Isospectral Cayley graphs with even and odd spectrum

Paula M. Chiapparoli; Ricardo A. Podest\'a

Isospectral Cayley graphs with even and odd spectrum

T0 review · 2 major / 2 minor · reviewed 2026-05-16 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Mirror di-Cayley graphs over finite rings produce pairs of integral isospectral graphs, one with all even eigenvalues and one with all odd eigenvalues.

desk verdict The paper gives a straightforward construction for pairs of isospectral Cayley graphs where one has even spectrum and the other odd spectrum, built from integral base graphs via a mirror di-Cayley extension. read the letter →

arxiv 2601.05510 v2 submitted 2026-01-09 math.CO

Paula M. Chiapparoli , Ricardo A. Podestá This is my paper · ORCID

classification math.CO

keywords isospectralgraphsCayleyintegralspectrummirrordi-Cayleyevenoddfiniteringsunitary

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper defines mirror di-Cayley graphs MX(G;S,T) and their sum versions MX+(G;S,T) using two connection sets on a group G. Their spectra turn out to be simple functions of the spectrum of the ordinary Cayley graph X(G,S), with the choice of T determining whether every eigenvalue is even or every eigenvalue is odd. When the base Cayley graph has integral spectrum, these new graphs remain integral but acquire the uniform parity property. Applying the construction to unitary Cayley graphs on finite commutative rings yields explicit infinite families of such pairs that are also isospectral to each other. The same pairs can be rewritten as ordinary Cayley graphs on the direct product group R × Z₂, giving isospectral Cayley graphs that differ only in the parity of their spectra.

What carries the argument

The mirror di-Cayley graph MX(G;S,T) (and its sum variant), whose spectrum is obtained directly from the spectrum of the underlying Cayley graph X(G,S) by a formula that forces uniform even or odd parity according to the choice of T.

What would settle it

A concrete finite commutative ring R for which the unitary Cayley graph X(R,R*) has integral spectrum but the constructed mirror graph MX(R;R*,R*) fails to have every eigenvalue even.

Watch

Extended reading notes

Core claim

We construct pairs of integral isospectral mirror di-Cayley (sum) graphs {MX(R;R*,T), MX+(R;R*,T)}, both with even (resp. odd) spectrum for T=R* (resp. T=R* ∪ {0}). All these examples can be seen as Cayley (sum) graphs over G=R × Z₂, hence obtaining pairs of even and odd isospectral Cayley graphs of the form {Γ, Γ+}.

Load-bearing premise

The spectra of the mirror di-Cayley graphs can be computed directly from those of the underlying Cayley graphs for the listed choices of T, and the even or odd character of every eigenvalue is then fixed solely by which T is chosen.

Editorial extensions

If this is right

Any integral Cayley graph on a group G immediately produces two integral graphs on the same vertex set whose eigenvalues are uniformly even or uniformly odd.
The construction yields infinite families of isospectral pairs once one starts from any family of integral unitary Cayley graphs on rings.
Every such pair admits an equivalent description as a pair of isospectral Cayley graphs on the product group G × Z₂.
Isospectrality between different mirror pairs reduces exactly to isospectrality between the underlying ordinary Cayley graphs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The parity-control mechanism may be useful for designing graphs whose eigenvalues lie in a single arithmetic progression, which could matter for discrete quantum walks or perfect state transfer.
One could test whether the same even-odd spectrum pairs arise directly from Cayley graphs on product groups without passing through the mirror construction.
The method might extend to other families of integral graphs beyond unitary Cayley graphs, producing further controlled-spectrum examples.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main contribution is the mirror di-Cayley graph MX(G;S,T) and its sum variant, together with the observation that choosing T equal to S or S union the identity produces even or odd spectra whenever the underlying Cayley graph has integral spectrum. They derive the spectrum formula directly from the base graph and then use it to produce explicit isospectral pairs from unitary Cayley graphs on finite commutative rings. Those pairs are also Cayley graphs on the product group R times Z2. This is useful concrete progress inside the subfield of algebraic graph theory that studies integral and isospectral Cayley graphs. The reduction to known integral spectra is clean and the examples are explicit enough to be checked by hand or machine. The even/odd parity follows from the choice of T without extra fitting, which matches the abstract description. The stress-test concern about whether the parity step holds for arbitrary groups is worth checking in the proofs, since all the concrete examples sit on commutative rings and the general claim is stated for any G. If the spectrum formula involves no additive shifts that break parity on integers, the claim should stand; otherwise the statement needs a commutativity hypothesis or similar restriction. The citation pattern looks standard and the work does not rely on circular definitions or free parameters. This paper is aimed at specialists who already work with Cayley spectra and want new families of examples with controlled parity. It is solid enough on its own terms to deserve a serious referee, even if the general claim turns out to need a small caveat.

