenvalues and Laplacian energy. We give exact results for those signed planar, cylindrical and toroidal grids which are Cartesian products of signed paths and cycles. We also treat the eigenvalues and energy of the line graphs of signed graphs, and the Laplacian eigenvalues and Laplacian energy in the regular case, with application to the line graphs of signed grids that are Cartesian products and to the line graphs of all-positive and all-negative complete graphs.