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Signed graphs with exactly two main eigenvalues: The unicyclic case

Zenan Du, Fenjin Liu, Hechao Liu, Jifu Lin, Wenxu Yang

Abstract

An eigenvalue $λ$ of a signed graph $S$ of order $n$ is called a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector $j$. Characterizing signed graphs with exactly $k$ $(1\le k\le n)$ distinct main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Du et al. (2024, 2026) characterized a class of signed graphs with exactly two main eigenvalues by analyzing a type of multigraph whose base graph is a tree. In this paper, we extend this study to the case where the associated multigraph has a unicyclic base graph, and we conclude by proposing several open problems.

Signed graphs with exactly two main eigenvalues: The unicyclic case

Abstract

An eigenvalue of a signed graph of order is called a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector . Characterizing signed graphs with exactly distinct main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Du et al. (2024, 2026) characterized a class of signed graphs with exactly two main eigenvalues by analyzing a type of multigraph whose base graph is a tree. In this paper, we extend this study to the case where the associated multigraph has a unicyclic base graph, and we conclude by proposing several open problems.
Paper Structure (12 sections, 15 theorems, 9 equations, 4 figures, 2 tables)

This paper contains 12 sections, 15 theorems, 9 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. $|V(F)|=0$
  5. $w(u_1,u_2)=1$ and $w(v_1,v_2)=1$
  6. $w(u_1,u_2)=1$ and $w(v_1,v_2)=2$
  7. $w(u_1,u_2)=2$ and $w(v_1,v_2)=1$
  8. $w(u_1,u_2)=w(v_1,v_2)=2$
  9. $|V(F)|\neq0$
  10. $a=0$ and $b\neq0$
  11. $a=1$ and $b\neq0$
  12. Conclusion

Key Result

Theorem 1.2

(Du-4) Let $H$ be a multigraph of order $n$$(\geq2)$ and $A$$(=A(H))$ be its adjacency matrix, $s(v)$ be the number of walks of length $2$ in $H$ that starting at $v\in V(H)$. Then the following three conditions are equivalent. (1)$H$ has exactly two distinct main eigenvalues. (2)$j$ is not an eig holds for any vertex $v\in V(H)$ and $j$ is not an eigenvector of $A$. In particular, if $H$ is a $

Figures (4)

  • Figure 1: The tree $T^*$.
  • Figure 2: The graphs $U^1_{4t},U^2_{3t},U^3_{3t},U^4_{5t},U^5_{4t},U^6_{2t},U^7_{t}$ with $t\geq1$.
  • Figure 3: The graphs $H_1,H_2,H_3,H_4$ and $H_5$.
  • Figure 4: The graphs $D_1$, $D_2$, $\mathcal{H}^2(b,t)$ and $\mathcal{H}^3(b,t)$, where $b\geq2$ and $t\geq3$.

Theorems & Definitions (18)

  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Remark 2.5
  • Lemma 2.6
  • Remark 3.2
  • Lemma 3.3
  • Theorem 3.5
  • ...and 8 more