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Generalized block diagonal Laplacian spectrum of graphs

Yanrui Xu, Da Zhao

TL;DR

This work refines the generalized block Laplacian spectrum by introducing the block-diagonal variant (GBDLS) that uses only $p$ diagonal block projections, ensuring a real spectrum. It proves an almost identical orthogonal-similarity characterization to the full multivariate spectrum: two graphs share the same GBDLS if and only if there exists a fixed orthogonal matrix $Q$ with $Q^\top A_1 Q = B_1$ and $Q^\top e_i = e_i$, which also yields $Q^\top \widetilde{W}_{A_1} = \widetilde{W}_{B_1}$. The paper extends the characterization to the Hermitian adjacency matrix of digraphs, highlighting that off-diagonal blocks can be discarded in the undirected graph setting but not for digraphs. Together, these results provide a tractable, real-spectrum criterion for generalized cospectrality in block-structured graph spectra and offer insights into graph isomorphism contexts within this framework.

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.

Generalized block diagonal Laplacian spectrum of graphs

TL;DR

This work refines the generalized block Laplacian spectrum by introducing the block-diagonal variant (GBDLS) that uses only diagonal block projections, ensuring a real spectrum. It proves an almost identical orthogonal-similarity characterization to the full multivariate spectrum: two graphs share the same GBDLS if and only if there exists a fixed orthogonal matrix with and , which also yields . The paper extends the characterization to the Hermitian adjacency matrix of digraphs, highlighting that off-diagonal blocks can be discarded in the undirected graph setting but not for digraphs. Together, these results provide a tractable, real-spectrum criterion for generalized cospectrality in block-structured graph spectra and offer insights into graph isomorphism contexts within this framework.

Abstract

We reduce the block all-one matrices in the generalized block Laplacian spectrum of graphs to 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.

Paper Structure

This paper contains 2 sections, 4 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Proof

Key Result

Theorem 1.1

Let $G$ and $H$ be two non-isomorphic graphs. Let $A$ and $B$ be their adjacency matrices respectively. If $G$ and $H$ are generalized cospectral, then there exists a regular orthogonal matrix $Q$ such that $Q^\top A Q = B$. In particular, if $G$ is controllable, then $Q$ is unique and rational and

Theorems & Definitions (8)

  • Theorem 1.1: johnson1980NoteCospectralGraphs wang2006SufficientConditionFamily
  • Theorem 1.2: wang2026GraphIsomorphismMultivariate
  • Theorem 1.3
  • Theorem 1.4
  • proof : Proof of \ref{['thm:GBDLS']}
  • Claim 2.1
  • proof
  • proof : Proof of \ref{['thm:GBLS_complex']}