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Are sparse graphs typically determined by their spectrum?

Nils Van de Berg, Alexander Van Werde

Abstract

We investigate whether it is typical for a sparse graph to be uniquely characterized by its adjacency spectrum up to isomorphism. Our first result shows that the giant component of an Erdős-Rényi graph is cospectral when the average degree is sufficiently small. The proof relies on the existence of a specific pendant tree, combined with a method by Schwenk that swaps trees to construct a cospectral mate. It seems possible that pendant trees are essentially the only obstruction, meaning that the giant should become characterized by spectrum with high probability if one prunes these by considering the 2-core. The majority of the paper is devoted to theoretical and numerical evidence supporting this concept. Our main theorem in this direction establishes that local switching methods can not cause the 2-core to be cospectral. We also discuss R-cospectrality and rational cospectrality at fixed level.

Are sparse graphs typically determined by their spectrum?

Abstract

We investigate whether it is typical for a sparse graph to be uniquely characterized by its adjacency spectrum up to isomorphism. Our first result shows that the giant component of an Erdős-Rényi graph is cospectral when the average degree is sufficiently small. The proof relies on the existence of a specific pendant tree, combined with a method by Schwenk that swaps trees to construct a cospectral mate. It seems possible that pendant trees are essentially the only obstruction, meaning that the giant should become characterized by spectrum with high probability if one prunes these by considering the 2-core. The majority of the paper is devoted to theoretical and numerical evidence supporting this concept. Our main theorem in this direction establishes that local switching methods can not cause the 2-core to be cospectral. We also discuss R-cospectrality and rational cospectrality at fixed level.
Paper Structure (15 sections, 15 theorems, 27 equations, 6 figures, 1 table)

This paper contains 15 sections, 15 theorems, 27 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. The giant is cospectral
  3. Are pendant trees the only obstruction?
  4. Further theoretical evidence and open questions
  5. Outline
  6. The giant is cospectral --- Proofs of Theorem and Proposition
  7. A switching method in a 2-core needs at least two cycles -- Proof of Theorem
  8. Switching methods induce cospectrality on subgraphs
  9. Disjoint union of cyles results in isomorphism
  10. Generating cospectrality for graphs with maximum degree 2
  11. No non-isomorphic switching if GX is a union of cycles
  12. Proofs of Theorems ? and ?
  13. Numerical evidence
  14. Main findings
  15. Algorithmic optimizations

Key Result

Theorem 1.1

Consider a $G(n,p)$ random graph with $p = \lambda/n$ where $\lambda = \lambda_n >1$ remains bounded away from one as $n\to \infty$. Further, consider an arbitrary sequence $h(n)$ with $h(n)\to \infty$. Then,

Figures (6)

  • Figure 1: The giant can be decomposed as a skeleton consisting of cycles with pendant trees attached to some nodes. (See ding2014anatomy for precise results of this nature.) To show that it is cospectral, it suffices to find a specific rooted tree $T_1$ among these pendant trees. A non-isomorphic cospectral graph is then found by replacing $T_1$ with a different rooted tree $T_2$.
  • Figure 2: The $2$-core of the giant includes a few small cycles with nonzero probability. It is possible for a switching method to act on such a cycle, but we show in the proof of \ref{['thm: GiantCospectral']} that it is impossible to obtain a non-isomorphic graph in this fashion.
  • Figure 3: The neighborhood graph of $X$ includes the edges with at least one endpoint in $X$.
  • Figure 4: The proof of \ref{['lem:switchingcyclesiso']} constructs an isomorphism $G\cong H$ based on a permutation $\phi$ of the paths in $G_X[X]$ and $H_X[X]$. This permutation has the property that for every $y,z \in N(X)$ adjacent to a path $A \subseteq G_X[X]$ in $G$, it holds that the path $\phi(A) \subseteq H_X[X]$ is adjacent to $y,z$ in $H$.
  • Figure 5: The proof of \ref{['thm:noswitching2core']} constructs the required subgraph by considering a cycle $C_0 \subseteq G_X$ with a vertex $v_0$ of degree $\geq 3$ and creates a second cycle by considering maximal path in $G_X\setminus C_0$ starting from a neighbour $v_1$ of $v_0$. In the depicted example, such a path $v_1v_2\ldots v_6$ can be found by taking $v_6$ to be one of the circled vertices.
  • ...and 1 more figures

Theorems & Definitions (39)

  • Theorem 1.1
  • Proposition 1.2
  • Conjecture 1.3
  • Definition 1.4
  • Example 1.5: Godsil--McKay switching godsil1982constructing
  • Example 1.6: Swapping pending graphs
  • Theorem 1.7
  • Theorem 1.8: Wang and Zhao wang2025graphscospectralmatefixed
  • proof
  • Lemma 2.1
  • ...and 29 more