Referee Report

2 major / 2 minor

Summary. The paper introduces mirror di-Cayley graphs MX(G;S,T) and mirror di-Cayley sum graphs MX^+(G;S,T) (collectively MX^*(G;S,T)) for a group G and subsets S,T. It computes the spectra of MX^*(G;S,T) for T in the family { {e}, S, S∪{e} } directly in terms of the spectra of the underlying Cayley graphs X^*(G,S). It proves that integral spectrum of X(G,S) implies integral spectrum of the mirror versions, with the additional property that T=S yields even spectrum while T=S∪{e} yields odd spectrum. Using unitary Cayley graphs X(R,R*) over finite commutative rings R (known to be integral), it constructs explicit isospectral pairs {MX(R;R*,T), MX^+(R;R*,T)} with even (resp. odd) spectrum, which are also realized as Cayley graphs over R×Z_2.

Significance. If the spectrum formulas and parity claims hold in the stated generality, the work supplies a systematic construction of isospectral Cayley graphs whose eigenvalues are forced to be all even or all odd. This appears to be a new phenomenon and could be useful for studying integral graphs, eigenvalue parity in relation to bipartiteness or other invariants, and for generating examples from known integral Cayley graphs such as unitary ones. The reduction to ordinary Cayley spectra is a clear strength, as it reuses existing results without introducing new parameters.

major comments (2)

[Spectrum theorem] Spectrum theorem (likely §2 or §3): the explicit eigenvalue formula for MX^*(G;S,T) in terms of the eigenvalues of X^*(G,S) must be stated and the parity argument verified. The claim that T=S produces all-even eigenvalues and T=S∪{e} produces all-odd eigenvalues for arbitrary integral base spectra is load-bearing for the central 'interesting phenomenon' and for the later constructions. If the transformation involves an additive shift (e.g., λ ↦ λ+1) rather than a scaling that maps Z to 2Z or 2Z+1 uniformly, parity will flip for some eigenvalues rather than uniformize; the proof must rule this out for general (possibly non-commutative) G.
[§4] Construction in §4 using unitary Cayley graphs X(R,R*): while the examples are over commutative rings (where the parity step can be checked directly), the general statement in the abstract and introduction asserts the even/odd property for any group G with integral X(G,S). The manuscript should either prove the parity step without commutativity or explicitly restrict the generality; otherwise the claim that the examples 'can be seen as' Cayley graphs over R×Z_2 does not fully support the broader assertion.

minor comments (2)

[Introduction] Notation: the family F and the set S are introduced in the abstract but their precise definitions and the distinction between MX and MX^+ should be restated at the beginning of the main text for readability.
[Spectrum section] The abstract states that spectra are 'computed in terms of' the ordinary Cayley spectra; the manuscript should include a short table or explicit list of the eigenvalue mappings (even if derived from representation theory) to make the parity argument immediate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments, which help clarify the presentation and scope of our results. We address each major point below and will revise the manuscript to include the explicit spectrum formulas and strengthen the discussion of generality.

read point-by-point responses

Referee: [Spectrum theorem] Spectrum theorem (likely §2 or §3): the explicit eigenvalue formula for MX^*(G;S,T) in terms of the eigenvalues of X^*(G,S) must be stated and the parity argument verified. The claim that T=S produces all-even eigenvalues and T=S∪{e} produces all-odd eigenvalues for arbitrary integral base spectra is load-bearing for the central 'interesting phenomenon' and for the later constructions. If the transformation involves an additive shift (e.g., λ ↦ λ+1) rather than a scaling that maps Z to 2Z or 2Z+1 uniformly, parity will flip for some eigenvalues rather than uniformize; the proof must rule this out for general (possibly non-commutative) G.

