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Spectral integral variation of signed graphs

Jungho Ahn, Cheolwon Heo, Sunyo Moon

TL;DR

This work addresses when adding an edge to a signed graph yields integral changes in the spectrum of the signed Laplacian $L(G,Σ)$ and extends the notion of integral completable graphs to the signed setting. It develops explicit type-1 and type-2 characterizations for spectral variation under edge addition, using interlacing, eigenvector structure, and switching to derive necessary and sufficient conditions. The authors then fully classify integrally $Σ$-completable graphs on $K_n$ by introducing structural sets $X(Σ)$ and $Y(Σ)$ and a vertex-substitution framework that governs spectrum under graph expansions. The results generalize Kirkland’s graph-theoretic integrality criteria to signed graphs, providing constructive criteria and new tools (e.g., substitution, centered reductions) to build Laplacian-integral signed graphs with prescribed complete targets.

Abstract

We characterize when the spectral variation of the signed Laplacian matrices is integral after a new edge is added to a signed graph. As an application, for every fixed signed complete graph, we fully characterize the class of signed graphs to which one can recursively add new edges keeping spectral integral variation to make the signed complete graph.

Spectral integral variation of signed graphs

TL;DR

This work addresses when adding an edge to a signed graph yields integral changes in the spectrum of the signed Laplacian and extends the notion of integral completable graphs to the signed setting. It develops explicit type-1 and type-2 characterizations for spectral variation under edge addition, using interlacing, eigenvector structure, and switching to derive necessary and sufficient conditions. The authors then fully classify integrally -completable graphs on by introducing structural sets and and a vertex-substitution framework that governs spectrum under graph expansions. The results generalize Kirkland’s graph-theoretic integrality criteria to signed graphs, providing constructive criteria and new tools (e.g., substitution, centered reductions) to build Laplacian-integral signed graphs with prescribed complete targets.

Abstract

We characterize when the spectral variation of the signed Laplacian matrices is integral after a new edge is added to a signed graph. As an application, for every fixed signed complete graph, we fully characterize the class of signed graphs to which one can recursively add new edges keeping spectral integral variation to make the signed complete graph.
Paper Structure (9 sections, 17 theorems, 35 equations)

This paper contains 9 sections, 17 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Spectral integral variation for signed graphs
  3. Spectral integral variation of type 1
  4. Spectral integral variation of type 2
  5. Integrally $\Sigma$-completable signed graphs
  6. Structures of $X(\Sigma)$ and $Y(\Sigma)$
  7. Vertex substitution
  8. Spectral integral variation not captured by $X(\Sigma)$
  9. Proof of \ref{['thm:main3']}

Key Result

Theorem 1.1

For a signed graph $(G,\Sigma)$ and non-adjacent vertices $v$ and $w$ of $G$, spectral integral variation of type 1 occurs under the addition of an even edge $vw$ if and only if $N^-(v)=N^-(w)$ and $N^+(v)=N^+(w)$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: So so99
  • proof : Proof of \ref{['thm:main1 even']}
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 26 more