Cospectral Cubic Graphs

• Cospectral cubic graphs are 3-regular graphs sharing the same eigenvalue multiset despite being non-isomorphic, highlighting that spectra do not uniquely determine structure.
• Constructive methods such as GM-switching, vertex/edge composition, and truncation generate explicit cospectral pairs and infinite families with varying combinatorial properties.
• The unique 16-vertex pair, where one graph is 3-edge-colorable and the other isn’t, illustrates the spectrum's inability to capture NP-hard invariants like chromatic index.

A cospectral cubic graph is a 3-regular graph whose adjacency spectrum coincides with that of another non-isomorphic 3-regular graph. Cospectrality in this context refers to equality of the multisets of eigenvalues of the adjacency matrices. This property highlights the non-uniqueness of the spectrum in determining graph structure and combinatorial invariants, especially those of NP-hard complexity, such as chromatic and matching properties. Recent research resolves foundational questions in the area and provides explicit constructions, classification theorems, and infinite families of cospectral cubic graphs.

1. Definitions and Fundamental Properties

Let $G = (V, E)$ be a simple cubic graph of order $n$; that is, every vertex has degree 3. The adjacency matrix $A(G)$ is a real symmetric $n \times n$ matrix with entries $a_{ij} = 1$ if $\{i, j\} \in E$ and 0 otherwise. The spectrum of $G$ is the multiset of real eigenvalues of $A(G)$.

Two graphs $G_1, G_2$ of order $n$ are cospectral (isospectral) if their adjacency spectra are identical as multisets, i.e., $\mathrm{spec}(A(G_1)) = \mathrm{spec}(A(G_2))$. If $G_1$ and $G_2$ are not isomorphic, they form a cospectral pair.

Cubic graphs are a fertile domain for spectral analysis due to their structural regularity and the NP-completeness of their edge-coloring problems—a fact established by Holyer (1981). Notably, many parameters for regular graphs are determined by the spectrum (e.g., regularity, bipartiteness), while others are not (e.g., Hamiltonicity, perfect matching existence, chromatic index) (Yan et al., 2021, Haemers, 7 Jan 2026).

2. Existence and Construction of Cospectral Cubic Graphs

The earliest questions surrounding cospectral cubic graphs involved the search for non-isomorphic pairs that differed with respect to a combinatorial invariant not detectable by the spectrum. For decades, the existence of a cospectral pair of cubic graphs with different chromatic indices remained open. Comprehensive computational studies, exploiting algorithms for isomorph-free generation of cubic graphs (notably nauty's canonical labeling) and characteristic polynomial computation, established the existence and enumerative landscape of such cospectral pairs up to moderate order (Yan et al., 2021).

Constructive Methods

There are several methodologies for constructing cospectral cubic graphs:

• Godsil–McKay (GM) Switching: This classical switching operation yields cospectral pairs through local modifications, always corresponding to rational orthogonal similarities (Yan et al., 2021).
• Vertex- and Edge-Composition Techniques: The Haythorpe–Newcombe vertex-composition method applies when "replaceable vertices" (those whose neighborhood sets are removal-cospectral) exist in the initial graphs. This construction produces cospectral pairs and extends to regular graphs of arbitrary degree (Haythorpe et al., 2021).
• Truncation Operation: Truncation replaces each vertex in a cubic graph with a triangle, connecting each original incident edge to a distinct vertex of the new triangle. This operation, which algebraically corresponds to the line graph of the complete subdivision, preserves cospectrality and chromatic index, and can be iterated to produce infinite families (Haemers, 7 Jan 2026).

3. Notable Results and Infinite Families

An exhaustive search of connected cubic graphs up to 16 vertices established:

• For $n \leq 12$, no nontrivial cospectral pairs exist.
• At $n = 14$, three cospectral pairs exist (all are 3-edge-colorable).
• At $n = 16$, there are 41 pairs and one triple; among these, exactly one pair with differing chromatic index ($3$ and $4$) was found (Yan et al., 2021, Haemers, 7 Jan 2026).

This unique 16-vertex pair demonstrates that the chromatic index of a regular graph is not a spectral invariant, directly answering the question of Etesami and Haemers. The construction of infinite families leverages the preservation of spectrum and chromatic index under truncation. Iterated truncation applied to suitable seed pairs yields arbitrarily large cospectral pairs with distinct edge-chromatic numbers (Haemers, 7 Jan 2026).

