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A sufficient condition for generalized spectral characterization of graphs with loops

Alexander Van Werde

TL;DR

The paper addresses when graphs with loops are determined by their generalized spectrum, introducing a square-free determinant condition on the walk matrix as a sufficient criterion. It develops a general framework using the invariant $\Phi_{\mathbf{X},\zeta}(\lambda,t)$ for symmetric integral pairs and shows $\Phi$-cospectrality is equivalent to an orthogonal similarity $\mathbf{Q}$ with $\mathbf{Q}\zeta=\eta$ and $\mathbf{Q}\mathbf{X}\mathbf{Q}^{\top}=\mathbf{Y}$. The main result proves that if $\det(\mathbf{W}_{\mathbf{X},\zeta})$ is square-free, then $(\mathbf{X},\zeta)$ is characterized by its $\Phi$-spectrum up to signed permutation, yielding a loop-graph variant of Wang–Xu’s condition. The approach also relates to discriminant-based refinements and highlights how loops mitigate 2-adic obstacles in spectral characterization.

Abstract

Sufficient conditions for a simple graph to be characterized up to isomorphism given its spectrum and the spectrum of its complement graph are known due to Wang and Xu. This note establishes a related sufficient condition in the presence of loops: if the walk matrix has square-free determinant, then the graph is characterized by its generalized spectrum. The proof includes a general result about symmetric integral matrices.

A sufficient condition for generalized spectral characterization of graphs with loops

TL;DR

The paper addresses when graphs with loops are determined by their generalized spectrum, introducing a square-free determinant condition on the walk matrix as a sufficient criterion. It develops a general framework using the invariant for symmetric integral pairs and shows -cospectrality is equivalent to an orthogonal similarity with and . The main result proves that if is square-free, then is characterized by its -spectrum up to signed permutation, yielding a loop-graph variant of Wang–Xu’s condition. The approach also relates to discriminant-based refinements and highlights how loops mitigate 2-adic obstacles in spectral characterization.

Abstract

Sufficient conditions for a simple graph to be characterized up to isomorphism given its spectrum and the spectrum of its complement graph are known due to Wang and Xu. This note establishes a related sufficient condition in the presence of loops: if the walk matrix has square-free determinant, then the graph is characterized by its generalized spectrum. The proof includes a general result about symmetric integral matrices.

Paper Structure

This paper contains 6 sections, 14 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem
  3. Characterization in terms of orthogonal matrices
  4. Constraints on the orthogonal matrix
  5. Comparing the kernels of the walk matrices
  6. Proof of the sufficient condition

Key Result

Theorem 1.1

Suppose that $G$ is simple and that $\det(\mathbf{W})/2^{\lfloor n/2 \rfloor}$ is odd and square-free integer. Then, $G$ is characterized by its $\mathbb{R}$-spectrum.

Figures (1)

  • Figure 1: An example of a graph with loops and associated matrices. Note that the $ij$-th entry of the walk matrix counts walks of length $j-1$ starting at vertex $i$, explaining the terminology. It here holds that $\det(\mathbf{W}) = -3$. Thus, the graph is characterized by its $\mathbb{R}$-spectrum due to Theorem \ref{['thm: WithLoops']}.

Theorems & Definitions (27)

  • Theorem 1.1: Wang wang2017simple
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 17 more