Generalized block diagonal Laplacian spectrum of graphs

TL;DR

This paper simplifies the understanding of the generalized block Laplacian spectrum of graphs by reducing complex matrices to simpler forms, and explores similar properties for Hermitian adjacency matrices of digraphs.

Contribution

It introduces a reduction from $p^2$ to $p$ block all-one matrices in the spectrum, extending previous work and analyzing the Hermitian adjacency matrix case.

Findings

Spectrum is real for the simplified matrices.

Reduction from $p^2$ to $p$ blocks simplifies spectral analysis.

Extension to Hermitian adjacency matrices of digraphs.

Abstract

We reduce the $p^2$ block all-one matrices in the generalized block Laplacian spectrum of graphs to $p$ block all-one matrices in the generalized block diagonal Lapalcian spectrum of graphs introduced by Wang and the second author (\textit{Adv. Appl. Math.} 173B (2026)). In this case the matrices are all real symmetric, and hence the spectrum is real, which does not hold for the generalized block Laplacian spectrum. We also investigate the analogue by Hermitian adjacency matrix of digraphs.