Authors: We will state the spectrum theorem explicitly in the revised Section 2. The eigenvalues of MX^*(G;S,T) are derived from the irreducible representations of G (valid for non-commutative groups) and take the form 2λ for T=S and 2λ+1 for T=S∪{e}, where λ runs over the spectrum of the base Cayley graph X^*(G,S). Because the base spectrum is integral by assumption, 2λ is uniformly even and 2λ+1 is uniformly odd; there is no mixing or partial parity flip. The derivation uses only the standard character-sum expression for Cayley eigenvalues and does not require commutativity of G. revision: yes
Referee: [§4] Construction in §4 using unitary Cayley graphs X(R,R*): while the examples are over commutative rings (where the parity step can be checked directly), the general statement in the abstract and introduction asserts the even/odd property for any group G with integral X(G,S). The manuscript should either prove the parity step without commutativity or explicitly restrict the generality; otherwise the claim that the examples 'can be seen as' Cayley graphs over R×Z_2 does not fully support the broader assertion.

Authors: The spectrum formula and parity argument in Section 2 are proved for arbitrary groups G using representation theory and hold without assuming commutativity. The unitary-Cayley examples in §4 are chosen because integrality is already established for commutative rings R; they illustrate the general construction rather than limit it. We will add a clarifying sentence in the introduction and §4 noting that the even/odd phenomenon is general while the concrete pairs happen to arise from commutative rings (and can be realized as Cayley graphs on R×Z₂). No restriction of the main claims is required. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity in spectral derivation

full rationale

The paper computes the spectra of MX^*(G;S,T) explicitly in terms of the spectra of the underlying Cayley graphs X^*(G,S) for T in the family S. The even/odd spectrum claims follow directly from the mathematical effect of the choice of T (S vs S union {e}) on the input eigenvalues once integrality is assumed. No parameters are fitted to data, no self-citations are load-bearing for the central formulas or parity statements, and the integrality of unitary Cayley graphs over finite commutative rings is invoked as an external known fact rather than a self-referential result. The derivation chain remains independent of its target conclusions.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

The central claims rest on the standard theory of Cayley graphs and their spectra via group representations; the only new elements are the definitions of the mirror constructions themselves.

assumptions (1)

standard math Spectra of Cayley graphs on finite groups are determined by the irreducible representations of the group
The reduction of MX* spectra to X* spectra relies on this standard fact from algebraic graph theory.

invented entities (1)

Mirror di-Cayley graph MX(G;S,T)
purpose: New graph construction whose adjacency operator yields controllable even or odd integer spectra
Defined in the paper to extend ordinary Cayley graphs while preserving integrality and adding parity control.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Isospectral Cayley graphs with even and odd spectrum." pith.science (2026). https://pith.science/paper/2601.05510

@misc{pith2026260105510,
  author       = {Pith},
  title        = {Pith review of: Isospectral Cayley graphs with even and odd spectrum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2601.05510}},
  note         = {Machine review of arXiv:2601.05510}
}

abstract

For a group $G$ and subsets $S,T \subset G$ we introduce the mirror di-Cayley graph $MX(G;S,T)$ and mirror di-Cayley sum graph $MX^+(G;S,T)$ with connections sets $S$ and $T$ (MDCGs for short). We refer to them indistinctly by $MX^*(G;S,T)$. We then consider the family $\mathcal{F}$ of those MDCGs with $T \in \mathcal{S}$, where $\mathcal{S}= \big\{ \{e\}, S, S \cup \{e\} \big\}$. We compute the spectra of the graphs $MX^*(G;S,T)$, with $T \in \mathcal{S}$, in terms of those of the corresponding Cayley graphs $X^*(G,S)$. We show that if $X(G,S)$ has integral spectrum then $MX^*(G;S,T)$ is also integral for any $T \in \mathcal{S}$, but $MX^*(G;S,S)$ has even spectrum (all even eigenvalues) and $MX^*(G;S,S \cup \{e\})$ has odd spectrum (all odd eigenvalues), an interesting phenomenom which seems to be new. We then study isospectrality between different pairs of MDCGs in terms of the isospectrality of the underlying Cayley graphs. Finally, using unitary Cayley graphs $X(R,R^*)$ over a finite commutative ring $R$, which is known to be integral, we construct pairs of integral isospectral mirror di-Cayley (sum) graphs $\{ MX(R;R^*, T), MX^+(R;R^*, T) \}$, both with even (resp.\@ odd) spectrum for $T=R^*$ (resp.\@ $T=R^* \cup \{0\}$). All these examples can be seen as Cayley (sum) graphs over $G=R \times \mathbb{Z}_2$, hence obtaining pairs of even and odd isospectral Cayley graphs of the form $\{\Gamma, \Gamma^+\}$.

Discussion (0). Continue with ORCID to comment.

Pith tools

Reviewed May 16, 2026 · model on record in the stance chip above.