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The signed graphs with symmetric spectra

Deqiong Li, Qiongxiang Huang

TL;DR

The paper addresses the problem of characterizing spectrally symmetric and sign-symmetric signed graphs, including non-bipartite underlying graphs. It develops a Sachs-type framework for the characteristic polynomial $\phi(Γ,x)$ and proves that spectral symmetry is equivalent to the vanishing of all odd coefficients, formalized through a relation between the matching polynomials $M(G-x,x)$ and 2-regular subgraphs; it also provides an automorphism-based criterion for sign symmetry via weak automorphisms and base cycles. The main contributions are a complete criterion for spectral symmetry, a combinatorial and automorphic description of sign symmetry, and a complete constructive classification of sign-symmetric graphs in $\mathbb{S}_1(G)$ (including new families and unifying prior results), with open questions extending to higher-order classes. These results advance the understanding of the interplay between cycle structure, switching, and spectral properties in signed graphs, offering concrete constructions of expansive families with prescribed symmetry properties.

Abstract

It is well known that a graph $G$ has a symmetric spectrum if and only if it is bipartite, a signed graph $Γ=(G,σ)$ has a symmetric spectrum if $G$ is bipartite. However, there exists a spectrally symmetric signed graph $Γ=(G,σ)$ such that $G$ is not bipartite. In this paper, we focus to characterize the signed graphs with symmetric spectra. Some necessary and (or) sufficient conditions for spectrally symmetric signed graphs are given. Moreover, some methods to construct signed graphs with symmetric spectra are found and infinite families of these signed graphs are produced.

The signed graphs with symmetric spectra

TL;DR

The paper addresses the problem of characterizing spectrally symmetric and sign-symmetric signed graphs, including non-bipartite underlying graphs. It develops a Sachs-type framework for the characteristic polynomial and proves that spectral symmetry is equivalent to the vanishing of all odd coefficients, formalized through a relation between the matching polynomials and 2-regular subgraphs; it also provides an automorphism-based criterion for sign symmetry via weak automorphisms and base cycles. The main contributions are a complete criterion for spectral symmetry, a combinatorial and automorphic description of sign symmetry, and a complete constructive classification of sign-symmetric graphs in (including new families and unifying prior results), with open questions extending to higher-order classes. These results advance the understanding of the interplay between cycle structure, switching, and spectral properties in signed graphs, offering concrete constructions of expansive families with prescribed symmetry properties.

Abstract

It is well known that a graph has a symmetric spectrum if and only if it is bipartite, a signed graph has a symmetric spectrum if is bipartite. However, there exists a spectrally symmetric signed graph such that is not bipartite. In this paper, we focus to characterize the signed graphs with symmetric spectra. Some necessary and (or) sufficient conditions for spectrally symmetric signed graphs are given. Moreover, some methods to construct signed graphs with symmetric spectra are found and infinite families of these signed graphs are produced.
Paper Structure (4 sections, 32 theorems, 32 equations, 5 figures)

This paper contains 4 sections, 32 theorems, 32 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Signed graphs with symmetric spectra
  3. Signed graphs with sign symmetry
  4. Complete characterization of the signed graphs in $\mathbb{S}_1(G)$

Key Result

Theorem 2.1

The signed graph $\Gamma=(G,\sigma)$ has a symmetric spectrum if and only if its characteristic polynomial $\phi(\Gamma, x) =\sum_{i=0}^na_ix^{n-i}$ satisfies $a_i =0$ for all odd indices $i$.

Figures (5)

  • Figure 1: $\Gamma$ is spectrally symmetric but not sign-symmetric. Here and in the remaining figures negative edges are represented by dash lines.
  • Figure 2: $\Gamma_1$ and $\Gamma_2$ are two signed graphs that are spectrally symmetric but not sign-symmetric.
  • Figure 3: $\Gamma_1\in \mathbb{S}_1(G_1)$, $\Gamma_2\in \mathbb{S}_2(K_5),$ and $\Gamma_3\in \mathbb{S}_1(G_3)$
  • Figure 4: $\Gamma=(G,\sigma)\in \mathbb{S}_1(G)$ has two types: $F=\emptyset$ and $F\not=\emptyset$.
  • Figure 5: $\Gamma=(G, \sigma)\in \mathbb{FS}_1(G)$

Theorems & Definitions (53)

  • Theorem 2.1
  • Theorem 2.2: $\!\!$Belardo2
  • Theorem 2.3: $\!\!$Belardo2
  • Theorem 2.4
  • proof
  • Example 2.1
  • Corollary 2.1
  • Corollary 2.2
  • Example 2.2
  • Lemma 3.1: $\!$Zaslavsky1
  • ...and 43 more