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On token signed graphs

C. Dalfó, M. A. Fiol, E. Steffen

Abstract

We introduce the concept of a $k$-token signed graph and study some of its combinatorial and algebraic properties. We prove that two switching isomorphic signed graphs have switching isomorphic token graphs. Moreover, we show that the Laplacian spectrum of a balanced signed graph is contained in the Laplacian spectra of its $k$-token signed graph. Besides, we introduce and study the unbalance level of a signed graph, which is a new parameter that measures how far a signed graph is from being balanced. Moreover, we study the relation between the frustration index and the unbalance level of signed graphs and their token signed graphs.

On token signed graphs

Abstract

We introduce the concept of a -token signed graph and study some of its combinatorial and algebraic properties. We prove that two switching isomorphic signed graphs have switching isomorphic token graphs. Moreover, we show that the Laplacian spectrum of a balanced signed graph is contained in the Laplacian spectra of its -token signed graph. Besides, we introduce and study the unbalance level of a signed graph, which is a new parameter that measures how far a signed graph is from being balanced. Moreover, we study the relation between the frustration index and the unbalance level of signed graphs and their token signed graphs.
Paper Structure (10 sections, 17 theorems, 26 equations, 5 figures, 6 tables)

This paper contains 10 sections, 17 theorems, 26 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Basic concepts and results
  3. Token signed graphs
  4. Frustration index and unbalance level
  5. Sign-symmetric token graphs
  6. The Laplacian spectrum
  7. The complement signed graph and its $k$-token graph
  8. Declarations
  9. Data availability
  10. Funding and/or Conflicts of interests/Competing interests

Key Result

Theorem 2.1

A signed graph $(G,\sigma)$ is balanced if and only if $V(G)$ can be partitioned into two sets, one of which might be empty, such that every positive edge connects two vertices of the same set and every negative edge connects two vertices of different sets.

Figures (5)

  • Figure 1: A signed $K_5$ and its 2-token signed graph.
  • Figure 2: Token graphs preserve switching equivalence.
  • Figure 3: The sign-symmetric signed graph $K_4^-$ and its sign-symmetric 2-token signed graph.
  • Figure 4: A sign-symmetric graph and its sign-symmetric 2-token graph (the 'bird graph').
  • Figure 5: A balanced signed graph $\Gamma$ and its balanced 2-token signed graph $F_2(\Gamma)$.

Theorems & Definitions (30)

  • Theorem 2.1: Harary53
  • Corollary 2.2
  • Theorem 2.3: ddffhtz21
  • Theorem 2.4: ddffhtz21
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Proposition 4.1
  • ...and 20 more