The Haythorpe–Newcombe method systematically constructs families of cospectral cubic graphs. Empirical enumeration shows that a large and increasing proportion of all large cospectral cubic graphs are constructible by this method, with tabled statistics up to $n = 22$. For example, of 83 non-DS (non-uniquely spectrally determined) cubic graphs of order 16, 48.2% are constructible by vertex/edge-composition (Haythorpe et al., 2021).

4. The 16-Vertex Cospectral Pair with Distinct Chromatic Indices

The canonical example comprises two 16-vertex cubic graphs $G$ and $H$ with the same spectrum given by the characteristic polynomial: $$\chi(x) = (x-3)x(x+2)(x^2-2)(x^2-x-3)(x^3-4x-2)(x^3-4x+1)(x^3+2x^2-2x-2)$$ Both spectra consist of 16 real eigenvalues. $G$ contains a Hamiltonian cycle and thus has chromatic index 3; $H$ is constructed from the Petersen graph via local replacement of a 3-path by triangles and inherits chromatic index 4, paralleling the Petersen graph’s class II behavior (Yan et al., 2021).

Crucially, application of the Liu–Wang irrational-norm lemma establishes that any orthogonal similarity connecting the adjacency matrices of $G$ and $H$ must be irrational (since the ratio of integral eigenvector norms associated with a simple eigenvalue is $\sqrt{3}$). Therefore, these graphs are not related by any GM-switching or rational-matrix based cospectral construction (Yan et al., 2021).

5. Spectral Characterizations and Limitations

Key findings concerning spectral determination in cubic graphs include:

• The chromatic index is not a DS-property (i.e., it is not uniquely determined by the spectrum) (Yan et al., 2021, Haemers, 7 Jan 2026).
• The operation of truncation preserves the spectrum and chromatic index, enabling the construction of infinite cospectral families with invariant or differing combinatorial properties (Haemers, 7 Jan 2026).
• Sufficient conditions for matching properties, such as perfect matchings: For cubic graphs of even order $n > 76$, if the second largest eigenvalue $\lambda_2 < \theta' \approx 2.94272$ (the spectral radius of the graph $F'$), then the graph admits a perfect matching (Haemers, 7 Jan 2026).

By contrast, for connected cubic graphs, the vertex-chromatic number is spectrally determined, as only $K_4$ and bipartite graphs attain extremal chromatic numbers, and these classes are recognizable by their spectra via Brooks’ theorem and spectral signatures (Haemers, 7 Jan 2026).

6. Special Cases: Cayley Cubic Graphs and Spectral Uniqueness

A sharp contrast to the general prevalence of cospectral cubic graphs appears in the family of cubic Cayley graphs on the dihedral group $D_{2p}$ with $p$ odd. In this class, it has been established that cospectrality coincides with isomorphism. That is, every two cospectral cubic Cayley graphs on $D_{2p}$ are in fact isomorphic, making all such graphs determined by their spectrum. This follows from character-theoretic analysis and group automorphism actions. Enumeration of these graphs, up to isomorphism, is governed by a formula involving quadratic reciprocity (Huang et al., 2016).

Family	DS?	Infinite Families	Constructibility by Known Methods
General cubic graphs	No	Yes	Yes (GM-switching, truncation, vertex/edge composition)
Cubic Cayley graphs on $D_{2p}$	Yes	Yes	Not by switching; determined by spectra

7. Open Problems and Research Directions

Current research continues to target exhaustive classification of cospectral cubic (and higher-regularity) graphs, identification of further spectrum-blind invariants, and exploration of analytical techniques for certifying the necessity of irrational orthogonal similarity in cospectral constructions. The question remains whether the existence of a perfect matching is spectrally characterizable (for all cubic graphs) and which other NP-hard invariants remain invisible to the spectrum (Yan et al., 2021, Haemers, 7 Jan 2026). Empirical and theoretical evidence supports the conjecture that almost all large cospectral cubic graphs can be constructed via vertex- or edge-composition, but a rigorous asymptotic proof remains open (Haythorpe et al., 2021).

A plausible implication is that, except for highly symmetric or arithmetic classes (e.g., dihedral Cayley graphs), the phenomenon of cospectral but non-isomorphic graphs is ubiquitous among cubic graphs, with distinct combinatorial properties not determined by the adjacency spectrum. This suggests new benchmarks for the limit of spectral determination in discrete mathematics and algebraic graph theory.