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On perfect matchings, edge-colourings and eigenvalues of cubic graphs

Willem H. Haemers

TL;DR

This work investigates whether graph properties of cubic graphs can be inferred from the adjacency spectrum. By using a truncation operation, the authors generate infinite families of cospectral cubic graphs with different edge-chromatic numbers and show that three-edge-disjoint perfect matchings are spectrally indistinguishable, while introducing a new spectral condition that guarantees the existence of a perfect matching for large enough graphs. They further show that spectral characterizations for connected cubic graphs can determine chromatic number (and related properties like bipartiteness), and discuss how truncation and cospectrality interact with line graphs. Overall, the paper highlights both the limitations and the potential of spectral methods in cubic graphs, providing concrete constructions and a practical PM criterion.

Abstract

We discuss the question whether the existence of perfect matchings in a cubic graph can be seen from the spectrum of its adjacency matrix. For regular graphs in general and for three edge-disjoint perfect matchings in a cubic graph (that is, an edge colouring with three colors) the answer is known to be negative. In the latter case, a few counter examples (found by computer) are known. Here we show that these counter examples can be extended to an infinite family by use of truncation. Thus we obtain infinitely many pairs of cospectral cubic graphs with different edge-chromatic number. For all these pairs both graphs have a perfect matching, and the mentioned question is still open. But we do find a new sufficient condition for a perfect matching in a cubic graphs in terms of its spectrum. In addition we obtain a few more results concerning spectral characterizations of cubic graphs.

On perfect matchings, edge-colourings and eigenvalues of cubic graphs

TL;DR

This work investigates whether graph properties of cubic graphs can be inferred from the adjacency spectrum. By using a truncation operation, the authors generate infinite families of cospectral cubic graphs with different edge-chromatic numbers and show that three-edge-disjoint perfect matchings are spectrally indistinguishable, while introducing a new spectral condition that guarantees the existence of a perfect matching for large enough graphs. They further show that spectral characterizations for connected cubic graphs can determine chromatic number (and related properties like bipartiteness), and discuss how truncation and cospectrality interact with line graphs. Overall, the paper highlights both the limitations and the potential of spectral methods in cubic graphs, providing concrete constructions and a practical PM criterion.

Abstract

We discuss the question whether the existence of perfect matchings in a cubic graph can be seen from the spectrum of its adjacency matrix. For regular graphs in general and for three edge-disjoint perfect matchings in a cubic graph (that is, an edge colouring with three colors) the answer is known to be negative. In the latter case, a few counter examples (found by computer) are known. Here we show that these counter examples can be extended to an infinite family by use of truncation. Thus we obtain infinitely many pairs of cospectral cubic graphs with different edge-chromatic number. For all these pairs both graphs have a perfect matching, and the mentioned question is still open. But we do find a new sufficient condition for a perfect matching in a cubic graphs in terms of its spectrum. In addition we obtain a few more results concerning spectral characterizations of cubic graphs.
Paper Structure (4 sections, 8 theorems, 1 equation, 3 figures)

This paper contains 4 sections, 8 theorems, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. Truncation
  3. Perfect matching
  4. Final remarks

Key Result

Theorem 1

ZCC Let $G$ be a cubic graph of order $n$ with spectrum $\lambda_1=3\geq\ldots\geq\lambda_n$. Then the eigenvalues of the Truncated graph $T(G)$ are $\frac{1}{2}\pm\frac{1}{2}\sqrt{13+4\lambda_i}$ for $i=1,\ldots,n$, $-2$ and $0$, both with multiplicity $n/2$.

Figures (3)

  • Figure 1: Cospectral cubic graphs with chromatic index 3 and 4, respectively, and characteristic polynomial $x(x-3)(x+2)(x^2-2)(x^2-x-3)(x^2-4x-2)(x^2-4x+1)(x^3+2x^2-2x-2)$
  • Figure 2: Truncation
  • Figure 3: Graphs with largest eigenvalue $\theta\approx 2.85577$ and $\theta'\approx 2.94272$, respectively

Theorems & Definitions (12)

  • Theorem 1
  • Proposition 1
  • proof
  • Theorem 2
  • Corollary 1
  • Proposition 2
  • Proposition 3
  • proof
  • Theorem 3
  • proof
  • ...and 2 more